The Lie Theory of Connected Pro-Lie Groups

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EMS Tracts in Mathematics 2

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EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions

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Karl H. Hofmann Sidney A. Morris

The Lie Theory of Connected Pro-Lie Groups A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups

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S E M E S

S E M E S

European Mathematical Society

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Authors: Karl Heinrich Hofmann Fachbereich Mathematik, AG 5 Technische Universität Darmstadt Schloßgartenstraße 7 64289 Darmstadt Germany E-Mail: hofmann @mathematik.tu-darmstadt.de

Sidney A. Morris School of Information Technology and Mathematical Sciences University of Ballarat P.O. Box 663 Ballarat, Victoria 3353 Australia E-mail: S.Morris @ballarat.edu.au

2000 Mathematics Subject Classification: Primary 22-02; secondary 17B65, 22D05, 22E20, 22E65, 44A13, 44M40, 58B25.

Key words: pro-Lie groups, pro-Lie algebras, Lie theory of connected pro-Lie groups, exponential function, structure theory of locally compact groups, completeness, quotient groups, open mapping theorem, Levi–Malcev splitting, local Iwasawa splitting.

ISBN 978-3-03719-032-6 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Preface

Sophus Marius Lie (1842–1899) laid the foundation of the theory named Lie theory in honor of its creator. Several mathematicians, likewise prominent in the history of modern mathematics, contributed to its inception in the decades following 1873, which was the year in which Lie started to occupy himself intensively in the study of what he called continuous groups, notably: Friedrich Engel, Wilhelm Killing, Élie Cartan, Henri Poincaré, and Hermann Weyl. From the beginning, however, the advance of Lie theory bifurcated into two separate major highways, which is the reason why the words Lie theory mean different things to different people. Lie himself aimed at accomplishing for the solution of differential equations (in the widest sense) what Évariste Galois and Lie’s countryman Niels Henrik Abel achieved for the solution of algebraic equations: A profound understanding and, to the best extent possible, a classification in terms of groups. Even though Lie considered himself a “geometer,” he created a territory of analysis that is called “Lie theory” by those working in it, and that is represented by the well-known text by Peter J. Olver entitled “Applications of Lie Groups to Differential Equations” [Springer-Verlag, Berlin, New York, etc., 1986]. We should say in the beginning, that the project of Lie theory which we shall discuss in this book, in philosophy and thrust, does not belong to this line. A second highway was taken by Killing and Cartan. It led to a study of what soon became known as Lie algebras, of the group and structure theory of Lie groups, and to the geometry of homogeneous spaces. The latter notably yielded the classification of symmetric spaces by Élie Cartan. At long last it merged into the encyclopedic attempt by Nicolas Bourbaki of the nineteen hundred sixties and seventies, to summarize what had been achieved, and to the emergence of an immense collection of textbooks at all levels. In 1973 Jean Dieudonné quipped “Les groupes de Lie sont devenus le centre des Mathématiques; on ne peut rien faire de sérieux sans eux.” (Lie groups have moved to the center of mathematics. One cannot seriously undertake anything without them. Gazette des Mathématiciens, Société Mathématique de France, Octobre 1974, p. 77.) By and large, in this line of “Lie theory” the words meant the structure theory of Lie algebras and Lie groups, and in particular how the latter is based on the former. The term ‘Lie group’ originally meant ‘finite-dimensional Lie group’ and most people understand the words in this sense today. However even Sophus Lie spoke of “unendliche Gruppen” by which he meant something like infinite-dimensional Lie groups. But reasonable concepts of dimension were not yet available in the 19th century before topology was on its way. And indeed Lie’s attempts in this direction did not appear to have gotten off the ground. The significance of Lie’s discoveries was emphasized by David Hilbert by raising the question in 1900 whether (in later terminology) a locally euclidean topological group is in fact an analytic group in the sense of Lie. This was the fifth of his famous 23 problems which foreshadowed so much of the mathematical creativity

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of the 20th century. It required half a century of effort on the part of several generations of eminent mathematicians until it was settled in the affirmative. Partial solutions came along as the structure of topological groups was understood better and better: Hermann Weyl and his student F. Peter in 1923 laid the foundations of the representation and structure theory of compact groups, and a positive answer to Hilbert’s Fifth Problem for compact groups was a consequence, drawn by John von Neumann in 1932. Lev Semyonovich Pontryagin and Egbert Rudolf van Kampen developed in 1932, respectively, 1936, the duality theory of locally compact abelian groups laying the foundations for an abstract harmonic analysis flourishing throughout the second half of the 20th century and providing the central method for attacking the structure theory of compact abelian groups via duality. Again a positive response to Hilbert’s question for locally euclidean abelian groups followed in the wash. One of the most significant and seminal papers in topological group theory was published in 1949 by Kenkichi Iwasawa, some three years before Hilbert’s Problem was finally settled by the concerted contribution of Andrew Mattei Gleason, Dean Montgomery, Leon Zippin, and Hidehiko Yamabe. It was Iwasawa who clearly recognized for the first time that the structure theory of locally compact groups reduced to that of compact groups and finite-dimensional Lie groups provided one knew that they happen to be approximated by finite-dimensional Lie groups in the sense of projective limits, in other words, if they were pro-Lie groups in our parlance. And this is what Yamabe established in 1953 for all locally compact groups which have a compact factor group modulo their identity component – almost connected locally compact groups as we shall say. The most influential monograph collecting these results was the book by Montgomery and Zippin of 1955 with the title “Topological Transformation Groups”. The theories of compact groups and of abelian locally compact groups had introduced in the first half of the century classes of groups with an explicit structure theory without the restriction of finite-dimensionality, and in the middle of the century these results opened up an explicit development for numerous results on the structure theory of locally compact groups. What are the coordinates of our book in this historical thread? It was recognized in 1957 by Richard Kenneth Lashof that any locally compact group G has a Lie algebra g. If g is appropriately defined, then the exponential function exp : g → G is supplied along with the definition. Yet the fact that these observations are the nucleus of a complete and rich, although infinite-dimensional Lie theory was never exploited. The present book is devoted to the foundations, and the exploitation of such a Lie theory. At a point in the overall historical development where infinite-dimensional Lie theories gain increasing acceptance and attract much interest, this appears to be timely. The Lie theory we unfold is based on projective limits, both on the group level and on the Lie algebra level. We shall find it very helpful that category theory, as a tool for the “working mathematician” as Saunders Mac Lane formulated it, is so well developed that we see immediately what we need, and we shall exploit it. In our case, we need the theory of limits in a complete category, that is, in a category in which all limits exist, and we need the theory of pairs of adjoint functors, which is closely linked with limits.

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The machinery of projective limits is familiar to mathematicians dealing with profinite groups in their work on Galois theory and arithmetics, quite generally. But the apparatus of projective limits is also familiar to mathematicians dealing with compact groups, their representation theory and abstract harmonic analysis. Indeed all group theoreticians working on the structure theory of locally compact groups encounter projective limits sooner or later. In this book we shall call projective limits of projective systems (or, as some authors say, inverse systems) of finite-dimensional Lie groups pro-Lie groups. That is, pro-Lie groups relate to finite-dimensional Lie groups exactly as profinite groups relate to finite groups. However, in the theory of locally compact groups, one encounters a special kind of projective limit, namely, limit situations where limit maps and bonding maps are proper, that is, are closed continuous homomorphisms between locally compact groups having compact kernels. Some authors call such maps perfect. This type of projective limit has a significant element of compactness already built into its definition, and it is this type of limit that has shaped the intuitions of group theoreticians for fifty years or more. From the vantage point of category theory, however, such a restriction is entirely unnatural, as is indeed the entire focus on locally compact pro-Lie groups: The class of locally compact groups is not even closed under the formation of products – as the example of the groups RN or ZN shows immediately. Mathematicians will be naturally attracted to the problem of eliminating the focus on locally compact groups. As one proceeds in the direction of pro-Lie groups in general, however, one comes to realize that the restriction to locally compact groups is unnatural also for reasons that are entirely interior to the mathematics of topological groups and Lie groups. For several years we have been engaged in the laying of the foundations of a general theory of the category of pro-Lie groups. The results are presented in this book. On the first 60 pages, the reader will find a panoramic overview of what is contained in its 14 chapters, and the user of the book should get a more compact overview by perusing its table of contents. The Lie theory of finite-dimensional Lie groups works because for a connected Lie group G, its Lie algebra g and its exponential function exp : g → G largely determine the structure of G. We hasten to add that, except for the case that G is simply connected, they do not do so completely. As the title of our book indicates, we focus on a Lie theory for connected pro-Lie groups. As a consequence, our structure theory is one that is mainly concerned with connected pro-Lie groups, sometimes going a bit further, but rarely much beyond almost connected groups. In view of Yamabe’s Theorem, the structure theory of connected or almost connected pro-Lie groups applies at once to connected or even almost connected locally compact groups. There are several key elements to the structure theory of pro-Lie groups. Firstly, a thorough understanding of the working of projective limits is needed without the crutch of thinking in terms of proper maps all the time. Chapter 1 deals with many facets of this issue. But only after Chapter 3 will we have understood all aspects of what this means for the very definition of pro-Lie groups itself.

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Secondly, the entire theory depends on our accepting that pro-Lie groups, even though not being Lie groups, nevertheless have a working Lie theory, complete with the appropriate Lie algebras which we shall call pro-Lie algebras and working exponential functions that mediate between pro-Lie groups and their Lie algebras. Indeed we must become aware at an early stage that there is a good Lie algebra functor from the category of pro-Lie groups to the category of pro-Lie algebras. One of the very positive side effects of facing wider categories than the conventional ones in developing a Lie theory is that this enlargement of scope forces us to realize in great clarity that the Lie algebra functor is opposed by a Lie group functor that encapsulates lucidly the contents of Lie’s Third Fundamental Theorem. This applies to the classical situation as well, but it is not recognized there because the theory of universal covering Lie groups, while providing topologically satisfying results in general, tends to obscure the precise functorial set-up. Since for pro-Lie groups a classical covering theory is impossible as one knows from the theory of compact connected abelian groups, it is mandatory that one understands the functorial background of a more general universal covering theory. We shall discuss this in Chapters 2, 4, 6 and 8. Thirdly, the success of the structure theory of pro-Lie groups depends in a large measure on our success in dealing with the structure theory of pro-Lie algebras. This pervades the whole book, but most of this is done in our rather long Chapter 7. The point is that the topological vector spaces underlying pro-Lie algebras are what we call weakly complete topological vector spaces, because they are exactly the duals of real vector spaces given the weak ∗-topology, that is, the topology of pointwise convergence of linear functionals. Since the vector space duality is crucial for this class of topological vector spaces and hence for the structure theory of pro-Lie algebras we present the essential features of the linear algebra of weakly complete topological vector spaces in an appendix, namely, Appendix 2. The relevance of weakly complete topological vector spaces in the structure theory of pro-Lie groups themselves is evidenced in that chapter in which we discuss the structure of commutative pro-Lie groups, and that is Chapter 5. With all of these foundations done, the Lie and structure theory of pro-Lie groups can proceed, as it does in Chapters 9 through 13. This preface is not the place to go into the details, but we shall present to our readers in the beginning of the book, in our panoramic overview, the results which we obtain. One of the lead motives of our structure theory is to reduce the structure of connected pro-Lie groups in the optimal extent possible to the structure theory of compact connected groups, weakly complete topological vector spaces, and finite-dimensional Lie groups. We will prove some major structure theorems which expose that we, in essence, achieve this goal. Acknowledgements. Our mathematical collaboration in the area of compact and locally compact groups originated in the late seventies. We record with deep gratitude the steady support of our families that we enjoyed throughout this time, particularly on the part of our wives Isolde Hofmann and Elizabeth Morris. Our work on the structure of pro-Lie groups began in the fall of the year 2000; we are grateful to our academic home institutions, notably to the University of Ballarat in Victoria, Australia, the Darmstadt

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University of Technology in Germany, the University of South Australia in Adelaide, Australia, and Tulane University in New Orleans, USA. We thank Dr. Manfred Karbe who accompanied the project from its beginning with great interest and finally published it at the European Mathematical Society Publishing House. Our special thanks go to Dr. Irene Zimmermann who acted as our copy editor and typsetter. Only a mathematician such as Dr. Zimmermann could have identified errors and inconsistencies as she did. On the top of this she polished rough spots in the text and asked questions which resulted in improving the presentation. In short we have been blessed to have such a professional handling of our TEX script. Darmstadt and Ballarat, March 2007

Karl. H. Hofmann Sidney A. Morris

Contents

Preface Panoramic Overview Part 1. The Base Theory of Pro-Lie Groups . . . . . . . . . . Part 2. The Algebra of Pro-Lie Algebras . . . . . . . . . . . . Part 3. The Fine Lie Theory of Pro-Lie Groups . . . . . . . . Part 4. Global Structure Theory of Connected Pro-Lie Groups Part 5. The Role of Compactness on the Pro-Lie Algebra Level Part 6. The Role of Compact Subgroups of Pro-Lie Groups . . Part 7. Local Splitting According to Iwasawa . . . . . . . . . 1

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Limits of Topological Groups Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The External Approach to Projective Limits . . . . . . . . . . Projective Limits and Local Compactness . . . . . . . . . . . The Fundamental Theorem on Projective Limits . . . . . . . . The Internal Approach to Projective Limits . . . . . . . . . . . Projective Limits and Completeness . . . . . . . . . . . . . . The Closed Subgroup Theorem . . . . . . . . . . . . . . . . . The Role of Local Compactness . . . . . . . . . . . . . . . . The Role of Closed Full Subcategories in Complete Categories Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Lie Groups and the Lie Theory of Topological Groups The General Definition of a Lie Group . . . . . . . . . . . . . . . . . The Exponential Function of Topological Groups . . . . . . . . . . . The Lie Algebra of a Topological Group . . . . . . . . . . . . . . . . The Category of Topological Groups with Lie Algebras . . . . . . . . The Lie Algebra Functor Has a Left Adjoint Functor . . . . . . . . . . Sophus Lie’s Third Fundamental Theorem . . . . . . . . . . . . . . . The Adjoint Representation of a Topological Group with a Lie Algebra Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Pro-Lie Groups Projective Limits of Lie Groups . . . . . . . . . . . . . . . . . The Lie Algebras of Projective Limits of Lie Groups . . . . . . Pro-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . Weakly Complete Topological Vector Spaces and Lie Algebras Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . .

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Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 An Overview of the Definitions of a Pro-Lie Group . . . . . . . . . . . . . 160 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4

Quotients of Pro-Lie Groups Quotient Objects in Categories . . . . . . . . . . . . . Quotient Groups of Pro-Lie Groups . . . . . . . . . . The Exponential Function of Compact Abelian Groups phisms . . . . . . . . . . . . . . . . . . . . . . The One Parameter Subgroup Lifting Theorem . . . . Sufficient Conditions for Quotients to be Complete . . Quotients and Quotient Maps between Pro-Lie Groups Postscript . . . . . . . . . . . . . . . . . . . . . . . .

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5 Abelian Pro-Lie Groups Examples of Abelian Pro-Lie Groups . . . . . . . . . . . . . . Weil’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Group Splitting Theorems . . . . . . . . . . . . . . . . Compactly Generated Abelian Pro-Lie Groups . . . . . . . . . Weakly Complete Topological Vector Spaces Revisited . . . . The Duality Theory of Abelian Pro-Lie Groups . . . . . . . . The Toral Homomorphic Images of an Abelian Pro-Lie Group Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 169 170 173 182 194 208 210

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Lie’s Third Fundamental Theorem Lie’s Third Fundamental Theorem for Pro-Lie Groups . . . . . . . . . . . . Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 264 266

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Profinite-Dimensional Modules and Lie Algebras Modules over a Lie Algebra . . . . . . . . . . . . . Duality of Modules . . . . . . . . . . . . . . . . . Semisimple and Reductive Modules . . . . . . . . Reductive Pro-Lie Algebras . . . . . . . . . . . . . Transfinitely Solvable Lie Algebras . . . . . . . . . The Radical and Levi–Mal’cev: Existence . . . . . Transfinitely Nilpotent Lie Algebras . . . . . . . . The Nilpotent Radicals . . . . . . . . . . . . . . . Special Endomorphisms of Pro-Lie Algebras . . . . Levi–Mal’cev: Uniqueness . . . . . . . . . . . . . Direct and Semidirect Sums Revisited . . . . . . . Cartan Subalgebras of Pro-Lie Algebras . . . . . . Theorem of Ado . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . .

269 269 272 277 281 284 291 296 300 305 309 313 315 330 332

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8 The Structure of Simply Connected Pro-Lie Groups The Adjoint Action . . . . . . . . . . . . . . . . . . . . . Simply Connected Pronilpotent Pro-Lie Groups . . . . . . The Topological Splitting Technique . . . . . . . . . . . . Simple Connectivity . . . . . . . . . . . . . . . . . . . . . Universal Morphism versus Universal Covering Morphism Postscript . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups The Exponential Function on the Inner Derivation Algebra . . Analytic Subgroups . . . . . . . . . . . . . . . . . . . . . . . Automorphisms and Invariant Analytic Subgroups . . . . . . . Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . Subalgebras and Subgroups . . . . . . . . . . . . . . . . . . . The Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Commutator Subgroup . . . . . . . . . . . . . . . . . . . Finite-Dimensional Connected Pro-Lie Groups . . . . . . . . Compact Central Subgroups . . . . . . . . . . . . . . . . . . Divisibility of Groups and Connected Pro-Lie Groups . . . . . The Open Mapping Theorem . . . . . . . . . . . . . . . . . . Completing Proto-Lie Groups . . . . . . . . . . . . . . . . . Unitary Representations . . . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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356 356 360 369 370 372 374 376 376 385 402 404 409 413 415 416

10 The Global Structure of Connected Pro-Lie Groups Solvability of Pro-Lie Groups . . . . . . . . . . . . . . The Radical . . . . . . . . . . . . . . . . . . . . . . . Semisimple and Reductive Groups . . . . . . . . . . . Nilpotency of Pro-Lie Groups . . . . . . . . . . . . . The Nilradical and the Coreductive Radical . . . . . . The Structure of Reductive Pro-Lie Groups . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . .

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11 Splitting Theorems for Pro-Lie Groups Splitting Reductive Groups Semidirectly . . . . . . . Vector Group Splitting in Noncommutative Groups . The Structure of Pronilpotent and Prosolvable Groups Conjugacy Theorems . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . .

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12 Compact Subgroups of Pro-Lie Groups Procompact Modules and Lie Algebras . . . . . . . . . . . . . . . . . . . . Procompact Lie Algebras and Compactly Embedded Lie Subalgebras of ProLie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximal Compactly Embedded Subalgebras of Pro-Lie Algebras . . . . . . Conjugacy of Maximal Compactly Embedded Subalgebras . . . . . . . . . Compact Connected Groups . . . . . . . . . . . . . . . . . . . . . . . . . Compact Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potentially Compact Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . The Conjugacy of Maximal Compact Connected Subgroups . . . . . . . . . The Analytic Subgroups Having a Full Lie Algebra . . . . . . . . . . . . . Maximal Compact Subgroups of Connected Pro-Lie Groups . . . . . . . . An Alternative Open Mapping Theorem . . . . . . . . . . . . . . . . . . . On the Center of a Connected Pro-Lie Group . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500 504 507 516 519 521 524 532 544 556 558 561

13 Iwasawa’s Local Splitting Theorem Locally Splitting Lie Group Quotients of Pro-Lie Groups The Lie Algebra Theory of the Local Splitting . . . . . . Splitting on the Group Level . . . . . . . . . . . . . . . Some Comments on Connectedness . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Catalog of Examples Classification of the Examples in the Catalog . . . . . . . . . . Abelian Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . . A Simple Construction . . . . . . . . . . . . . . . . . . . . . . Pronilpotent Pro-Lie Groups . . . . . . . . . . . . . . . . . . . Prosolvable Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . Semisimple and Reductive Pro-Lie Groups . . . . . . . . . . . . Mixed Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples Concerning the Definition of Lie and Pro-Lie Groups . Analytic Subgroups of Pro-Lie Groups . . . . . . . . . . . . . . Examples Concerning Simple Connectivity . . . . . . . . . . . Example Concerning g-Module Theory . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Appendix 1 The Campbell–Hausdorff Formalism

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2 Appendix 2 Weakly Complete Topological Vector Spaces

629

3 Appendix 3 Various Pieces of Information on Semisimple Lie Algebras 651 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

Contents

xv

Bibliography

657

List of Symbols

667

Index

669

Panoramic Overview

Compactness is a core concept in general topology, because it introduces finiteness in otherwise infinite geometric objects. When we combine compactness with group theory and its enormous background we can expect a theory rich in results, varied in direction, and fertile in applications. And we get it as is evidenced through a sizeable body of monographs and texts having come about in the second half of last century. Naturally, we like to cite our book on compact groups [102] that appeared in 1998 and that experienced a second revised and augmented edition in 2006. The standard examples are linear groups such as the orthogonal and unitary groups, or the additive groups of p-adic integers, and this confirms that the concept of a compact group is natural. The class of compact groups is closed in the class of all topological groups under the formation of arbitrary products and the passage to closed subgroups. This makes it a closed category in its own right, and that in itself is a fact from which many desirable properties of this category follow. But there are before our eyes just as natural examples of groups that illustrate that there are many topological groups basic to analysis, geometry and algebra which are not compact; easily perceived examples are the additive groups of Rn or, more generally, finite-dimensional vector spaces over locally compact fields, and linear groups like the full linear groups Gl(n, R) and their closed subgroups. All of these groups, however, are locally compact. The most important locally compact groups are real Lie groups which are connected or have, at most, finitely many connected components. One definition of a Lie group is that it is a real analytic manifold with a group structure such that multiplication and inversion are analytic functions. A topological group which is isomorphic to a closed subgroup of the topological group Gl(n, R) is a Lie group, and we shall call such a Lie group a linear Lie group. Let us emphasize at this point that here, and in the following, when we speak about two topological groups as being isomorphic, we mean them to be isomorphic as topological groups; some writers like to stress this by saying that they are “isomorphic algebraically and topologically”. We give a definition of a general Lie group in Appendix 1 to this book which allows a quicker access to the group theoretical aspects of Lie group theory than one involving analytical manifolds. In our book [102] on compact groups we devote a whole chapter to an introduction to linear Lie groups. It is shown in that book that all compact Lie groups are matrix groups, that is, linear Lie groups. All these groups possess identity neighborhoods which are homeomorphic to Rn for a suitable dimension n: they are locally euclidean. In 1900 David Hilbert raised the question whether every locally euclidean group is a Lie group. It took half a century until this question was answered in the affirmative by the concerted joint efforts of Gleason [63], Montgomery and Zippin [144] published back to back in the Annals of Mathematics. The monograph [145] by Montgomery and Zippin appeared three years later and summarized the entire development including

2

Panoramic Overview

the important complements by Yamabe [206], [207] which followed one year later and to which we shall return presently. Montgomery and Zippin’s book became a classic which has not been replaced to this day, in spite of an excellent secondary source authored by Kaplansky [129] sixteen years later. Hilbert’s Problems numbered 23 in all; they were formulated in order to indicate the directions which research in mathematics was to take in the 20th century. The problem concerning Lie groups is number 5, and its difficulty as well as the sheer quantity of research that it fertilized was very indicative of Hilbert’s vision. So it is only natural that something even more influential came along with the affirmative solution of Hilbert’s Fifth Problem, namely, Yamabe’s Theorem. Yamabe’s Theorem tells us that every connected locally compact group G is approximated by a connected Lie group in the sense that G contains arbitrarily small normal subgroups N such that G/N is a Lie group ([206], [207]). In fact Yamabe’s Theorem applies to more than connected groups: it says that every locally compact group for which the group of connected components G/G0 is compact is approximated by Lie groups in the sense just explained. The concept of being approximated by Lie groups is so important that it certainly deserves a definition of its own. For this purpose let us first recall that a topological group is complete if every Cauchy filter (or every Cauchy net) converges; this aptly generalizes the concept of completeness of a metric space which is complete if every Cauchy sequence converges. Every locally compact group is complete and so no mention of completeness need be made when one deals with locally compact groups. Definition 1. (i) A topological group G is called a pro-Lie group if it is complete and if every identity neighborhood of G contains a normal subgroup N such that G/N is a Lie group. The category of all pro-Lie groups with continuous group morphisms between them is written proLieGr. (3.39) (ii) A topological group G is called almost connected if the factor group G/G0 of G modulo the connected component G0 of the identity is compact. Let us then reformulate Yamabe’s Theorem in this terminology: Every almost connected locally compact group is a pro-Lie group. It is a generally adopted notation that for a category A and objects A and B in A, the set of all morphisms A → B is denoted by A(A, B). For instance, if TopGr denotes the category of all topological groups and continuous group homomorphisms between them, then TopGr(G, H ) denotes the set of all continuous homomorphisms from the topological group G to the topological group H ; if G and H are pro-Lie groups, then we have proLieGr(G, H ) = TopGr(G, H ) by definition. This means that the category proLieGr is a full subcategory of the category TopGr of all topological groups. Algebraists, in particular ring theorists, are rather familiar with a concept similar to that of pro-Lie groups, namely, profinite groups. A group G is profinite if it is a complete topological group such that every identity neighborhood of G contains a normal open subgroup N such that G/N is finite. Profinite groups are compact, and

Panoramic Overview

3

they are pro-Lie groups. Profinite groups generalize finite groups in the exact same way as pro-Lie groups generalize Lie groups. Only three years before the solution of Hilbert’s Fifth Problem was found by Gleason, Montgomery and Zippin, a seminal paper by Iwasawa had appeared in the Annals of Mathematics [120]. In that paper he exposed fundamental properties of locally compact pro-Lie groups. So Yamabe’s result made all of this available for the study of the structure and the representation theory of almost connected locally compact groups. This was the culmination of half a century of research on topological groups following Hilbert’s vision in 1900. But at the same time, and certainly not less significant from the present vantage point, the work by Iwasawa, Gleason, Montgomery, Zippin and Yamabe provided motivation and incentive for another half a century’s worth of research on locally compact groups during the second half of the twentieth century. What went into this entire century of research naturally was the full body of highly developed structure and representation theory of finite-dimensional Lie groups and finite-dimensional Lie algebras. Let us briefly say what we mean by the Lie theory of a topological group on a very general level; after all, the words Lie theory appear in the title of this book. To each topological group G one can easily associate a topological space L(G), namely, the space Hom(R, G) of all continuous group homomorphisms from the additive topological group R of real numbers to the topological group G, endowed with the topology of uniform convergence on compact sets. We also have a continuous function exp : L(G) → G given by exp X = X(1) and a “scalar multiplication” (r, X) → r · X : R × L(G) → L(G) given by (r · X)(s) = X(sr). Whether these concepts are useful depends in large measure on the degree to which additional properties are satisfied. In Chapter 2 we shall elaborate on the following definition. Definition 2 (2.6ff.). A topological group G is said to have a Lie algebra, if L(G) has a continuous addition and bracket multiplication making it into a topological Lie algebra in such a fashion that   r   r  n (X + Y )(r) = lim X Y n→∞ n n and

  r   r   r −1  r −1 n2 X Y . Y X n→∞ n n n n If G has a Lie algebra, then L(G) is called the Lie algebra of G and exp : L(G) → G is called its exponential function. [X, Y ](r 2 ) = lim

Clearly a topological group G has a Lie algebra if and only if the connected component G0 of the identity has a Lie algebra G0 and L(G) = Hom(R, G) = Hom(R, G0 ) = L(G0 ). The image of the exponential function is contained in G0 . If we believe that L(G) and the exponential function encapsulate the Lie theory of G, then it is true that the identity component G0 already captures the Lie theory of G.

4

Panoramic Overview

We show in this book that every pro-Lie group G has a Lie algebra L(G) and the image exp L(G) of the exponential function algebraically generates a subgroup which is dense in the connected component G0 of the identity. We shall have much more to say about the topological Lie algebra L(G) that can arise in this fashion. But for the moment we observe this: In every totally disconnected locally compact group, the open (hence closed) subgroups form a basis of the neighborhood filter of the identity element. If G is a locally compact group, then G/G0 is a locally compact totally disconnected group, and so there is an open subgroup U of G containing G0 such that U/G0 is compact. Then U is almost connected and thus, by Yamabe’s Theorem, is a pro-Lie group. Therefore every locally compact group has an open subgroup which is a pro-Lie group and which captures the Lie theory of G. Apart from individual studies such as [134], [64], [103], [104], [106], the Lie theory of locally compact groups has never been systematically considered or exploited, although a start was made in [102] for the purpose of a structure theory of compact groups. One of the thrusts of this book is to change this situation with determination. In addition to the successful resolution of Hilbert’s Fifth Problem there is yet a second prime reason for the success of the structure and representation theory of locally compact groups: The 1932 proof by A. Haar of the existence and uniqueness of left invariant integration on a locally compact group G. Its full power for abstract harmonic analysis was recognized by A. Weil in his influential monograph [198] of 1941. Haar measure is the key to the representation theory of compact and locally compact groups on Hilbert space, and the wide field of abstract harmonic analysis with ever so many ramifications (including e.g. abstract probability theory on locally compact groups). A theorem due to A. Weil shows that, conversely, a complete topological group with a left- (or right-) invariant σ -finite measure is locally compact (see e.g. [76], [198]). Thus the category of locally compact groups is that which is exactly suited for real analysis resting on the existence of an invariant integral based on σ -additive measures. One cannot expect to extend this aspect of locally compact groups to larger classes without abandoning σ -additivity. (Bourbaki indicates in Chapter 9 of his “Intégration” [23], pp. 50–55, 70ff., how such an extension may be handled; however we shall not consider this aspect in this book.) In quiet moments of introspection one might even admire the small miracle inherent in the fact that measure theory carries as far as locally compact groups go. The proper domain for an invariant measure theory again appears to be the category of compact groups, where one has a unique invariant two sided invariant measure P with respect to which G is measurable with measure P (G) = 1. That is, P is a veritable probability measure that allows averaging over G as a remarkably simple but effective device ([102]). Yet there it is, Haar measure of locally compact groups, infinite but eminently useful making locally compact groups the analysts’ delight. However, from each of a group theoretical, of a Lie theoretical, and of a category theoretical point of view, the class of a locally compact groups has serious defects which go rather deep.

Panoramic Overview

5

Indeed, if we consider a family of Lie groups Gj , j ∈ J for an index set J , then its product j ∈J Gj is a perfectly good Hausdorff topological group with a lucid structure, but it fails to be locally compact whenever infinitely many of the Gj fail to be compact. Furthermore, while every locally compact group G does have a Lie algebra L(G), the additive group of the Lie algebra is never locally compact unless it is finitedimensional. Indeed even the additive topological group of the Lie algebra of a compact def

abelian group need not be locally compact; for example the product G = TJ of circle groups T = R/Z has a Lie algebra L(G) isomorphic to RJ and thus fails to be locally compact as soon as J is infinite, while the group TJ is comfortably within the realm of compact groups. Each Lie group G has a tangent bundle which is again a Lie group, namely, the semidirect product L(G) Ad G with G acting on its Lie algebra by adjoint action induced by inner automorphisms. Does a locally compact group have a tangent bundle? The answer is yes, it does, in fact every pro-Lie group has one (as we shall show in this book), but it is almost never a locally compact group except when the group itself is finite-dimensional. Thus the category of locally compact groups appears to have two major drawbacks: – The topological abelian group underlying the Lie algebra L(G) and the tangent bundle of a locally compact group fail to be locally compact unless L(G) is finitedimensional. In other words, the very Lie theory that makes the structure theory of locally compact groups interesting leads us outside the class. – The category of locally compact groups is not closed under the forming of products, even of copies of R; it is not closed under projective limits of projective systems of finite-dimensional Lie groups, let alone under arbitrary limits. In other words, the category of locally compact groups is badly incomplete. This book presents an argument for a shift in the vantage point of looking at locally compact groups. We plead for a structure theory of topological groups that places the focus squarely and systematically on pro-Lie groups. Recall that we denote the category of all (Hausdorff) topological groups and continuous group homomorphisms by TopGr. It turns out that the full subcategory proLieGr of TopGr consisting of all projective limits of finite-dimensional Lie groups avoids both of these difficulties. This would perhaps not yet be a sufficient reason for advocating this category if it were not for two facts: – Firstly, while not every locally compact group is a projective limit of Lie groups, every locally compact group has an open subgroup which is a projective limit of Lie groups, so that, in particular, every connected locally compact group is a pro-Lie group; also all compact groups and all locally compact abelian groups are pro-Lie groups. – Secondly, the category proLieGr is astonishingly well-behaved. Not only is it a complete category, it is closed under passing to closed subgroups and to those quotients which are complete, and it has a demonstrably good Lie theory.

6

Panoramic Overview

It is therefore indeed surprising that this class of groups has been little investigated in a systematic fashion. A serious attempt at such an investigation is made in this book where it is submitted that not only a general structure theory of locally compact groups can be based on a good understanding of the category proLieGr of pro-Lie groups, but that the category of pro-Lie groups is well worth a thorough study on its own account. In this book we will prove general structure theorems on pro-Lie groups which will include the best known general structure theorems on locally compact groups. Since the main strategy of the book is to provide a structure theory via Lie theory, en route we shall have to develop a full grown structure theory of those topological Lie algebras which occur as Lie algebras of pro-Lie groups. We shall call these pro-Lie algebras, because each of them is a complete topological Lie algebra such that every 0-neighborhood contains a closed ideal modulo which it is finite-dimensional (3.6).

Part 1. The Base Theory of Pro-Lie Groups For a description of some basic results on the theory of projective limits of Lie groups some technical background information appears inevitable even for an overview, long before we delve into the actual study of our topic.

Core Definitions and Facts on Pro-Lie Groups Definition 3. A projective system D of topological groups is a family of topological groups (Cj )j ∈J indexed by a directed set J and a family of morphisms {fj k : Ck → Cj | (j, k) ∈ J × J, j ≤ k}, such that fjj is always the identity morphism and i ≤ j ≤ k in J implies fik= fij  fj k . Then the projective limit of the system limj ∈J Cj is the subgroup of j ∈J Cj consisting of all J -tuples (xj )j ∈J for which the equation xj = fj k (xk ) holds for all j, k ∈ J such that j ≤ k. Every cartesian product of topological groups may be considered as a projective limit. Indeed, if (Gα )α∈A is an arbitrary family of topological groups indexed by an infinite set A, one obtains a projective system by considering J to be the set of finite  subsets of A directed by inclusion, by setting Cj = a∈j Ga for j ∈ J , and by letting fj k : Ck → Cj for j ≤ k in J be the projection onto the partial product. The projective  limit of this system is isomorphic to a∈A Ga . There are a few sample facts one should recall about the basic properties of projective limits (see e.g. [25], [64], [107], or this book 1.27 and 1.33): Let G = limj ∈J Gj be a projective limit of a projective system P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of topological groups with limit morphisms fj : G → Gj , and let Uj denote the filter of identity neighborhoods of Gj , U the filter of identity neighborhoods of G,

Panoramic Overview

7

and N the set {ker fj | j ∈ J }. Then U has a basis of identity neighborhoods {fk−1 (U ) | k ∈ J, U ∈ Uk } and N is a filter basis of closed normal subgroups converging to 1. If all bonding maps fj k : Gk → Gj are quotient morphisms and all limit maps fj are surjective, then the limit maps fj : G → Gj are quotient morphisms. The limit G is complete if all Gj are complete. Definition 4 (3.25). For a topological group G let N (G) denote the set of closed normal subgroups N such that all quotient groups G/N are finite-dimensional real Lie groups. Then G ∈ N (G) and G is said to be a proto-Lie group if every identity neighborhood contains a member of N (G). By our earlier Definition 1, if in addition, G is a complete topological group, then G is a pro-Lie group. While not every topological group can be embedded as a subgroup into a complete topological group, this is the case for proto-Lie groups, indeed every proto-Lie group has a completion which is a pro-Lie group. (See 4.1.)  Every product of a family of finite-dimensional Lie groups j ∈J Gj is a pro-Lie group. In particular, RJ is a pro-Lie group for any set J which is locally compact if and only if the set J is finite. The product ZN , accordingly, is a pro-Lie group. It is well known that the space ZN is homeomorphic to the space of irrational real numbers in the natural topology. We may formulate this by saying that the space of irrational numbers supports the structure of a pro-Lie group. It is a remarkable fact (which we discuss in Chapter 4) that the free abelian group Z(N) in countably many generators carries the structure of a nondiscrete pro-Lie group. The underlying topological space cannot be a Baire space and so certainly cannot be Polish (second countable completely metrizable), nor locally compact; indeed a countable homogeneous Baire space is necessarily discrete. These examples help us to realize from the beginning, that our general intuition of the topology of pro-Lie groups cannot be based on experience gathered from locally compact groups. If {Gj : j ∈ J } is a family of finite-dimensional real Lie groups then the subgroup    (gj )j ∈J ∈ Gj : {j ∈ J : gj = 1} is finite j ∈J

 of the direct product j ∈J Gj is a proto-Lie group which is not a pro-Lie group if J is infinite and the Gj are nonsingleton. We reiterate that a topological group G is called almost connected if the factor group G/G0 modulo the connected component G0 of the identity is compact. Everything that is proved for almost connected topological groups therefore is true for all connected groups and for all compact groups. One of the very weighty reasons why this concept is relevant for the theory of topological groups is the existence of Yamabe’s crucial result:

8

Panoramic Overview

Every almost connected locally compact group is a pro-Lie group. The group PSl(2, Qp ) of projective transformations of the p-adic projective line is locally compact, but has no nontrivial normal subgroups and is therefore a locally compact group which is not a pro-Lie group in our sense, while it is, of course, a p-adic Lie group. Every pro-Lie group G gives rise to a projective system {pN M : G/M → G/N : M ⊇ N in N (G)} whose projective limit it is (up to isomorphism). The converse is a difficult issue, but it is true. Theorem 5 (3.34, 3.35 (The Closed Subgroup Theorem)). Every projective limit of pro-Lie groups is a pro-Lie group. Every closed subgroup of a pro-Lie group is a pro-Lie group. A topological group is a pro-Lie group if and only if it is isomorphic to a closed subgroup of a product of Lie groups. In fact in simple category theoretical parlance the following theorem holds. Theorem 6 (3.3, 3.36). The category proLieGr of pro-Lie groups is closed in TopGr under the formation of all limits and is therefore complete. It is the smallest full subcategory of TopGr that contains all finite-dimensional Lie groups and is closed under the formation of all limits. This shows that the category proLieGr does not have some of the shortcomings of the category of locally compact groups which is obviously incomplete. It remains yet to be seen how good the Lie theory of the category proLieGr is and we shall say good things about it shortly. However, one must, at an early stage, admit that the category of pro-Lie groups has certain problems which are invisible as long as one stays inside the subcategory of locally compact pro-Lie groups. Indeed, every quotient group of a locally compact group is locally compact (which is a consequence of the fairly elementary observation that in any topological group, the product H K of a closed subset H and a compact subset K is closed, and the application of this fact to the case that H is a closed (normal) subgroup and K a compact identity neighborhood of G). It is one of the less elementary facts of Lie group theory that a quotient of a Lie group is a Lie group. Indeed the quotient of a linear Lie group need not be a linear Lie group, but is a Lie group nevertheless. The simplest example is the group of upper triangular matrices ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 x z def ⎝ 0 1 y ⎠ : x, y, z ∈ R G = ⎭ ⎩ 0 0 1 and the discrete central subgroup ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 0 n def Z = ⎝0 1 0 ⎠ : n ∈ Z ; ⎭ ⎩ 0 0 1

Panoramic Overview

9

here G is clearly a linear Lie group but the factor group G/Z, which is even locally isomorphic to G is not a linear Lie group. This was proved by Garret Birkhoff in 1936 [9] by astute but elementary linear algebra. (See also [102], p. 169ff.) It is a much more debilitating fact for the study of pro-Lie groups that quotient groups of pro-Lie groups need not be pro-Lie groups. (Corollary 4.11) Still, every quotient group of a pro-Lie group is a proto-Lie group and has a completion which is a pro-Lie group. (4.1) So the defect here arises from a phenomenon that is well observed and studied, that quotients of complete topological groups may fail to be complete. (See [176].) We shall explain in Chapter 4 that the additive group of the topological vector space R[0,1] has a nondiscrete closed subgroup K algebraically isomorphic to the free abelian group Z(N) in countably many generators such that R[0,1] /K is an abelian proto-Lie group which is dense in a compact connected and locally connected group (Corollary 4.11). We use this example in various places in the book to construct counterexamples. In this sense, this example is very helpful to build up our intuition on certain aspects of pro-Lie group theory that are invisible as long as we stay in the locally compact domain. Curiously, the counterexample itself arises from the theory of compact abelian groups, and it was discovered not so long ago ([106]). The defect of proLieGr of not being closed under the passing to quotients is, as we have said, debilitating, because passing to quotient groups is an extremely helpful device of reduction to simple situations in many proofs; therefore it is a handicap not having this tool available at all times inside proLieGr. Fortunately, we shall see that, even regarding quotients, the category proLieGr has its redeeming features. Theorem 7 (The Quotient Theorem; 4.28). Let G be an almost connected pro-Lie group and N a closed normal subgroup. Then G/N is a pro-Lie group provided at least one of the following conditions are satisfied by N : (i) N is almost connected. (ii) N is the kernel of a morphism from G onto some pro-Lie group. (iii) N is locally compact or Polish. Part (iii) of this theorem arises from general topological group theory, and we refer to sources like the book [176] of Dierolf and Roelke for such pieces of information. Parts (i) and (ii) belong to the proper substance of this book, and neither of the two is a trivial matter (See Theorem 4.28 and Corollary 9.58.) In fact, Part (ii) is a consequence of another core result concerning pro-Lie groups, namely, the Open Mapping Theorem that is well known to functional analysts as applying to a variety of operators between suitable topological vector spaces, and that is equally well known to people working with locally compact or Polish topological groups. If conditions are right, then the surjectivity of a continuous group homomorphism f : G → H from a topological

10

Panoramic Overview

group onto another implies already that f is an open function, or, in equivalent terms that the canonical decomposition G ⏐ ⏐ quot G/ ker f

f

−−−−−→ H  ⏐id ⏐ H −−−−−→ H F

produces an isomorphism of topological groups F : G/ ker f → H . If we let G be the additive group of real numbers Rd with the discrete topology, H the same group R but considered with its natural topology, then the identity function f : G → H is a bijective morphism between locally compact metric groups that is not open. We mentioned earlier that we shall expose a nondiscrete pro-Lie structure on the countable free abelian group H with infinitely many generators. So the identity morphism f from the discrete countable (hence locally compact Polish) group G = Z(N) to H is a continuous morphism between pro-Lie groups which is not open. These examples show that the following theorem is not likely to be either obvious or trivial: Theorem 8 (Open Mapping Theorem for Pro-Lie Groups; 9.60). Let G be an almost connected pro-Lie group and f : G → H a continuous group homomorphism onto a pro-Lie group. Then f is an open mapping. That is, under these circumstances, f is equivalent to a quotient homomorphism. One of the major impediments in the group theory of topological groups is the unavailability of the Second Isomorphism Theorem. The so called First Isomorphism Theorem says that if G is a topological group and M ⊆ N are normal subgroups of G then the morphism gM  → gN : G/M → G/N factors through an isomorphism of topological groups (G/M)/(N/M) → G/N. This is a very robust theorem belonging to universal algebra. The environment of the so-called Second Isomorphism Theorem is as follows: Assume that G is a topological group with a closed normal subgroup N and a closed subgroup H such that G = H N = N H . Then the surjective morphism h  → hN : H → G/N factors through a bijective continuous group homomorphism H /(H ∩ N) → H N/N . This may fail to be open even if H ∩ N = {1}. In [108] there is an example of a topological abelian group G and two (isomorphic) closed subgroups H and N such that G is algebraically the direct sum of H and N and G/H and G/N are (isomorphic) compact groups, while G blatantly fails to be compact. However, if G is a pro-Lie group, then a closed subgroup H is a pro-Lie group by the Closed Subgroup Theorem. If N is an almost connected closed normal subgroup of G and G is almost connected, then G/N is a pro-Lie group by the Quotient Theorem. Therefore, from the Open Mapping Theorem we get the next theorem. Theorem 9 (The Second Isomorphism Theorem for Pro-Lie Groups; 9.62). Let N be an almost connected normal subgroup and H an almost connected subgroup of a topological group G and assume that H , N , and H N are pro-Lie groups. Then N/(H ∩ N) and H N/N are naturally isomorphic.

Panoramic Overview

11

The Coarse Lie Theory of Pro-Lie Groups Let us consider a topological Lie algebra g and on it the filter basis of closed ideals j such that dim g/j < ∞; we shall denote it by  (g). Definition 10 (3.6). A topological Lie algebra g is called a pro-Lie algebra (short for profinite-dimensional Lie algebra) if  (g) converges to 0 and if g is a complete topological vector space. Under these circumstances, g ∼ = limj∈ (g) g/j, and the underlying vector space is a weakly complete topological vector space, that is, it is the algebraic dual of a real vector space with the weak ∗-topology. We give a systematic treatment of the duality of vector spaces and weakly complete topological vector spaces in Appendix 2 of this book. The category of pro-Lie algebras and continuous Lie algebra morphisms is denoted proLieAlg. Proposition 11 (3.3, 3.36). The category proLieAlg of pro-Lie algebras is closed in the category of topological Lie algebras and continuous Lie algebra morphisms under the formation of all limits and is therefore complete. It is the smallest category that contains all finite-dimensional Lie algebras and is closed under the formation of all limits. See also [104]. Our demonstration that Lie theory is applicable to pro-Lie groups begins with our showing results like the following: Theorem 12 (3.12, 2.25). Every pro-Lie group G has a pro-Lie algebra g as Liealgebra, and the assignment L which associates with a pro-Lie group G its pro-Lie algebra is a limit preserving functor. These matters will be shown in Chapters 2 and 3. In fact, a portion of this set-up allows a considerable improvement which we summarize in the next section.

The Category Theoretical Version of Lie’s Third Theorem Theorem 13 (Lie’s Third Theorem for Pro-Lie groups; 6.5, 6.6, 8.15). The Lie algebra functor L : proLieGr → proLieAlg has a left adjoint . It associates with every pro-Lie algebra g a unique simply connected pro-Lie group (g) and a natural isomorphism ηg : g → L((g)) such that for every morphism ϕ : g → L(G) there is a unique morphism ϕ  : (g) → G such that ϕ = L(ϕ  )  ηg . A good portion of this theorem we shall prove in Chapter 6, but we find it necessary to introduce a preliminary concept of simple connectivity. Indeed we shall call a proLie group prosimply connected if every member of N (G) contains a member N of N (G) such that G/N is a simply connected Lie group. This turns out to be, for a while, a very useful concept of simple connectivity for pro-Lie groups in all respects, and it

12

Panoramic Overview

reduces correctly to simple connectivity in the case of finite-dimensional Lie groups. Once we have developed enough structure theory we will be able in Chapter 8 to show that a pro-Lie group is prosimply connected if and only if it is simply connected. (See Theorem 8.15.) Indeed, for each pro-Lie algebra g, the group (g) is a projective limit of a projective system of simply connected Lie groups. The fact that L is a right adjoint confirms its property of preserving all limits. There is more to Theorem 13 than meets the eye, and we should alert the reader to these circumstances because they shed new light on the situation even when everything is restricted to the classical situation of finite-dimensional Lie groups. The adjointness of the two functors L and  may be expressed in terms of universal properties as follows. There is a natural isomorphism ηg : g → L((g)) such that for any morphism f : g → L(H ) of topological Lie algebras there is a unique morphism f  : (g) → H such that f = L(f  )  ηg . In diagrams: proLieAlg

proLieGr

ηg

g ⏐ ⏐ ∀f 

−−−−−→ L((g)) ⏐ ⏐  L(f )

L(H ) −−−−−→ idL(H )

(g) ⏐ ⏐  ∃!f

L(H )

H.

In fact, the natural isomorphism really allows us to identify g with the Lie algebra of (g). Sophus Lie’s Third Fundamental Theorem (in his own terminology) says that for every finite-dimensional Lie algebra there is a Lie group having as Lie algebra the given one. So this theorem persists for pro-Lie groups. The natural morphism η is what category theoreticians call the front adjunction or the unit of the adjunction. But any adjoint situation between two functors also has a back adjunction or counit with an appropriate version of the universal property. In the case of the present adjoint situation between L and , the back adjunction set-up is as follows. There is a natural morphism πG : (L(G)) → G of pro-Lie groups with the following universal property: Given a pro-Lie group G and any morphism f : (h) → G for some pro-Lie algebra h, there is a unique morphism f  : h → L(G) of pro-Lie algebras such that f = πG  (f  ). proLieAlg

L(G)  ⏐ ∃!f  ⏐ h

proLieGr πG

(L(G)) −−−−−→  ⏐  ⏐(f ) (h)

G  ⏐∀f ⏐

−−−−−→ (h) id(h)

Panoramic Overview

13

 and call the morphism πG : G  → G the We shall abbreviate (L(G)) by G, universal morphism of G. If G happens to be a pro-Lie group which has a universal covering group in the topological sense (in particular, if G is a finite dimensional Lie  → G is the universal covering morphism (8.21). In general the group), then πG : G universal morphism is neither surjective nor a local isomorphism. This is best realized at an early stage by considering any connected compact abelian group G together with  R), its exponential function expG : L(G) → G, L(G) = Hom(R, G) ∼ = Hom(G,  where G = Hom(G, T), T = R/Z is the discrete character group of G. These things  may be equated are explained in great detail in [102], Chapters 7 and 8. In this case G with the additive group of L(G) and πG with expG : L(G) → G. The world of compact abelian groups of course is full of examples for which the exponential function fails to be surjective, beginning with the one-dimensional examples that are different from the circle group, that is, the solenoids, the character groups of which are the noncyclic infinite subgroups of Q. Let us consider within the complete category proLieGr the full subcategory proSimpConLieGr of all simply connected pro-Lie groups. Then we have the following corollary. Corollary 14 (6.6(vi)). The restrictions and corestrictions of the functors L and  implement an equivalence of categories proLieAlg

L

−−−−−→ ←−−−−−

proSimpConLieGr.



Therefore, the category of pro-Lie algebras has a faithful copy inside the category of all pro-Lie groups, namely, the full subcategory of all simply connected pro-Lie groups.  → G is a group theoretical substitute for In this light, the universal morphism πG : G the exponential function expG : g → G; indeed for abelian pro-Lie groups the two functions agree for all practical intents and purposes. These matters are discussed in Chapter 6 but thereafter will pervade the whole book. Considering the problems we have encountered with quotients in the category of pro-Lie groups, it is remarkable that the functor L behaves well with regard to quotient morphisms. Indeed we see next that L not only preserves all limits, but some colimits as well.

Conservation Laws for L and  Theorem 15 (4.20). The functor L preserves quotients. Specifically, assume that G is a pro-Lie group and N a closed normal subgroup and denote by q : G → G/N the quotient morphism. Then G/N is a proto-Lie group whose Lie algebra L(G/N ) is a pro-Lie algebra and the induced morphism of pro-Lie algebras L(q) : L(G) → L(G/N) is a quotient morphism. The exact sequence 0 → L(N ) → L(G) → L(G/N ) → 0

14

Panoramic Overview

induces an isomorphism X + L(N ) → L(f )(X) : L(G)/L(N ) → L(G/N ). The core of Theorem 15 is proved by showing that for every quotient morphism f : G → H of topological groups, where G is a pro-Lie group, every one parameter subgroup Y : R → H lifts to one of G, that is, there is a one parameter subgroup σ of G such that Y = f  σ . (See 4.19, 4.20.) This requires the Axiom of Choice. It should be emphasized that, according to Theorem 15, a quotient group of a pro-Lie group always has a complete Lie algebra even if it is itself incomplete. Therefore, a proto-Lie group with an incomplete Lie algebra such as R(N) cannot be a quotient of a pro-Lie group. Corollary 16 (4.21). Let G be a pro-Lie group. Then {L(N ) | N ∈ N (G)} converges to zero and every closed ideal i of L(G) such that L(G)/i is finite-dimensional contains an L(N) for some N ∈ N (G). Furthermore, L(G) is the projective limit limN ∈N (G) L(G)/L(N ) of a projective system of bonding morphisms and limit maps all of which are quotient morphisms, and there is a commutative diagram L(γG ) G ∼ L(G) −−−−−→ L(limN ∈N ⏐ ⏐ (G) N ) = limn∈N (G) ⏐ ⏐ expG  L(limN∈N (G) expG/N ) G −−−−−→ limN ∈N (G) G/N.

L(G) L(N )

γG

Theorem 15 expresses a version of exactness of L. But there is also an exactness theorem for , the left adjoint of L. Theorem 17 (6.7, 6.8, 6.9). If h is a closed ideal of a pro-Lie algebra g, then the exact sequence q i 0 → h −−−→ g −−−→ g/h → 0 induces an exact sequence (j )

(q)

1 → (h) −−−→ (g) −−−→ (g/h) → 1, in which (j ) is an algebraic and topological embedding and (q) is a quotient morphism. There are some other noteworthy consequences of Theorem 15. Proposition 18 (4.22 (iv)). Any quotient morphism f : G → H of pro-Lie groups onto a finite-dimensional Lie group is a locally trivial fibration. Proposition 19 (4.22 (i)). For a pro-Lie group G, the subgroup exp g generated by the image of the exponential function, is dense in G0 , that is, exp g = G0 . In particular, a connected nonsingleton pro-Lie group has nontrivial one parameter subgroups. This may be viewed as an Existence Theorem for one parameter subgroups in proLie groups, indeed of an abundance of them – unless of course, the group in question is totally disconnected. So, as an illuminating consequence we get the following characterisation of a pro-Lie group to be totally disconnected.

Panoramic Overview

15

Corollary 20 (4.23). For a pro-Lie group G the following statements are equivalent: (a) G is totally disconnected. (b) L(G) = {0}. (c) The filter basis of open normal subgroups of G converges to 1. In this book we shall call topological groups satisfying these equivalent conditions prodiscrete groups. So every prodiscrete group is, in particular, a pro-Lie group. As we already mentioned and will observe again later, there are locally compact totally disconnected groups which are not prodiscrete. The group ZN in the product topology is prodiscrete but not locally compact. It is, as we remarked earlier, homeomorphic to the space of irrational numbers. Semidirect products of two topological groups (and semidirect sums of topological Lie algebras) permeate the whole book, beginning from Chapter 1 where we remind the reader of its definition in Exercise E1.5 through Chapter 11, that is specifically devoted to the splitting of pro-Lie groups, that is to results that assert that, under suitable circumstances, a given pro-Lie group may be represented as a semidirect product. If π : H → Aut(N ) is a representation of a topological group in the group def

of automorphisms of a topological group N such that the function (h, n)  → h · n = π(n)(h) : H × N → N is continuous, then the semidirect product N π H of N by H is the topological product N × H endowed with the multiplication (m, h)(n, k) = (m(h · n), hk). That N π H is a topological group is straightforwardly verified. Very simple examples show that semidirect products of pro-Lie groups need not be pro-Lie groups (see Examples 4.29). We shall demonstrate in this book that every proLie group acts under what will be called the adjoint action or adjoint representation Ad : G → Gl(L(G)) on L(G) (see 2.27ff.). So we can form the semidirect product L(G) Ad G,

(X, g)(Y, h) = (X + Ad(g)Y, gh),

and obtain this result. def

Proposition 21 (4.29 (iii)). For each pro-Lie group G, the semidirect product T (G) = L(G) Ad G is a well-defined pro-Lie group.

We call T (G) the tangent bundle of G. Thus pro-Lie groups have tangent bundles that are pro-Lie groups. In particular, all (almost) connected locally compact groups have tangent bundles within the category of pro-Lie groups. However, for a locally compact group G its tangent bundle T (G) is locally compact only if G is finite-dimensional. We have seen that the category of pro-Lie groups – contains all finite-dimensional real Lie groups, – is closed in the category of topological groups under the formation of all limits and the passing to closed subgroups, – has a substantial Lie algebra functor that possesses a very reasonable left adjoint,

16

Panoramic Overview

– is closed under the passing from a group to the additive group of its Lie algebra and under the passing from a group to its tangent bundle. In other words, we have seen that the category of pro-Lie groups has none of the defects which plague the category of locally compact groups while it still contains all almost connected locally compact groups. That is, it still houses comfortably all those locally compact groups that support all the Lie theory there is for locally compact groups. But can we exhibit, one might ask, enough fine structure theory of pro-Lie algebras and pro-Lie groups so that at least the known structure theory of locally compact groups is recovered? Like with all categories of groups, the first test that a group theory has to face is how well it elucidates the structure of its abelian representatives.

Abelian Pro-Lie groups Apart from a territory far removed from the domain of connected or even almost connected commutative pro-Lie groups, the situation is very satisfactory and is, as a first coarse approximation to the general structure theory of almost connected proLie groups, rather representative and a good guide for one’s intuition. A weakly complete vector space is a real topological vector space V for which there is a real vector space E such that V ∼ = E ∗ , where E ∗ is the algebraic dual E ∗ HomR (E, R) ⊆ R endowed with the weak -topology, that is, the topology of pointwise convergence induced from RE given the product topology. (See Appendix 2, notably Theorem A2.8.) If the cardinal dim E is the linear dimension of E, that is, the cardinality of one, hence every basis of E, then E ∼ = R(dim E) and thus V ∼ = Rdim E . Therefore, an equivalent definition of a weakly complete topological vector space is the postulate that there be a set J such that V ∼ = RJ (see Corollary A2.9). If N S(V ) denotes the filter basis of all closed vector subspaces F of a locally convex Hausdorff topological vector space V such that dim V /F < ∞, then V is a weakly complete vector space if and only if the natural morphism λV : V → limF ∈N S(V ) V /F , λV (v) = (v + F )F ∈N S(V ) is an isomorphism of topological vector spaces. If an abelian topological group is isomorphic to the additive group of a weakly complete topological vector space, that is, to RJ for some set J , then we shall call it a weakly complete vector group. The abelian pro-Lie groups we know best are the compact abelian groups and the weakly complete vector groups. So it is very pleasing that we can state the following result. Lemma 22 (Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups; 5.12). Any abelian almost connected pro-Lie group is isomorphic to the direct product of a weakly complete vector group and a compact abelian group.

Panoramic Overview

17

This result is succinct and very lucid. It illustrates that abelian pro-Lie groups, at least if they are almost connected are built up from weakly complete vector groups and compact abelian groups in a certainly simple fashion. In reality, we have better and more accurate information. For the more accurate information we have to pay a price: the formulations get more complicated. First we have a clean cut intermediate result showing that weakly complete topological vector spaces and tori are injectives in the category of abelian pro-Lie groups. Theorem 23 (5.19). Assume that G is an abelian pro-Lie group with a closed subgroup G1 and assume that there are sets I and J such that G1 ∼ = RI × TJ . Then G1 is a homomorphic retract of G, that is, G1 is a direct summand algebraically and topologically. So G ∼ = G1 × G/G1 . This allows us to argue that every abelian pro-Lie G group has a weakly complete vector subgroup V such that G is isomorphic to the direct product V × (G/V ) where the factor G/V has no nontrivial vector subgroup. We call any such subgroup V a vector group complement. For a topological group G we let comp(G) denote the set of all elements which are contained in a compact subgroup. Theorem 24 (Vector Group Splitting Theorem for Abelian Pro-Lie Groups; 5.20). Let G be an abelian pro-Lie group and V a vector group complement. Then there is a closed subgroup H such that (i) (v, h) → v + h : V × H → G is an isomorphism of topological groups. (ii) H0 is compact and equals comp G0 and comp(H ) = comp(G); in particular, comp(G) ⊆ H . (iii) H /H0 ∼ = G/G0 , and this group is prodiscrete. (iv) G/ comp(G) ∼ = V × S for some prodiscrete abelian group S without nontrivial compact subgroups. (v) G has a characteristic closed subgroup G1 = G0 + comp(G) which is isomorphic to V × comp(H ) such that G/G1 is prodiscrete without nontrivial compact subgroups. (vi) The exponential function expG of G = V ⊕ H decomposes as expG = expV ⊕ expH where expV : L(V ) → V is an isomorphism of weakly complete vector groups and expH = expcomp(G0 ) : L((comp(G0 )) → comp(G0 ) is the exponential function of the unique largest compact connected subgroup; here L(comp(G0 )) = comp(L)(G) is the set of relatively compact one parameter subgroups of G. (vii) The arc component Ga of G is V ⊕Ha = V ⊕comp(G0 )a = im L(G). Moreover, if h is a closed vector subspace of L(G) such that exp h = Ga , then h = L(G). This theorem actually is the basis of a rather explicit structure theory of abelian pro-Lie groups. We recall that all locally compact abelian groups belong to this class. There are still some portions of an abelian pro-Lie group G which we do not fully control:

18

Panoramic Overview

– The factor group G/G1 ∼ = H / comp G is prodiscrete and has no compact subgroups but is otherwise uncharted. – comp(G) = comp(H ) is a pro-Lie group that is a directed union of compact subgroups. We do not know much more in the absence of local compactness. If we impose certain natural additional hypothesis that are traditionally invoked in topological group theory, the situation is at once much better. A topological group is called compactly generated if it is algebraically generated by a compact subset. Theorem 25 (The Compact Generation Theorem for Abelian Pro-Lie Groups; 5.32). (i) For a compactly generated abelian pro-Lie group G the characteristic closed subgroup comp(G) is compact and the characteristic closed subgroup G1 is locally compact. (ii) In particular, every vector group complement V is isomorphic to a euclidean group Rm for some m ∈ N0 = {0, 1, 2, . . . }. (iii) The factor group G/G1 is a compactly generated prodiscrete group without compact subgroups. If G/G1 is Polish, then G is locally compact and G∼ = Rm × comp(G) × Zn . (iv) If G is a pro-Lie group containing a finitely generated abelian dense subgroup, then comp(G) is compact and G ∼ = comp(G)× Zm . In particular, G is locally compact. (v) A finitely generated abelian pro-Lie group is discrete. The full subcategory of locally compact abelian groups in the category of abelian pro-Lie groups has a celebrated structure theory that is primarily due to the highly effective and elegant duality theory going back to L. S. Pontryagin [169] and E. R. van Kampen [125] in the early thirties of the 20th century. For any topological abelian group  = Hom(G, T) denote its dual with the compact open topology. (See e.g. G we let G   given [102, Chapter 7].) There is a natural morphism of abelian groups ηG : G → G by ηG (g)(χ) = χ (g) which may or may not be continuous; information regarding this issue is to be found for instance in [102, pp. 298ff], notably in Theorem 7.7 on p. 300.   is bijective and We shall call a topological abelian group semireflexive if ηG : G → G reflexive if ηG is an isomorphism of topological groups; in the latter case G is also said to have duality (see [102, p. 305]). In the direction of a duality theory of abelian pro-Lie groups we offer the following results. Proposition 26 (5.35). Let G be an abelian pro-Lie group and let V be a vector group complement. Then G is reflexive, respectively, semireflexive iff G/V is reflexive,  is isomorphic to a product E × A respectively, semireflexive. The character group G where E is the additive group of a real vector space with its finest locally convex topology and A is the character group of an abelian pro-Lie group whose identity component is compact. Theorem 27 (5.36). Every almost connected abelian pro-Lie group is reflexive, and its character group is a direct sum of the additive topological group of a real vector space

Panoramic Overview

19

endowed with the finest locally convex topology and a discrete abelian group. Pontryagin duality establishes a contravariant functorial bijection between the categories of almost connected abelian pro-Lie groups and the full subcategory of the category of topological abelian groups containing all direct sums of vector groups with the finest locally convex topology and discrete abelian groups.

Part 2. The Algebra of Pro-Lie Algebras The success of the Lie theory of classical Lie groups as well as in our case the Lie theory of pro-Lie groups depends on the effectiveness of the mechanism that allows us to translate problems of the topological group structure on the group level to algebraic problems on the Lie algebra level and back. Experience demonstrates that problems are more easily attacked in a purely algebraic environment. In the present case we know, however, that the Lie algebra of a pro-Lie group is a topological algebra itself. So we hope to repeat the classical success story only to the extent to which the topological algebra and the representation theory of pro-Lie groups themselves reduce to pure algebra – more or less. We shall see that this is largely the case for pro-Lie algebras due to the fact that the underlying topological vector spaces are weakly complete vector spaces and that these have a perfect duality theory that allows us to translate their topological linear algebra to pure linear algebra upon passing to the vector space duals. (See Appendix 2.)

The Module Theory of Pro-Lie Algebras We saw that for every pro-Lie group G there exists a simply connected pro-Lie group  → G. Thus the structure of  and a natural morphism with dense image πG : G G simply connected pro-Lie groups has no small influence on the structure of pro-Lie groups in general. We further saw that the structure of simply connected pro-Lie groups, in a well-understood sense, is completely determined by the structure of their Lie algebra. The lesson learned from Lie Theory of finite-dimensional Lie groups is that one must first study the structure of Lie algebras carefully and then apply the information gathered in this fashion to the group theory of Lie groups. It is no different with pro-Lie groups even though the connection between pro-Lie algebras and pro-Lie groups is more tenuous than in the finite-dimensional case. We develop the representation theory and structure theory of pro-Lie groups simultaneously. Elementary module theory is usually preceded by a rush of simple definitions which still turn out to be very effective. We record some to the extent they are necessary for the reader to follow this overview. Let L be a Lie algebra and E a vector space. Then E is an L-module if there is a bilinear map (x, v)  → x · v : L × E → E

satisfying

[x, y] · v = x · (y · v) − y · (x · v)

20

Panoramic Overview

for all x, y ∈ L and v ∈ E. A function f : E1 → E2 between L-modules is said to be a morphism of L-modules if it is linear and satisfies (∀x ∈ L, v ∈ E1 )

f (x · v) = x · f (v).

A submodule F of an L-module E is a vector subspace such that L · F ⊆ F . An L-module E is said to be simple if {0} and E = {0} are its only submodules. An L-module E is called semisimple if every submodule is a direct module summand. If L is a topological Lie algebra, then a topological vector space V is said to be a topological L-module if (x, v)  → x · v : L × V → V is continuous in each variable separately. If the topological vector space V is weakly complete, and if the filter basis of closed submodules W such that dim V /W < ∞ converges to 0, then V is said to be a profinite-dimensional L-module. The profinite-dimensional modules have a perfect duality; indeed if E is the topological dual of a profinite-dimensional L- module, then E is an L-module with respect to the module operation defined by x · ω, v = −ω, x · v for x ∈ L, v ∈ V and ω ∈ E. Duality permits us to transfer concepts from algebraic module theory to topological module theory. For instance, let V be a profinite-dimensional topological vector space and an L-module. Then the module is said to be reductive if its dual module is semisimple. Duality then permits us to prove theorems like the following: Theorem 28 (7.18). (a) Let V be a profinite-dimensional L-module for a Lie algebra L. Then the following statements are equivalent: (i) (ii) (iii) (iv)

V is reductive. Every finite-dimensional quotient module of V is reductive. V is the projective limit of finite-dimensional reductive module quotients. V is isomorphic to a product of finite-dimensional simple modules.

(b) Every profinite-dimensional L-module has a unique smallest submodule V ss such that V /V ss is reductive. The theory and duality of L-modules are discussed in great detail, among many other things, in Chapter 7. Now these module theoretical concepts apply to the structure theory of pro-Lie algebras. The key is the following remark. If g is a pro-Lie algebra, then the underlying weakly complete topological vector space |g| is a topological L-module with respect to the module operation defined by x · v = [x, v] for x ∈ g and v ∈ |g|. This module is called the adjoint module gad . A pro-Lie algebra g is called reductive if its adjoint module gad is a reductive g-module. It is called semisimple if it is reductive and its center z(g) is zero.

Panoramic Overview

21

While the duality theory of profinite-dimensional L-modules works perfectly, the theory of pro-Lie algebras has no duality theory in the sense that a pro-Lie algebra g could have a Lie algebra as a dual object. However, its adjoint g-module gad has a dual g-module, also called its coadjoint module gcoad . This module duality, however, attaches to each pro-Lie algebra g an almost purely algebraic object, the coadjoint module gcoad , and that is extremely helpful for the structure theory of pro-Lie algebras as the following results will show. Theorem 29 (The Structure Theorem of Reductive and Semisimple Pro-Lie Algebras). (a) For a pro-Lie algebra g the following conditions are equivalent. (i) g is reductive. (ii) g is the product of a family of finite-dimensional simple or one-dimensional ideals of g. (b) Let g be a reductive pro-Lie algebra. Then the commutator algebra [g, g] is closed and is a product of finite simple real Lie algebras. Further g ∼ = z(g) ⊕ [g, g] I ∼ algebraically and topologically, and z(g) = R for some set I . (c) A pro-Lie algebra is semisimple iff it is a product of finite simple real Lie algebras. (d) Every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. Consequently, for a pro-Lie algebra, the following statements are equivalent. (I) g is semisimple. (II) g is the product of a family of finite-dimensional simple ideals of g. (7.27, 7.29) In the light of the fact that pro-Lie algebras g arise as the Lie algebras of pro-Lie groups G, the very appealing duality theory of profinite-dimensional g-modules is a surprisingly effective tool for making the structure theory of pro-Lie groups algebraic.

Pro-Lie Algebras and Solvability Recalling in the structure theory of finite-dimensional Lie algebras that there is always a unique largest solvable ideal, called the radical, we cannot hope to be able to bypass the question of solvability in any structure theory of pro-Lie algebras that is deserving of this name. The fact that the underlying vector spaces of pro-Lie algebras are infinitedimensional as soon as the theory begins to be new and interesting is an ominous warning that solvability is going to be a delicate matter likely to involve set theory including well-ordering and ordinals. Firstly, on a purely algebraic basis, in any Lie algebra we must define a transfinite commutator series and use this transfinite series to define a general concept of solvability. This proceeds as follows. Let g be a Lie algebra. Set g(0) = g and define sequences of ideals g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g(α) is defined for α < β.

22

Panoramic Overview

 (i) If β is a limit ordinal, set g(β) = α 0 such that Z(] − ε, ε[) ⊆ V . Thus Z ∈ Hom(R, G) = L(G). Finally show that F converges to Z in the topology of L(G).] Among other things we have shown that L(G) is a uniform space which is complete if G is complete. The following remarks are still quite elementary. Remark 2.8. (i) Every partial one parameter subgroup I → G of a group extends uniquely to a group morphism R → G. (ii) Every local one parameter subgroup of a topological group extends uniquely to a one parameter subgroup, i.e. to a member of L(G). def

(iii) If X, Y ∈ L(G), then the equalizer E = {r ∈ R : X(r) = Y (r)} is a closed subgroup of R, and thus is either R (in which case X = Y ) or is of the form Zr for some r ∈ Z. (For a category theoretical definition of equalizers, see Definition 1.4 in the previous chapter.) In particular, if two one parameter subgroups agree on some nonempty open set, then they are equal.

The Exponential Function of Topological Groups

113

Exercise E2.4. Prove Remark 2.8. [Hint. (ii) is a consequence of (i) in view of the fact that a group homomorphism between topological groups is continuous iff it is continuous at the identity. For a proof of (i) consider a partial one parameter subgroup f : I → G. For r ∈ R find a natural number m such that m1 · r ∈ I and show that for two such numbers m and n m   n   = f n1 · r ; define this group element to be F (r) and show we have f m1 · r that F : R → G is the desired extension. See [102, Lemma 5.8]. Condition (iii) is readily verified; one must know the elementary fact that a closed proper subgroup of R is cyclic.] Lemma 2.9. Let G be a Lie group and let g, B, e : B → U be as in Definition 2.1. (i) For each x ∈ g, set Ix = {r ∈ R | r · x ∈ B}. Then ex : Ix → G, ex (r) = e(r · x) is a local one parameter subgroup. (ii) For each x ∈ g there is a unique one parameter subgroup η(x) : R → G such that η(x)|Ix = ex . Proof. (i) Since B is a convex open symmetric neighborhood of 0 in g and r  → r · x : R → g is a continuous linear map, the set Ix is an open symmetric interval about 0. If r, s, r + s ∈ Ix , then ex (r + s) = e((r + s) · x) = e(r · x ∗ s · x) = e(r · x)e(s · x) = ex (r)ex (s). (ii) For x ∈ g, let η(x) : R → G be the unique extension of ex according to 2.8 (ii). Proposition 2.10. Let G be a Lie group and let g, B, e : B → U be as in Definition 2.1 for a concomitant Dynkin algebra g. Then η : g → L(G) is a homeomorphism satisfying t · η(x) = η(t · x) for all t ∈ R. Proof. Choose η(x) = ex as in 2.9 (ii). Recall that (∀r ∈ Ix )

η(x)(r) = e(r · x).

(1)

Firstly, we show that t · η(x) = η(t · x) for t ∈ R. For all sufficiently small r ∈ R we have (t · η(x))(r) = η(x)(rt) = e(rt · x) and η(t · x)(r) = e(r · (t · x)) = e(rt · x) and this proves the claim in view of 2.8 (iii). Secondly, we show that η is injective. Assume η(x) = η(y). Then e(r ·x) = e(r ·y) for all sufficiently small r ∈ R. Since e : B → U is a homeomorphism, r · x = r · y follows for all sufficiently small r, which implies x = y. Thirdly, we show that η is surjective. Let X ∈ L(G). We must show that there is an x ∈ g such that X = η(x). Since X : R → G is continuous, there is an open symmetric interval J around 0 in R such that X(J ) ⊆ U . Let j : B → g denote the def

inclusion map. Then f = j  e−1  X|J : J → g is a continuous function satisfying f (r + s) = f (r) ∗ f (s) for r, s ∈ J . We fix a positive t ∈ J . For 0 < m ≤ n ∈ N we set q = m n and compute −1 f (q · t) = f (n · · + n−1 · +t ) = f (n−1 · t) ∗ · · · ∗ f (n−1 · t) = m · f (n−1 · t). ( · t + ·)* )* + ( m times

m times

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2 Lie Groups and the Lie Theory of Topological Groups

The case m = n shows f (t) = n · f (n−1 · t) i.e. f (n−1 · t) = n−1 · f (t). Thus f (q · t) = m n f (t) = q · f (t). Since f is continuous, we have f (rt) = r · f (t) for all r ∈ [0, 1]. Hence e(r · f (t)) = X(rt) = t · X(r) for all 0 ≤ r ≤ 1. On the other hand η(f (t))(r) = e(r · f (t)) for all r ∈ If (t) . Thus η(f (t)) = t · X and therefore X = t −1 · η(f (t)) = η(t −1 · f (t)). The claim follows with x = t −1 · f (t). Thus η is a bijection. It now remains to show that it is a homeomorphism. Since both R and G satisfy the First Axiom of Countability (that is the statement that each point has a countable basis for the filter of its neighborhoods) it suffices to show that for each sequence (xn )n∈N in g and each x ∈ g the relations x = lim xn and

(2)

n∈N

η(x) = lim η(xn )

(3)

n∈N

are equivalent. Statement (3) is equivalent to (∀p, q ∈ N)(∃N ∈ N)(∀n > N, |r| ≤ p)

η(xn )(r) ∈ η(x)(r)e

1 q

 ·B .

(3 )

  1 Assume   1that (3 ) is satisfied. Remember (1) above and let p0 and q0 be such that p0 · x ∗ q0 · B ⊆ B. Then for all q ≥ q0             η(x) p10 e q1 · B = e p10 · x e q1 · B = e p10 · x ∗ q1 · B ⊆ e(B).       So for all n > N with η(xn ) p10 ∈ η(x) p10 e q1 · B and q ≥ q0       e p10 · xn ∈ e p10 · x ∗ q1 · B ,

i.e. 1 p0

We conclude that limn∈N

1 p0

· xn ∈

· xn =

1 p0



1 p0

   · x ∗ q1 · B

· x and thus x = limn∈N xn , proving (2).

Conversely, assume (2) holds. Pick p0 ∈ N so that p10 · x ∈ B. Then pr0 · xn converges to pr0 · x uniformly for r ∈ [0, 1] on B ⊆ g (where we consider only     sufficiently large n). Then p10 · η(xn )(r) = η p10 · xn (r) = e pr0 · xn converges   to e pr0 · x = · · · = p10 · η(x)(r) uniformly for r ∈ [0, 1] on G. Since scalar   multiplication is continuous on L(G) by 2.7 (ii), η(xn )(r) = pp0 · p10 · η(xn ) pr   converges to η(x)(r) = pp0 · p10 · η(x) pr uniformly for r ∈ [0, p]. The same argument shows that convergence is uniform for r ∈ [−p, 0] and hence is uniform on [−p, p]. This completes the proof. If X and Y are two topological spaces with scalar multiplication, then a function f : X → Y is called a scalar morphism if it is continuous and satisfies f (t ·x) = t ·f (x) for all t ∈ R and x ∈ X. With this concept we can phrase Proposition 2.10 as follows: For each Lie group G with a concomitant Dynkin algebra g there is a scalar isomorphism from g to L(G) as spaces with scalar multiplication.

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The Lie Algebra of a Topological Group In any group G we define the commutator comm(g, h) of two elements g, h ∈ G to be ghg −1 h−1 . For a topological group G and one parameter subgroups X, Y ∈ L(G) we define continuous functions XY, comm(X, Y ) : R → G by (XY )(r) = X(r)Y (r) and comm(X, Y )(r) = comm(X(r), Y (r)). Of course these functions are not one parameter subgroups in general except when comm(X(r), Y (r)) = 1 for all r, which is certainly the case if G is commutative. For the following theorem let us denote by  : R → R the function given by (t) = t 2 . For a subset A of a (not necessarily topological) group G we shall write A for the subgroup algebraically generated by A in G. The Definition of a Topological Group with a Lie Algebra and of a Generating Lie Algebra Definition 2.11. Let G be a topological group. Then it is said that G has a Lie algebra or, equivalently, that G is a topological group with a Lie algebra if the following conditions hold: (i) For all X, Y ∈ L(G), the following limits exist pointwise: ,, -, --n 1 1 def X + Y = lim , ·X ·Y n→∞ n n , -n2 1 1 def [X, Y ]   = lim comm · X, · Y n→∞ n n

(4) (5)

and X + Y, [X, Y ] ∈ L(G). (ii) Addition (X, Y )  → X + Y : L(G) × L(G) → L(G) and bracket multiplication (X, Y )  → [X, Y ] : L(G) × L(G) → L(G) are continuous. (iii) With respect to scalar multiplication ·, addition +, and bracket multiplication [· , ·] the set L(G) is a real Lie algebra. The Lie algebra L(G) of a topological group is said to be generating, if the closed subgroup exp L(G) generated by the image is the identity component G0 of G. In particular, if G has a Lie algebra, then L(G) is a topological Lie algebra. Note that a topological group G has a Lie algebra if and only if G0 has a Lie algebra. We shall amplify this remark later in Proposition 3.5. Theorem 2.12. Every Lie group has a Lie algebra, and the Lie algebra of a Lie group is a Dynkin algebra. Proof. First we verify condition 2.11 (i). Let G be a Lie group, g a Dynkin algebra and B a Campbell–Hausdorff neighborhood such that there is a homeomorphism e : B → U

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onto an open identity neighborhood U of G as in Definition 2.1. Let X, Y ∈ L(G) and determine x, y ∈ g with the help of Proposition 2.10 in such a fashion, that η(x) = X and η(y) = Y . By the Fundamental Theorem on Dynkin Algebras A1.7 (iii), for each t ∈ R we have , t t ·x∗ ·y , (6) t · (x + y) = lim n · n→∞ n n , t t 2 2 t · [x, y] = lim n · comm∗ · x, · y . (7) n→∞ n n Since η is continuous and preserves scalar multiplication by Proposition 2.10, we have η(x + y)(t) = η(t · x + t · y)(1) -n , , t t t t · x ∗ · y (n) = lim e ·x∗ ·y = lim η n→∞ n→∞ n n n n , , , , - , --n , --n t t t t = lim e = lim η(x) ·x e ·y η(y) n→∞ n→∞ n n n n , , - , --n ,, - , - -n t 1 t 1 = lim X = lim Y · X (t) · Y (t) n→∞ n→∞ n n n n ,., -, -/ -n 1 1 = lim ·X ·Y (t) , n→∞ n n and this limit is (X + Y )(t) by definition. An analogous computation shows , , - - n2 1 1 2 , η([x, y])(t ) = lim comm · X, · Y (t) n→∞ n n and this limit is [X, Y ](t 2 ) = ([X, Y ] )(t). Thus (i) is proved and we have established (∀x, y ∈ g) η(x + y) = η(x) + η(y), (∀x, y ∈ g) η([x, y]) = [η(x), η(y)].

(8) (9)

Condition 2.11 (ii) and (iii) now follow from the fact that η is a homeomorphism and from (8) and (9) above which show that η transports the topological Lie algebra structure from g to L(G). This completes the proof of the theorem. Let G be a Lie group and let g, B, e : B → U be as in Definition 2.1 with a concomitant Dynkin algebra g. Let η be as in 2.9 and 2.10. In summing up what has been said we find the following commutative diagram helpful: B ⏐ ⏐ e V

incl

−−−−−→

g

η

−−−−−→ ⏐ L(G) = Hom(R, G) ⏐exp  G −−−−−→ G −−−−−→ G. incl

idG

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Theorem 2.13. A Lie group G determines the Dynkin algebra g which occurs in Definition 2.1 uniquely up to isomorphism. Indeed every such Dynkin algebra is isomorphic to L(G). The exponential function expG : L(G) → G maps a suitable Campbell– Hausdorff neighborhood B of L(G) homeomorphically onto an open identity neighborhood of G in such a fashion that (∀X, Y ∈ B)

exp(X ∗ Y ) = (exp X)(exp Y ).

The image of the exponential function of a Lie group is an identity neighborhood of G; in particular, the subgroup expG L(G) algebraically generated by this image agrees with the identity component G0 of G. Proof. The Dynkin algebra L(G) is uniquely determined by G through its definition and by Theorem 2.12. That η : g → L(G) is the desired isomorphism of Dynkin algebras also follows from 2.12. From A1.9 we deduce that η(x ∗ y) = η(x) ∗ η(y) for all sufficiently small x, y ∈ g. For all sufficiently small x ∈ g we recall η(x)(1) = e(x) and so, setting X = η(x), Y = η(y) we have exp X = X(1) = e(x), exp Y = Y (1) = e(y) and thus exp(X ∗Y ) = (X ∗Y )(1) = (η(x)∗η(y))(1) = η(x ∗y)(1) = e(x ∗y) = e(x) ∗ e(y) = η(x)(1) ∗ η(y)(1) = (exp X) ∗ (exp Y ) for all sufficiently small x and y. Since the identity neighborhood e(B) = expG (B) is contained in the image of expG , the latter is an identity neighborhood. Hence expG L(G) is an open subgroup of G, and is therefore closed (see [102, A4.25 (ii), (iii)]). Thus it contains the connected component G0 of the identity. On the other hand, the set expG L(G) is arcwise connected, and so is the group it generates; therefore expG L(G) ⊆ G0 . Theorem 2.14. Every abelian topological group G has a Lie algebra L(G), and L(G) is commutative. If G is a complete topological group, then L(G) is a complete topological vector space. Proof. If X, Y ∈ L(G) and comm(X(r), Y (r)) = 0 for all r ∈ R, then , , - , --n t t X = X(t)Y (t) Y n n and thus 2.11 (i) is clearly satisfied with X + Y being the pointwise product. Obviously the bracket product vanishes. The assertion about completeness follows from Remark 2.7 (iv). Proposition 2.15. If G is the additive group of a topological vector space E, then the function ϕ : E → L(G) which is defined by ϕ(x)(t) = t · x is a linear map which is the inverse of the exponential function exp : L(G) → G. Therefore, L(G) ∼ = E as topological vector spaces. In particular, every topological vector space occurs (up to isomorphism of topological vector spaces) as the Lie algebra of a topological group, and every complete topological vector space occurs as the Lie algebra of a complete topological group. Proof. Exercise E2.5.

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Exercise E2.5. Prove Proposition 2.15. Next we observe that Lie groups have a property which is expressible in terms of simple group theoretical and simple topological concepts. Definition 2.16. A topological group G is said to have no small subgroups if it has an identity neighborhood U such that for any subgroup H of G the relation H ⊆ U implies H = {1}. A subgroup of a group having no small subgroups has no small subgroups. Proposition 2.17. A Lie group has no small subgroups. Proof. Let G be a Lie group and B an open convex symmetric zero neighborhood of L(G) which is mapped homeomorphically onto an open identity neighborhood V of G   def by the exponential function. Set U = expG 21 · B and let H be a subgroup of G  with H ⊆ U . Now assume that h ∈ H . Then since expG  21 · B : 21 · B → U is a homeomorphism, there is a unique X ∈ 21 · B such that h = expG X. Suppose now that h = 1. Then X = 0 and there is a smallest natural number n > 1 such that n · X ∈ B \ 21 · B. It follows that hn = (expG X)n = expG n · X ∈ H ∩ (V \ U ) = Ø. This contradiction shows h = 1 and thereby H = {1}. In [102, Exercise E7.17] we have seen a closed arcwise connected subgroup G of the additive group of a separable Banach space E such that L(G) = {0}; in particular, G is an arcwise connected complete group without small subgroups which fails to be a Lie group. We also saw in [102, Proposition 5.33 (iv)], that a compact group without small subgroups is a (linear) Lie group; this remains true for locally compact groups as we shall see in due course, but the proof is much harder. At this point we observe, however, that it is easy to see that a locally compact group with no small subgroups is metrizable. Exercise E2.6. Let G be a topological group with no small subgroups. Then G has a metrizable topology which is coarser than or equal to the given one, and if G is locally compact, then they are equal. [Hint. Let U0 be a symmetric identity neighborhood of G containing no subgroups other than {1}. Construct inductively a descending sequence Un of symmetricidentity neighborhoods Un such that Un Un ⊆ Un−1 , n = 1, 2, . . . . Then D = ∞ n=1 Un is a group contained in U0 and thus is {1}. If U0 can be chosen to be compact, then the filter basis {Un | n ∈ N} converges to 1 and thus forms a basis of identity neighborhoods of 1; i.e. the topology of G is first countable and thus metrizable (see e.g. [102, Lemma A4.10ff., notably Theorem A4.16]). If G is not locally compact, the filter basis {Un | n ∈ N} defines a left (or right) Hausdorff uniform structure with a countable basis which is coarser than or equal to the left (respectively right) uniform structure determined by the given group topology. Uniform spaces with a countable basis are metrizable.]

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Let U be the group of unitary operators on a separable infinite-dimensional Hilbert space H . With respect to the operator norm topology, U is a Lie group, and its Lie algebra is (up to a natural identification) the Lie algebra of all skewadjoint bounded operators with the bracket [X, Y ] = XY − Y X and the operator norm topology. However, if we give U the strong operator topology, that is, the topology of pointwise convergence, then U is a topological group in which L(U ) may be identified with the set of all not necessarily bounded, but closed skewadjoint operators X (for which the domains of definition of X and X∗ agree). This is the content of Stone’s Theorem (see e.g. [69, p. 32, Theorem, 4.7]). However, L(G) is not a vector space because X + Y in general fails to be a closed operator with dense domain of definition. The unitary groups in the strong operator topology thus provide examples of topological groups which do not have a Lie algebra. However, if the closure X + Y is a closed skewadjoint operator, then the formula (4) does hold (Trotter Product Formula); see e.g. [69, p. 53, Theorem 8.12]. A central tool for the analysis of the structure of connected pro-Lie groups will be the associated Lie algebras or, more specifically, topological Lie algebras. Thus we use the machinery of Lie algebras and exponential functions on the background of topological groups in general. Accordingly we study the category LieAlgGr of topological groups which have Lie algebras and a number of its interesting subcategories.

The Category of Topological Groups with Lie Algebras We shall review and record the names of several categories of interest to us. Definition 2.18. (i) The category of topological groups (all of which are assumed to be Hausdorff) and continuous group homomorphisms as morphisms is denoted by TopGr. (ii) If C is a category, a subcategory A is called full if A(A1 , A2 ) = C(A1 , A2 ) for each pair  of objects A1 , A2 in ob(A). If C is any class of objects of a category C, then A = (A1 ,A2 )∈C×C C(A1 , A2 ) is a full subcategory of C called the full subcategory of C-objects. (iii) The full subcategory of TopGr of all topological groups having a Lie algebra (in the sense of Definition 2.11) is denoted by LieAlgGr. The full subcategory of LieAlgGr of all Lie groups will be called LieGr. (iv) The full subcategory of TopGr of abelian topological groups is denoted by AbTopGr. (v) The category of Hausdorff topological spaces with continuous functions as morphisms is denoted by Top, and the category of Hausdorff topological spaces with base point and base point preserving continuous functions as morphisms is denoted by Top∗ . (vi) The category of Hausdorff topological spaces with scalar multiplication and scalar morphisms between them is denoted by Scal (see discussion following 2.6).

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It is worth taking note of the fact that a morphism f : G → H between two groups having a Lie algebra is simply a continuous group homomorphism. Theorem 2.19. Let f : G → H be a TopGr-morphism. Then there is a unique scalar morphism L(f ) : L(G) → L(H ) such that L(f )

L(G) −−−−−→ L(H ⏐ ⏐ ) ⏐ ⏐exp expG   H G −−−−−→ H

(10)

f

commutes. It is defined as L(f )(X) = f  X. Proof. Since f is continuous, if U is an identity neighborhood of G we find an identity neighborhood V of G such that f (V ) ⊆ U ; if now C is a compact subset of R, then for X, Y ∈ L(G), the relation Y (r)−1 X(r) ∈ V for all r ∈ C implies that L(f )(Y )(r)−1 L(f )(X)(r) = (f Y )(r)−1 (f X)(r) = f (Y (r)−1 X(r)) ∈ f (V ) ⊆ U for all r ∈ C. It follows that L(f ) is continuous. Let r ∈ R. Then L(f )(r · X)(t) = f ((r · X)(t)) = f (X(tr)) = (f  X)(tr) = (r · (f  X))(t) = (r · L(f )(X))(t). Thus L(f ) is a scalar morphism. If x ∈ L(G), then expH L(f )(X) = L(f )(X)(1) = f (X(1)) = f (expG X). Thus (10) is commutative. It remains to show uniqueness. Assume that  : L(G) → L(H ) is a scalar morphism such that expH  = f  expG . Let X ∈ L(G). Then expH ((r · X)) = expH (r · (X)) = (X)(r) on the one hand and f (expG r · X) = f (X(r)), that is (X) = f  X = L(f )(X). Let I : Scal → Top∗ and J : TopGr → Top∗ be the forgetful functors which assign to a space with scalar multiplication the underlying pointed space, respectively, to a topological group the underlying pointed space. Theorem 2.20. The assignments G  → L(G) and f  → L(f ) define a continuous functor L : TopGr → Scal, and exp : I L → J is a natural transformation of functors: TopGr → Top∗ . Proof. We have to verify several statements. (a) L is a functor: If f : G → G is the identity map, then L(f )(X) = f  X = X shows that L(f ) is the identity morphism. If f1 : G1 → G2 and f2 : G2 → G3 are morphisms of topological groups, then L(f2 f1 )(X) = (f2 f1 )X = f2 (f1 X) = (L(f2 )  L(f1 ))(X). Thus L(f2  f1 ) = L(f2 )  L(f1 ). (b) exp is a natural transformation I L → J : This is immediate from the definition of a natural transformation of functors in 1.1 and the commutativity of the diagram (10) above. (c) The functor L is continuous, i.e., preserves limits: (See 1.2 and 1.16.) We shall apply Theorem 1.17 and show that L preserves products and equalizers. def

groups. Let pr j : P =  Products. Let {Gj | j ∈ J } be a family of topological  G → G denote the projections. Define ϕ : L(P ) → j k∈J k j ∈J L(Gj ) by ϕ(X) = (pr j X)j ∈J ; conversely, if (Xj )j ∈J ∈ P , then the universal property of the product gives a unique X : R → P such that Xj = pr j X. If we write X = ψ((Xj )j ∈J ),

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121

 then ψ : j ∈J L(Gj ) → P is an inverse of ϕ. Since both ϕ and ψ preserve scalar multiplication (which is componentwise on products) we have    L Gj ∼ L(Gj ). = j ∈J

j ∈J

Equalizers. Let f1 , f2 : G → H be two morphisms of topological groups, set E = {g ∈ G | f1 (g) = f2 (g)} and e : E → G the inclusion, then E is a closed subgroup of G and e is the equalizer of f1 and f2 . We claim that L(e) : L(E) → L(G) is the equalizer of L(f1 ), L(f2 ) : L(G) → L(H ). For this purpose, let X ∈ L(E), then L(f1 )(X)(t) = f1 (X(t)) = f2 (X(t)) = L(f2 )(X)(t). Thus X equalizes L(f1 ) and L(f2 ). Conversely assume that X ∈ L(G) equalizes L(f1 ) and L(f2 ). Then f1 (X(t)) = L(f1 )(X)(t) = L(f2 )(X)(t) for all t ∈ R and thus X(t) ∈ E for all t, and thus X ∈ L(E). This concludes the proof that L preserves products and equalizers and thus arbitrary limits by 1.17. To the category theorists, the continuity of L is not a surprise because L = Hom(R, −) : TopGr → Scal is a hom-functor. Definition 2.21. Let G be a topological group, let G0 denote the connected component of 1 in G, called the identity component of G, and let Ga denote the arc component of 1 in G, called the identity arc component of G. Further, denote by E(G) the subgroup expG L(G) algebraically generated by all points lying on a one parameter subgroup. Finally, let E(G) denote the smallest closed subgroup of G containing the images of all one parameter subgroups; that is 1 0 E(G) = expG (L(G)) = E(G).

(11)

Observe that G0 and Ga are mapped into themselves by any continuous map G → G which fixes 1, and that G0 , Ga , E(G), and E(G) are mapped into themselves by all endomorphisms of G. That is, each of these four groups is a fully characteristic subgroup of G (see [102, Exercise E1.12 following Theorem 1.34]). Notice that Ga and E(G) may not be closed as the example of the p-adic solenoid (in 1.4(A) (ii)) shows; see also [102], 1.28, 1.38, and Chapter 8 of [102] in general. In the paragraph following Proposition 2.5 we described an arcwise connected closed additive subgroup G of a Banach space where G has no nontrivial one-parameter subgroup, whence E(G) = E(G) = {0} = Ga = G0 = G. Recall from Definition 2.11 (iii) that we say that the Lie algebra L(G) is generating if G has a Lie algebra and G0 = E(G). Since expG L(G) is arcwise connected, so is expG L(G) and hence the closure of this group is connected. If G is a Lie group, then by Theorem 2.13 we have E(G) = G0 and thus the Lie algebra of any Lie group is generating. The universe in which we shall work is that of topological groups which have a generating Lie algebra. Later in this chapter we will identify a smaller universe that is just right for us.

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In [102], Exercise E7.17 following Example 7.39, we encountered a closed contractible (hence arcwise connected) nonsingleton subgroup G of the additive Lie group of a Banach space which has no one parameter subgroups, hence satisfies E(G) = {0}. If X is an arcwise connected compact pointed space and F (X) is the free compact abelian group on X (see [102, Definition 8.51ff.]) the subgroup X of F (X) is free as an abelian group (see [102, Proposition 8.52]) and thus as a topological group satisfies E(X) = {0} while being arcwise connected. Similar comments apply to the free (nonabelian) compact group. Thus there is an abundance of connected topological groups (even abelian ones) which have a Lie algebra due to the fact that they have no nontrivial one parameter subgroups. This confirms that no structural information via the exponential function is to be obtained unless the Lie algebra is generating. We note that for every one parameter subgroup X and every endomorphism f of G the composition f  X is again a one parameter subgroup. Thus f (E(G)) ⊆ E(G) and so again E(G) is a fully characteristic subgroup. If G has a Lie algebra, the group E(G) has a generating Lie algebra by definition. If H is a subgroup of a topological group G, then the coextension of any one parameter subgroup X : R → H to G gives a one parameter subgroup R → G which we shall also write as X by a slight abuse of notation; that is, if incl : H → G is the inclusion, we identify X and incl  X. Thus we write L(H ) ⊆ L(G). Proposition 2.22. For any topological group G, one has L(expG L(G)) = L(Ga ) = L(G0 ) = L(G) = L(E(G)).

(12)

The following statements are equivalent: (i) (ii) (iii) (iv)

G has a Lie algebra. G0 has a Lie algebra. E(G) has a Lie algebra. Ga has a Lie algebra.

Proof. Since R is arcwise connected and every one parameter subgroup X : R → G is a continuous map mapping 0 to 1 we have X(R) ⊆ expG L(G) ⊆ Ga ⊆ G0 and X(R) ⊆ E(G). Hence L(G) ⊆ L(expG L(G)) ⊆ L(Ga ) ⊆ L(G0 ) and L(G) ⊆ L(E(G)) ⊆ L(G0 ). But since trivially L(G0 ) ⊆ L(G), all of these containments are equalities. We shall see later that for locally compact groups G we have Ga = exp L(G) and G0 = E(G), but that is not obvious at this point. In [102] it was shown that for any locally compact abelian group G one has Ga = exp L(G) (see [102, Theorem 8.30 (ii)]) and that for a compact abelian group  Z) G, the factor group G/Ga , as an abstract abelian group, is isomorphic to Ext(G, (loc. cit. (iii)). In the case of the p-adic solenoid Tp of Example 1.20 (A)(ii) we have L(Tp ) ∼ = R, and expTp : L(Tp ) → Tp is injective. Thus (Tp )a is a copy of R endowed with a properly coarser topology. By the preceding proposition we have

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123

R∼ = L(Tp ) = L((Tp )a ). In particular, (Tp )a is a topological group with a Lie algebra due to Theorem 2.14 and its Lie algebra is R. It should be clear that, in the spirit of universal topological algebra, a topological Lie algebra is a real Lie algebra L which is at the same time a Hausdorff topological space such that scalar multiplication R × L → L, addition L × L → L and Lie bracket [·, ·] : L × L → L are continuous. Thus a Dynkin algebra is a topological Lie algebra.

Definition 2.23. The category of topological Lie algebras and continuous Lie algebra morphisms is denoted by LieAlg. Proposition 2.24. (i) The category LieAlg of topological Lie algebras is complete. (ii) The functor L : TopGr → Scal maps the category LieAlgGr of topological groups with Lie algebras into the category LieAlg of topological Lie algebras. Proof. Part (i) is Exercise E2.7. (ii) If G is a topological group with a Lie algebra, then by Definition 2.11, L(G) ∈ ob(Scal) is a topological Lie algebra and thus belongs to the subcategory LieAlg. Now let f : G → H be a LieAlgGr-morphism. We know that L(f ) : L(G) → L(H ) is a Scal-morphism. We must show that for X, Y ∈ L(G) we have f (X + Y ) = f (X) + f (Y ) and f [X, Y ] = [f (X), f (Y )]. Firstly we deal with addition: By 2.11(4) we have , , - , --n t t . Y (X + Y )(t) = lim X n→∞ n n Since f is continuous and a group morphism we have , , , -- , , ---n t t L(f )(X + Y )(t) = f ((X + Y )(t)) = lim f X f Y n→∞ n n , -n 1 1 = lim · (f  X) · (f  Y ) (t) = (L(f )(X) + L(f )(Y ))(t). n→∞ n n Next we treat the Lie bracket: By 2.11 (5) we have , , , --n2 t t [X, Y ](t ) = lim comm X . ,Y n→∞ n n 2

Accordingly, we get this time , , , -, , ---n2 t t L(f )[X, Y ](t) = f ([X, Y ](t)) = lim comm f X ,f Y n→∞ n n - n2 , 1 1 · (f  X), · (f  Y ) = lim comm (t) n→∞ n n = [L(f )(X), L(f )(Y )](t). Thus L(f ) is a Lie algebra morphism, and the proof of the proposition is complete.

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Exercise E2.7. Prove Proposition 2.24 (i). [Hint. Prove that the cartesian product of topological Lie algebras with the product topology is a topological Lie algebra and that the equalizer E = {x ∈ L | f1 (x) = f2 (x)} of two morphisms fj : L → K, j = 1, 2 of topological Lie algebras is a topological Lie algebra.] The Completeness Theorem of the Category of Groups with Lie Algebras Theorem 2.25. (i) The category LieAlgGr of topological groups having a Lie algebra is closed in the category TopGr of topological groups under the formation of arbitrary limits and passage to closed subgroups. In particular, LieAlgGr is a complete category. The full subcategory LieAlgGenGr of all topological groups having a generating Lie algebra is closed under the formation of arbitrary products and passage to retracts. (ii) The functor L : LieAlgGr → LieAlg is continuous, i.e. preserves all limits. (iii) There exists a functor L : TopGr → LieAlgGr such that given any topological group G, there is a topological group LG with Lie algebra and a morphism ηG : G → LG with dense image, such that for every morphism f : G → H into a topological group H with Lie algebra, there is a unique morphism f  : LG → H such that f = f   ηG . TopGr

G ⏐ ⏐ ∀f  H

LieAlgGr

ηG

−−−−−→ LG ⏐ ⏐  f

LG ⏐ ⏐  ∃!f

−−−−−→

H

idH

H

Proof. Once it is shown that LieAlgGr is closed under the formation of limits, from Theorem 2.20 and Proposition 2.24 we conclude that L : LieAlgGr → LieAlg is a continuous functor. Thus, by Theorem 1.11 (ii) it suffices to show that LieAlgGr and LieAlgGenGr are closed under the formation of products and the passing to intersections of coherent retracts in TopGr. def  Products. Let {Gj | j ∈ J } be a family of topological groups and P = j ∈J Gj its product. From Theorem 2.20 we know that we can write 

L

j ∈J

  Gj = L(Gj ) j ∈J

such that expP (Xj )j ∈J = (expGj Xj )j ∈J .

The Category of Topological Groups with Lie Algebras

Since each L(Gj ) is a topological Lie algebra, so is



j ∈J

125

L(Gj ) by 2.24 (i). Moreover,

(Xj )j ∈J + (Yj )j ∈J = (Xj + Yj )j ∈J ,, -n 1 1 = lim · Xj · Yj n→∞ n n j ∈J , -n 1 1 = lim · (Xj )j ∈J · (Yj )j ∈J . n→∞ n n This shows that condition 2.11(i)(4) holds. Analogously one   proves condition 2.11(i)(5). Gj 0 = j ∈J (Gj )0 (Exercise E2.8.) One observes straightforwardly that j ∈J    and E = j ∈J Gj j ∈J E(Gj ) and concludes readily from these facts that LieAlgGenGr is closed in TopGr under the formation of products. Passage to closed subgroups. Let H be a closed subgroup of G. We may assume that L(H ) ⊆ L(G), as every one parameter subgroup of H may be considered as one of G. The relations ,, -, --n 1 1 def X + Y = lim ·X ·Y , (∗) n→∞ n n , - n2 1 1 def [X, Y ]   = lim comm (∗∗) · X, · Y n→∞ n n and X + Y, [X, Y ] ∈ L(H ) hold as they hold in G and since H is closed. Thus H is a topological group with Lie algebra and thus belongs to LieAlgGr. Retracts. Let p : G → H be a retraction in TopGr, i.e. there is a morphism j : H → G in TopGr such that pj = idH . There is no loss in generality in assuming that H is a subgroup of G and that j : H → G is the inclusion map (see E1.5), and since H is a retract, it is in fact a closed subgroup. Assume that G belongs to LieAlgGenGr. Then H belongs to LieAlgGr by the preceding arguments. Functors preserve retractions and coretractions; specifically, L(p) : L(G) → L(H ) is a retraction with L(p)  L(j ) = L(pj ) = L(idH ) = idL(H ) . Thus L(H ) is a retract of L(G). We notice that p(E(G)) ⊆ E(H ). Hence, if G ∈ ob LieAlgGenGr, then p(G0 ) = p(E(G)) ⊆ E(H ). Now p|H = pj = idH , whence H0 = p(H0 ) ⊆ p(G0 ). Thus H0 ⊆ E(H ) ⊆ H0 , and equality follows. Thus LieAlgGenGr is closed in TopGr under passing to retracts, and this completes the proof. Finally, (iii) is an immediate consequence of the completeness of the subcategory LieAlgGr and the Retraction Theorem for Full Closed Subcategories of TopGr 1.41. In the proof we used the fact that a retraction p : G → H of topological groups maps G0 onto H0 . This is a special situation for retractions. In general, even quotient maps between locally compact abelian groups fail to map components onto components. In [102, E1.11 following Definition 1.30], one observes a quotient morphism Zp × R → Tp ;

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the identity component of the domain is {0}×R and the p-adic solenoid Tp is connected. (This is only a special case of a general theorem on abelian topological groups: see [102, Theorem 8.20].) We also recall that there are examples of additive groups of separable Banach spaces G having closed arcwise connected subgroups H with zero Lie algebra (see [102, Exercise E7.17 following Example 7.39]). Then G is in LieAlgGenGr while H is not. Since G and G/H are in LieAlgGenGr the subgroup H , as a kernel is a limit. Thus the category LieAlgGenGr is not closed under the passage to all limits and thus fails to be complete. Exercise E2.8. Fill in the steps left out in the proof of the claim that a product of topological groups with Lie algebras is a topological group with a Lie algebra. The category LieGr of Lie groups, which by Theorem 2.12 is a subcategory of LieAlgGr, has finite products and equalizers, hence finite limits. Theorem 2.25 shows that all limits, in particular, arbitrary products and projective limits of Lie groups, are topological groups with Lie algebras. In Exercise E4.4 in Chapter 4 we shall give an explicit construction for the functor L : TopGr → LieAlgGr.

The Lie Algebra Functor Has a Left Adjoint Functor In Theorem 2.25 (iii), which rests on the Retraction Theorem for Full Closed Subcategories 1.41, we have, more or less implicitly, established the existence of a left adjoint functor, but we did not talk about this fact at that time. Talking about left adjoint functors explicitly now becomes inevitable. We recall that a functor U : A → B is said to have a left adjoint functor F : B → A if there is a natural transformation ηB : B → U (F (B)) such that for each morphism f : B → U (A) in B there is a unique morphism f  : F (B) → A such that f = U (f  )  ηB . In diagram form: B B ⏐ ⏐ ∀f 

A

ηB

−−−−−→ U (F⏐(B)) ⏐  U (f )

U (A) −−−−−→ idU (A)

U (A)

F (B) ⏐ ⏐  ∃!f

()

A

For details we refer to any basic text on category theory or to [102, Appendix 3]. The relation between the two adjoint functors may be expressed by saying that there is a natural bijection between the sets B(B, U (A)) and A(F (B), A). In fact, in our notation, the bijection is implemented by the function f  → f  and its inverse g  → U (g)  ηB : A(F (B), A) → B(B, U (A)).

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127

The natural transformation η is called the front adjunction. There is, dually, a back adjunction πA : F (U (A)) such that for each morphism g : F (B) → A there is a unique morphism g  : B → U (A) such that g = πA  F (g  ) (see for instance [102, Proposition A3.36]). B U (A)  ⏐ ∃!g  ⏐ B

A πA

F (U(A)) −−−−−→ ⏐ F (g  )⏐ F (B)

A  ⏐∀g ⏐

−−−−−→ F (B) idF (B)

Among the characteristic properties of adjunctions the following one will be relevant in the present context: If B is an object in B, then πF (B)  F (ηB ) : F (B) → F (B) is the identity map of the A-object F (B). Similarly, if A is an object of A, then U (πA )  ηU (A) : U (A) → U (A) is the identity map of the B-object U (A). (See for instance [102, Proposition A3.38].) Let Cat be a complete full subcategory of the category LieAlgGr of topological groups with Lie algebra. Then we have a continuous functor L : Cat → LieAlg. The following theorem will provide a left adjoint functor Cat : LieAlg → Cat. We alert the reader beforehand that this functor will depend on the subcategory Cat. We can take Cat = LieAlgGr right away, but later we shall be interested in the category Cat of all projective limits of Lie groups, and the left adjoints Cat and LieAlgGr are likely to be quite different; the former we shall identify rather accurately and explicitly via projective limits. The Adjunction Theorem for L Theorem 2.26. (i) The functor L : Cat → LieAlg possesses a left adjoint functor Cat : LieAlg → Cat. (i ) In other words, for each topological Lie algebra g there is a functorially associated topological group Cat (g) with a Lie algebra and there is a natural transformation ηg : g → L(Cat (g)) such that for each morphism f : g → L(H ) of topological Lie algebras, there is a

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unique morphism f  : Cat (g) → H such that f = L(f  )  ηg . In diagram form: LieAlg

g ⏐ ⏐ ∀f 

Cat

ηg

−−−−−→ L(Cat ⏐ (g)) ⏐  L(f )

L(H ) −−−−−→ idL(H )

Cat ⏐(g) ⏐  ∃!f

L(H )

()

H

(i ) Let G be a group with a Lie algebra. Then there is a natural transformation  → G, πG : G

 def G = Cat (L(G)),

such that for each topological Lie algebra g and each morphism f : Cat (g) → G there is a unique morphism f  : g → L(G) of topological Lie algebras such that f = πG  Cat (f  ). LieAlg

L(G)  ⏐ ∃!f  ⏐ h

Cat πG

Cat (L(G)) −−−−−→  ⏐  Cat (f )⏐ Cat (h)

G  ⏐∀f ⏐

(⊥)

−−−−−→ Cat (h) idCat (h)

(ii) The group Cat (g) has a generating Lie algebra and is therefore a member of

LieAlgGenGr. Thus Cat maps LieAlg into LieAlgGenGr.

(iii) For a topological Lie algebra, g, abbreviate Cat (g) by G. Then there are two inverse isomorphisms πG : Cat (L(G)) → G and Cat (ηg ) : G → Cat (L(G)).

Proof. (i) The proof is almost pure category theory; we will use the core existence result for adjoint functors which we have already invoked in the proof of Theorem 1.40. By the Left Adjoint Functor Existence Theorem (see for instance [102, Theorem A3.60]), we have to verify that L satisfies the Solution Set Condition (see for instance [102, Definition A3.58]), which in our case reads as follows: For each topological Lie algebra g, there is a set S(g) (and not a proper class) of pairs (f, H ), where f : g → L(H ) is a morphism of topological Lie algebras, such that for any pair (F, K), F : g → L(K), there is a pair (f, H ) ∈ S(g) and a morphism f0 : H → K such that F = L(f0 )  f . As is usual in such a situation, this condition is verified by establishing cardinality estimates. (a) There are, up to equivalence, only a set of homomorphic surjective homomorphisms f : g → h of topological Lie algebras since there is a cardinality bound on the set of closed ideals i of g and a cardinality bound on the set of topologies on each quotient g/i.

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129

(b) Given a topological space T there is a cardinality bound on all equivalence classes of dense embeddings of T into some space T, because there is a cardinality bound on the set of all filters on T , and because every point in a space T, in which T is contained densely, is the limit of a filter on T . (c) Given a topological Lie algebra h, there is, up to equivalence, only a set of continuous functions e : h → S onto a Hausdorff space S up to equivalence; next there is for each space S, up to isomorphy, at most a set of groups H which are algebraically generated by S, and there is at most a set of group topologies on H . Hence there is at most a set of topological groups H which have h as their Lie algebra and satisfy H = expH h. Moreover, by (b) above, there is at most a set of topological groups in which expH h is dense. We say that two pairs (fj , Hj ), fj : g → L(Hj ), j = 1, 2 are equivalent if there is an isomorphism ϕ : H1 → H2 such that f2 = L(ϕ)  f1 . Now we consider the 0 1 class of all pairs (f, H ), f : g → L(H ) such that H = expH (f (g)) . The preceding cardinality considerations show that we have a set S(g) of representatives (f, H ) for the equivalence classes of such pairs. Now let K be a topological group with a Lie algebra and F : g → L(K) a morphism of topological Lie algebras. Let H = expK f (g). Then H is a closed subgroup of K and thus has a Lie algebra by Theorem 2.25 (i). Moreover, the corestriction f : g → L(H ) gives a pair (f, H ) which is equivalent to a member of S(g). If f0 : H → K is the inclusion morphism we have F = L(f0 )  f , and since (f, H ) is equivalent to a member of S(g), this proves that L satisfies the Solution Set Condition. Since L is continuous by Theorem 2.25 (ii), the LeftAdjoint Functor Existence Theorem applies and proves the existence of a left adjoint functor Cat for L. (i ) and (i ): The universal properties expressed in (i ) and (i ) are equivalent and express the fact that Cat is a left adjoint functor of L. See for instance [102, Proposition A3.36]. (ii) This assertion is a consequence of the selection of the solution set and the construction of the left adjoint functor from the solution set in the Left Adjoint Functor Existence Theorem; indeed, (ηg , Cat (g)) is a member of the solution set S(g). def

(iii) Let g be a topological Lie algebra and set G = Cat (g). By (ii), G =  def  = expG ηg (g). Set G = Cat (L(G)) and note that in a similar vein, we have G 1 0 expG˜ ηL(G) (L(G)) . The situation is described by the following diagram: ηg

ηL(G)

 −−−−−→ L(⏐G) g −−−−−→ L(G) ⏐ ⏐ ⏐ expG  expG˜  G. G −−−−−→ Cat (ηg ) 0 1 0 1 Thus expG˜ ηL(G) (L(G)) = Cat (ηg ) (expG L(G)) is dense in Cat (ηg )(G) on the  on the other. Hence Cat (ηg ) has a dense image. We recall from one hand and in G our review preceding the theorem, that πCat (g)  Cat (ηg ) : Cat (g) → Cat (g) is the identity of Cat (g) = G, that is πG  Cat (ηg ) = idG . We saw that the coretraction

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Cat (ηg ) has a dense image; but then it must be surjective and thus an isomorphism whose inverse is πG . According to this Theorem 2.26 (i) we have a natural bijection between the sets LieAlg(g, L(G))

and Cat(Cat (g), G).

 = Cat ◦ L does so Since Cat depends on the category Cat, the functor G  → G as well, but notation indicating this dependence would be clumsy.

Sophus Lie’s Third Fundamental Theorem Once a complete full subcategory Cat of LieAlgGr containing all finite-dimensional Lie groups is fixed we often simply write  = Cat and note that Theorem 2.26 (ii ) is a very general form of Lie’s Third Theorem. This becomes more evident if one restricts one’s attention to the class of topological Lie algebras, for which ηg : g → L((g)) is an isomorphism. In that case (g) realizes a group whose Lie algebra is the given Lie algebra g. If g is a finite-dimensional Lie algebra, then (g) is the simply connected Lie group whose Lie algebra is the given one. There are Banach Lie algebras g for which ηg fails to be an isomorphism. In Chapter 6 we shall show that for all Lie algebras g which are projective limits of finite-dimensional ones the morphism ηg is an isomorphism. For Lie groups G, the morphism πG : (L(G)) → G is the universal covering  def = (L(G)) is the simply connected morphism of the identity component, and G covering group of the identity component G0 of G. One may wish to consider the two adjunction theorems 2.25 (iii) and 2.26 side by side. If G is a topological group, then there is a left reflection LG, that is a universally attached topological group with a Lie algebra; its Lie algebra L(LG) is the image in LieAlg under a right adjoint functor, and (L(LG)) is the image of it under the left adjoint functor . We have natural maps

πG

(L(LG)) −−−−−→

G ⏐ ⏐η G LG.

On the lower level we are dealing with topological groups with Lie algebras, and the left lower corner is a group in LieAlgGenGr.

The Adjoint Representation of a Topological Group with a Lie Algebra

131

The Adjoint Representation of a Topological Group with a Lie Algebra Let G be a topological group with a Lie algebra and let expG : L(G) → G be its exponential function. Let g ∈ G and denote by Ig : G → G, Ig (x) = gxg −1 , the inner automorphism implemented by g. Let Aut(G) denote the group of automorphisms of G. In the absence of any special information on G we do not topologize Aut(G). The function g  → Ig : G → Aut(G) is a homomorphism of groups. Definition 2.27. Let G be a topological group. The morphism L(Ig ) : L(G) → L(G) of spaces with scalar multiplication induced by Ig according to Theorem 2.20 is called the adjoint morphism and is denoted by Ad(g). By Proposition 2.24 (ii), it is a morphism of topological Lie algebras if G has a Lie algebra. The morphism of (abstract) groups Ad : G → Aut(L(G)) is called the adjoint representation of G. We note that for each g ∈ G the following diagram is commutative: Ad(g)

L(G) −−−−−→ L(G) ⏐ ⏐ ⏐exp expG ⏐   G G −−−−−→ G.

(13)

Ig

This fact may be rephrased in terms of the following formula: (∀g ∈ G, X ∈ L(G))

g(expG X)g −1 = expG Ad(g)(X).

(14)

We note that if a topological group G acts on a topological vector space E by continuous linear operators, then we have g · X − g0 · X0 = g0 · X − g0 · X0 + g · X0 − g0 · X0 + g · X − g0 · X0 − g0 · X + g0 · X0 = g0 · (X − X0 ) + g0 · (g0−1 g · X0 − X0 ) + g0 · (g0−1 g · (X − X0 ) − (X − X0 )). Thus the action (g, X)  → g · X : G × E → E is continuous if it is continuous at (1, 0) and the function g  → g · X0 : G → E is continuous for each X0 . Proposition 2.28. For a topological group G, the function (g, X)  → Ad(g)(X) : G × L(G) → L(G) is continuous. Proof. First we consider an X0 ∈ L(G) and show that g  → g · X0 : G → L(G) is continuous. Let W (Ad(g0 )(X0 ); n, V ) = {X ∈ L(G) : (∀|r| ≤ n)X(r) ∈ V Ad(g0 )X0 (r)} be a basic neighborhood of Ad(g0 )(X0 ) in L(G). We have to find an identity neighborhood U of G such that Ad(Ug0 )(X0 ) ⊆ W (Ad(g0 )X0 ; n, V ). def

Now Ad(Ug0 ) = Ad(U ) Ad(g0 ). Setting Y = Ad(g0 )(X0 ) we have to find U such that Ad(U )Y ⊆ W (Y ; n, V ), i.e. that Ad(u)Y (t) ∈ V Y (t) for all t ∈ R with |t| ≤ n and all u ∈ U , and that is equivalent to (∀u ∈ U, t ∈ [−n, n])

uY (t)u−1 ∈ V Y (t).

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Since G is a topological group and Y is continuous, for each t ∈ [−n, n] there is an identity neighborhood U (t) of G and an open interval It around t in R such that U (t)Y (It )U (t)−1 ⊆ V Y (t). Since [−n, n] is compact, there is a finite sequence t1 , . . . , tk ∈ [−n, n] such that [−n, n] ⊆ It1 ∪ · · · ∪ Itk . Set U = U (t1 ) ∩ · · · ∩ U (tk ). Then U is an open identity neighborhood of G and u ∈ U implies uY (t)u−1 ∈ V Y (t) for all t ∈ [−n, n] which is what we had to show. Finally we have to show that (g, X)  → Ad(g)X : G × L(G) → L(G) is continuous at (1, 0). So let W (0; n, V ) = {X ∈ L(G) : (∀|r| ≤ n)X(r) ∈ V } be a basic neighborhood of Ad(1)(0) = 0 in L(G). We have to find an identity neighborhood U in G and a basic neighborhood W (0; n , V  ) of 0 in L(G) such that Ad(U )(W (0; n , V  )) ⊆ W (0; n, V ), that is, such that for all u ∈ U , the relation X(r) ∈ V  for all |r| ≤ n implies uX(r)u−1 ∈ V for all |r| ≤ n. This can clearly be satisfied by taking n = n and picking U and V  so that U V  U −1 ⊆ V . Corollary 2.29. Let G be a topological group with a Lie algebra. If the group Aut(L(G)) of linear operators on the topological vector space L(G) is given the topology of pointwise convergence, then the representation Ad : G → Aut(L(G)) is continuous. In the context of linear operators on Banach spaces or Hilbert spaces, the topology of pointwise convergence is called the strong operator topology, and Corollary 2.29 says that the adjoint representation of G is strongly continuous. Proposition 2.30. Assume that G is a topological group with a Lie algebra. Then the kernel of the adjoint representation Ad : G → Aut(L(G)) is the centralizer Z(E(G), G) of the closed subgroup generated by the image of the exponential function in G, and the adjoint representation of G induces a faithful representation of G/Z(E(G), G) into Aut(L(G)). Proof. An element g ∈ G is contained in ker Ad iff Ad(g)(X) = X for all X ∈ L(G), i.e. gX(t)g −1 = (Ad(g)(X))(t) = X(t) for all X ∈ L(G) and all t ∈ R. This is saying that gxg −1 = x for all x ∈ expG L(G), and it means that g ∈ Z(expG (L(G)), G). The centralizer Z(A, G) of any subset A of a topological group G is a closed subgroup as an intersection of centralizers of singletons. Hence it contains A ⊆ Z(Z(A, G), G), whence Z(A, G) ⊇ Z(Z(Z(A, G), G), G) = Z(A, G) ⊇ Z(A, G). Thus Z(A, G) = Z(A, G). Therefore we see ker Ad = Z(E(G), G). The canonical decomposition of the morphism Ad (see e.g. [102, passage preceding Definitions 1.9]) gives an injective morphism G/ ker Ad → Aut(L(G)) which is continuous since Ad is continuous. Notice that the kernel is closed, since the topology of pointwise convergence on Aut(L(G)) is Hausdorff. In the next chapter we shall observe that LieAlgGenGr is a rich category by proving that e.g. all compact groups have Lie algebras.

Postscript

133

Postscript There are many books on classical Lie groups. Their approach is quite different from ours primarily because their aims are different. We target the topological group structure of locally compact groups. As will become evident in later chapters (and was evident in [102]), the basic tools are the Lie algebra and the exponential map; this will be the case even when we are well outside the class of Lie groups. Therefore our definition of Lie group is not built around the concept of manifold, but rather that of Lie algebra and, in particular, the formalism of the Baker–Campbell–Dynkin–Hausdorff series. This formalism has been summarized for this purpose in Appendix 1. As we remarked after Definition 2.1, while our definition of a Lie group is not that commonly found in the literature, it is equivalent to other definitions in the literature. Our definition applies at once to linear Lie groups which permitted us to access the classical body of Lie group theory in [102]. The Lie algebra machinery will of course be useful only within a class of topological groups whose members have a Lie algebra. This class is introduced in Definition 2.11 and it is surprisingly large. From [102] we know that it includes all compact groups and all locally compact abelian groups; later we shall see that it includes all locally compact groups. Most importantly we shall see in Chapter 3 that it includes the class of pro-Lie groups; these groups are introduced in that chapter and will be shown to form a complete subcategory of LieAlgGr containing all finite-dimensional Lie groups. We readily see that the categories TopGr of all topological groups and continuous homomorphisms and its full subcategory CompGr of all compact groups are complete. However, the category LCGr of all locally compact groups and continuous homomorphisms is not complete as it fails to have arbitrary products (see 1.11 (iii)). So the category we seek should be better behaved than LCGr and should at least contain all connected locally compact groups. Chapters 8, 9 and 10 of [102] demonstrated how powerful a pair is the Lie algebra of a compact group and the exponential function mapping the Lie algebra of the compact group into the compact group. This pair will also turn out to be powerful in the context of locally compact groups. So we focus our attention on subcategories of LieAlgGr; this is the full subcategory of the category TopGr whose objects are topological groups which have Lie algebras. We showed that the full category LieAlgGr of all topological groups having Lie algebras is closed in the category TopGr of all topological groups, that the property of having a Lie algebra is preserved by passing to closed subgroups, and that the functor L : LieAlgGr → LieAlg assigning to a topological group having a Lie algebra its topological Lie algebra preserves all limits. We also showed that the restriction of L to any full complete subcategory Cat of LieAlgGr has a left adjoint functor  = Cat : LieAlg → Cat which attaches to a topological Lie algebra g in a universal def

fashion a topological group G = (g) in Cat, depending on Cat, such that for any morphism f : g → L(H ) of topological Lie algebras there is a unique morphism f  : G → H in Cat such that f = L(f  )  ηg . For the class of topological groups

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2 Lie Groups and the Lie Theory of Topological Groups

which we shall consider in the book, namely, projective limits of finite-dimensional Lie groups we shall show in Chapter 6, that ηg : g → L(G) is an isomorphism of topological Lie algebras. The existence of the functor  is therefore a novel expression of Lie’s Third Fundamental Theorem saying that every Lie algebra is realized as the Lie algebra of a group. The generality at which we approached Lie theory in the later portions of this chapter thus led us to the uncovering of a functorial set-up that escaped notice in the classical context where one rather straightforwardly aims for the universal  of a connected finite-dimensional Lie group H which is none other covering group H than (L(H )) with the universal covering morphism being none other than the back adjunction πH : (L(H )) → H . One should alert the reader to the possibility that a topological group having no one parameter subgroups whatsoever is a topological group with a Lie algebra, albeit trivial. Lie theory as introduced in this chapter gives information on a topological group only to the extent of the fully characteristic closed subgroup E(G) generated algebraically and topologically by all one parameter subgroups. If G is of the form (g) for some Lie algebra g, then indeed G = E(G).

Chapter 3

Pro-Lie Groups

With the concepts introduced in Chapter 1 we establish that within the category of topological groups TopGr, the full subcategory of projective limits of finite-dimensional Lie groups is closed under the formation of all limits and is therefore a complete category in its own right. We see that, in the spirit of Chapter 2, its members have a decent Lie theory. For a topological group G we define N (G) to be the set of all normal subgroups N of G such that G/N is a finite-dimensional Lie group. We shall call G a pro-Lie group if, firstly, G is a complete topological group, secondly, N (G) is a filter basis, and thirdly, every identity neighborhood of G contains some N ∈ N (G). It is then easy to see that every pro-Lie group G is a projective limit limN ∈N (G) G/N . The converse emerges as a difficult question, but it is shown in Theorem 3.34 of this chapter that any projective limit of finite-dimensional Lie groups is in fact a pro-Lie group. In particular then, the category of pro-Lie groups is complete. In Theorem 3.39 we shall summarize three equivalent ways of defining pro-Lie groups. Another result which sounds obvious, but is not, will be proved in Theorem 3.35; it says: A closed subgroup of a pro-Lie group is a pro-Lie group. Prerequisites. We shall use the Lie theory for topological groups developed in Chapter 2 and apply it to projective limits of Lie groups. Throughout the entire chapter we shall make frequent reference to the theory of limits presented in Chapter 1.

Projective Limits of Lie Groups The Definition of a Projective Limit of Lie Groups Definition 3.1. A topological group G is called a projective limit of Lie groups if there is a projective system of finite-dimensional Lie groups P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} whose projective limit limj ∈J Gj is isomorphic to G. The full subcategory in TopGr of projective limits of Lie groups will be called LieProGr. Proposition 3.2. For a topological group G, the following conditions are equivalent: (i) G is a projective limit of Lie groups. (ii) G is isomorphic to a closed subgroup of a product of a family of finite-dimensional Lie groups. (iii) G is isomorphic to a closed subgroup of a projective limit of Lie groups.

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Proof. (i) ⇒ (ii): This is an immediate consequence of the explicit definition of a projective limit (see Proposition 1.18 and the discussion preceding 1.18). (ii) ⇒ (iii): Every product is a projective limit of its finite partial products (see Theorem 1.5(b)). A finite product of finite-dimensional Lie groups is a finite-dimensional Lie group. Thus a product of a family of finite-dimensional Lie groups is a projective limit of Lie groups. (iii) ⇒ (i): A projective limit of Lie groups is a closed subgroup of a product of finite-dimensional Lie groups (see 1.18). Thus G is isomorphic to a closed subgroup H  of a product α∈A Gα of finite-dimensional Lie groups Gα . Then by Corollary 1.35 of ∼ the Closed Subgroup Theorem for Projective Limits 1.34 we know that H = limj ∈J Hj for a projective system of closed subgroups of finite products α∈j Gα of finite-dimensional Lie groups. Since closed subgroups of finite-dimensional Lie groups are Lie groups, H is a projective limit of Lie groups and since G ∼ = H , assertion (i) follows. Completeness Theorem for Projective Limits of Lie Groups Theorem 3.3. (i) The category LieProGr of projective limits of Lie groups is closed in TopGr under the formation of all limits and is therefore complete. (ii) LieProGr is the smallest full subcategory of TopGr that contains all finite-dimensional Lie groups and is closed under the formation of all limits. Proof. (i) By Proposition 3.2, the full subcategory LieProGr of the category TopGr of topological groups is closed under the passage to closed subgroups, hence the passing to kernels of continuous cocycles and under the formation of projective limits. Hence it is closed under the formation of all limits in TopGr and is therefore a complete category by Proposition 1.11 (b). (ii) Let C be any full subcategory of TopGr which contains all finite-dimensional Lie groups and is closed in TopGr under the formation of all limits and the passing to isomorphic objects. Let G be a projective limit of Lie groups. Then G = limj ∈J Gj for a projective system of finite-dimensional Lie groups Gj . Then all Gj are contained in C and since C is closed under the formation of all limits, G is in C. Thus LieProGr ⊆ C. By this result we have identified the smallest full subcategory of TopGr which is closed under the formation of all limits and contains the full subcategory of all finitedimensional Lie groups. This closure of the category of finite-dimensional Lie groups is precisely LieProGr. We shall show in Theorem 3.35 below that LieProGr is also closed under passing to closed subgroups. By Theorem 3.3, in particular every limit of projective limits of Lie groups is a projective limit of Lie groups. Exercise E3.1. Provide a direct proof of the fact that a product of projective limits of Lie groups is a projective limit of Lie groups.

The Lie Algebras of Projective Limits of Lie Groups

137

[Hint. Let {fjαk : Gαk → Gαj | (j, k) ∈ J α × J α , j ≤ k},

α∈A

be a family of projective systems of finite-dimensional Lie groups. Then we have a family {Gα : α ∈ A}, Gα = limj ∈J α Gαj of groups that are themselves projective def  limits of Lie groups. We shall show that G = α∈A Gα is a projective  limit of Lie groups. We let F be the poset of finite subsets of A and set J = F × α∈A J α . For j = (F1 , (j1α )α∈A ) and k = (F2 , (j2α )α∈A ) we set j ≤ k iff F1 ⊆ F2 and j1α ≤ j2α for  all α ∈ F1 . Then J is directed. For j = (F, (j α )α∈A ) we write Gj = α∈F Gαjα . For j ≤ k we define fj k : Gk → Gj by fj k ((gjαα )α∈F2 ) = (fj1α j2α (gjαα ))α∈F1 . 2

2

Then {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} is a projective system of finite-dimensional Lie groups. For j ∈ J , j = (F, (j α )α∈A ) we find a morphism pj : G → Gj , pj (((gjαα )j α ∈J α )α∈A ) = (gjαα )α∈F , giving us a morphism p : G → limj ∈J Gj . Show that p has an inverse.] Examples 3.4. If G belongs to any class of topological groups in the following list, then the set N (G) of all closed normal subgroups N such that G/N is a finite-dimensional Lie group is a filter basis converging to 1 and G ∼ = limN ∈N (G) G/N ; in particular, G is a projective limit of Lie groups, and G is indeed a limit of a very special projective system of finite-dimensional Lie groups with all bonding maps being quotient morphisms. (i) (ii) (iii) (iv)

All finite-dimensional Lie groups. Products of any family of finite-dimensional Lie groups. Compact groups. Locally compact abelian groups.

Exercise E3.2. Prove the assertions made in 3.4. [Hint. (iii) See Fundamental Theorem of Projective Limits 1.27 and [102, Corollary 2.43]. (iv) See Fundamental Theorem 1.27 and [102, Theorem 7.57]. Note that a locally compact abelian group H with a compact open subgroup K is a projective limit of Lie groups because K is a projective limit of Lie groups by (iii), and because a topological group having an open Lie subgroup is a Lie group.] In particular, for any set X the power RX is a projective limits of Lie groups.

The Lie Algebras of Projective Limits of Lie Groups We now merge the developments of topological groups with Lie algebras on the one hand and of projective limits of Lie groups on the other. In Definition 2.11 it was explained what it means when we say that a topological group has a Lie algebra.

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Theorem 3.5. (i) Every projective limit of Lie groups has a Lie algebra; that is, LieProGr ⊆ LieAlgGr.

(ii) The subcategory LieProGr is closed in LieAlgGr under the formation of all limits. (iii) The functor L : LieProGr → LieAlg is continuous, that is, preserves all limits. Proof. Let G ∈ ob LieProGr. Then G ∼ = limj ∈J Gj for a suitable projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of finite-dimensional Lie groups Gj . Theorem 2.12 shows that Gj ∈ LieAlgGenGr. Hence by the Completeness Theorem for Groups with Lie Algebras 2.25 we have G ∈ LieAlgGr. The remainder follows. By 3.4, every compact and every locally compact abelian group is a projective limit of Lie groups; all those groups have a generating Lie algebra, see [102, Chapter 9, Definition 9.44ff., and Chapter 7, Theorem 7.71].

Pro-Lie Algebras The Definition of Pro-Lie Algebras Definition 3.6. Let L be a topological Lie algebra and  (L) the filter basis of closed ideals I of L such that L/I is finite-dimensional. Then L is called a pro-Lie algebra if the following conditions are satisfied: (i)  (L) converges to 0. (ii) As a topological vector space, L is complete. The category of pro-Lie algebras and continuous Lie algebra morphisms is denoted by proLieAlg. It would have been a bit more accurate to call a pro-Lie algebra a profinite-dimensional Lie algebra, but in the end we opted for brevity. Recall from Appendix 2, Definition A2.5, that a real topological vector space L is called weakly complete iff it is the projective limit of the projective system of its finitedimensional quotient spaces. Weakly complete topological vector spaces are complete locally convex topological vector spaces. Thus we have the following proposition. Proposition 3.7. Let L be a real topological vector space, and consider it as a commutative topological Lie algebra. Then the following statements are equivalent. (i) L is a pro-Lie algebra. (ii) As a topological vector space, L is weakly complete. (iii) The natural evaluation map η : L → (L )∗ is an isomorphism of topological vector spaces, where L is the topological dual of L and (L )∗ is the algebraic dual of L endowed with the weak *-topology.

Pro-Lie Algebras

139

(iv) L is isomorphic as a topological vector space and as a topological Lie algebra with zero brackets to a product RJ for some set J . Proof. Exercise E3.3. Exercise E3.3. Prove Proposition 3.7. [Hint. Use Theorems 1.13 and 1.14 to establish the equivalence of (i) and (ii). The equivalence of (ii) and (iii) is established by applying Theorem A2.8 of the Appendix. (iii) implies (iv): The real vector space L has a basis and thus as a vector space is isomorphic to a vector space R(J ) . Now by (iii) we have the topological vector space isomorphisms L ∼ = (R(J ) )∗ ∼ = (R )J ∼ = RJ . = (L )∗ ∼ (iv) implies (ii): If L = RJ then L is the projective limit of the finite-dimensional partial products (see Example 1.20 (B)). The filter basis of partial products RJ \F with F finite in J and where R J \F is considered as a subspace of RJ in the obvious fashion is cofinal in the filter basis N0 (L) of all cofinite-dimensional closed vector subspaces. Thus L ∼ = limN ∈N0 (L) L/N. Alternative proof: The categories of real vector spaces and that of weakly complete topological vector spaces are dual to each other (see Appendix 2, Theorem A2.8.) The category of real vector spaces is cocomplete; by duality, the category of weakly complete topological vector spaces is complete, and thus is closed under the formation of products in the category of all topological vector spaces; thus, since R is weakly complete, so is RJ . See also Appendix 2, Corollary A2.9.] Proposition 3.8. The underlying topological vector space of a pro-Lie algebra is weakly complete. Proof. Let  (L) again denote the filter basis of closed ideals of the pro-Lie algebra L such that dim L/I < ∞ and let N denote the filter basis of all closed vector subspaces M such that dim L/M < ∞. Clearly,  (L) ⊆ N . In view of the Cofinality Lemma 1.21 it is sufficient to show that  is cofinal in N . For this purpose let N ∈ N . In the finite-dimensional vector space L/N find an open identity neighborhood V in which {0} is the only vector subspace. Let p : L → L/N be the quotient morphism of topological vector spaces and set U = p −1 (V ). Then U is a zero neighborhood of L. Since lim  (L) = 0 by 3.12 (i) there is an I ∈  such that I ⊆ U . Then p(I ) is a vector subspace contained in V and therefore is {0} by the selection of V . Thus I ⊆ M, and this shows that  (L) is cofinal in N . By Definition 3.12 and Theorem 1.30 we have L ∼ = limI ∈ (G) L/I . Then limI ∈ (L) L/I = limM∈N L/M by the Cofinality Lemma 1.21. Therefore L is weakly complete by definition. For an overview of the linear algebra of weakly complete topological vector spaces see Appendix 2, notably Theorems A2.8, A2.11, and A2.12. We immediately record an exact analog of Proposition 1.40 on topological groups. Proposition 3.9. Let L be a topological Lie algebra. Then the following conditions are equivalent. (i) L is a pro-Lie algebra.

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(ii) There is a filter basis J of ideals converging to 0 such that L/I is finite-dimensional for each I ∈ J. (iii) There is a filter basis J of ideals contained and cofinal in  (L). Proof. Since (i) ⇒ (ii) is trivial by Definition 3.6 of a pro-Lie algebra; (iii) ⇒ (i) is a consequence of the Cofinality Lemma 1.21. Therefore we must prove (ii) ⇒ (iii). Clearly, J ⊆  (L). We claim that (∀I ∈  (L))(∃J ∈ J) I ⊇ J.

(∗)

This means that J is cofinal in  (L). The proof of (∗) remains, and we leave it as Exercise E3.4. Exercise E3.4. Finish the proof of Proposition 3.9. [Hint. Emulate the last portion of the proof of 1.29 (vi), see also 3.27 below.]

The Definition of a Topological Group with a Pro-Lie Algebra and with a Lie projective component Definition 3.10. A topological group G is said to have a pro-Lie algebra if it has a Lie algebra (i.e. if it belongs to LieAlgGr) and if L(G) is a pro-Lie algebra. The full subcategory of LieAlgGr of all topological groups having pro-Lie algebras and morphisms of topological groups between them is denoted by proLieAlgGr. A topological group G is said to have a Lie projective component if G0 , the identity component, is a projective limit of Lie groups. The full subcategory of TopGr of all topological groups with a Lie projective component is called proLieComGr. Instead of using the very compact phrase that a topological group “has a pro-Lie algebra” we shall also say that a topological group “has a pro-Lie algebra as Lie algebra”. Proposition 2.22 yields at once a simple observation, which should nevertheless be kept in mind: Proposition 3.11. For a topological group G, the following statements are equivalent: (i) G has a pro-Lie algebra. (ii) The identity component, G0 , of G has a pro-Lie algebra. (iii) The identity arc component, Ga , of G has a pro-Lie algebra. As a first indication of the significance of this concept we record Theorem 3.12. Every topological group whose identity component is a projective limit of Lie groups has a pro-Lie algebra as Lie algebra. In particular, every projective limit of Lie groups has a pro-Lie algebra as Lie algebra. So proLieGr ⊆ LieProGr ⊆ proLieAlgGr.

Pro-Lie Algebras

141

Proof. In view of 3.11 it suffices to show that a projective limit of Lie groups has a pro-Lie algebra. Let G be a projective limit of Lie groups. Then G ∼ = limj ∈J Gj for a suitable projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of finitedimensional Lie groups Gj . By the continuity of L from 2.25 (ii) we have L(G) ∼ = limj ∈J L(Gj ). Let pj : L(G) → L(Gj ) denote the limit morphisms and set M = {ker pj | j ∈ J }. Since each ker pj is an ideal, and dim L(G)/ ker pj ≤ dim L(Gj ), the set M is a filter basis of ideals i of L(G) such that L(G)/i is finite-dimensional, and lim M = 0 by the Fundamental Theorem of Projective Limits 1.27 (ii). Now Proposition 3.9 shows that L(G) is a pro-Lie algebra. From 3.4 we get immediately: Corollary 3.13. Every compact group and every locally compact abelian group is a topological group with pro-Lie algebra. Proof. Exercise E3.5. Exercise E3.5. Prove Corollary 3.13. [Hint. Every compact group and every locally compact abelian group is a projective limit of Lie groups.] So the category proLieComGr (and thus also the category proLieAlgGr) contains CompGr and LCAbGr. Theorem 3.14 (Completeness Theorem for the Category of Topological Groups with Pro-Lie Algebras). The categories proLieAlgGr and proLieAlg are complete, and the functor L : proLieAlgGr → proLieAlg is continuous. Proof. From 3.8 we know that the category LieAlgGr of topological groups with Lie algebras and the category LieAlg of topological Lie algebras are complete, and that L : LieAlgGr → LieAlg is a continuous functor. We claim that an arbitrary product def  L = j ∈J Lj of pro-Lie algebras Lj is a pro-Lie algebra. Let j denote the filter basis of all closed ideals of Lj such that Lj /I is finite-dimensional for all I ∈ Jj . def  Now we let F denote all products I = j ∈J Ij such that Ij ∈ j and all but finitely many of the Ij agree with Lj . Then each member I of  is a closed ideal and L/I is finite-dimensional. Thus F is contained in the filter basis  of all closed ideals I of L such that L/I is finite-dimensional. Since all filter bases j converge to 0, the filter basis F and thus the finer filter basis  converge to 0. Since L is complete as the product of complete topological vector spaces, L is a pro-Lie algebra, as asserted. Now let L be a pro-Lie algebra and  the filter basis of ideals I such that L/I is finite-dimensional. Let E be a closed subalgebra of L; then {K ∩ I | I ∈  } is a filter basis of ideals of K which converges to 0, and K is complete as a closed vector subspace of a complete topological vector space. Finally, let I ∈  . Then K/(K ∩ I ) ∼ = (K + I )/I ⊆ L/I . Hence K/(K ∩ I ) is finite-dimensional. This shows that K is a pro-Lie algebra. In particular, this applies to the equalizer E ⊆ L1 of two morphisms L1 → L2 of pro-Lie algebras. Thus proLieAlg has products and equalizers

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3 Pro-Lie Groups

and is therefore complete (see Theorem  1.10). Since L is continuous on LieAlgGr we conclude that for a product G =  j ∈J Gj of Lie groups Gj with pro-Lie algebras L(Gj ) has a Lie algebra L(G) ∼ = j ∈J L(Gj ) which is a pro-Lie algebra. A similar argument applies to equalizers. Thus proLieAlgGr has products and equalizers and thus is complete by 1.8. Theorem 3.15 (Completeness Theorem for the Category of Topological Groups with Lie Projective Component). The category proLieComGr of topological groups whose identity component is a projective limit of Lie groups is complete, and the functor L : proLieComGr → proLieAlg is continuous. Proof. From 2.24 and 2.25 we know that the categories LieAlg of topological Lie algebras and LieAlgGr of topological groups with Lie algebras are complete, and that L : LieAlgGr → LieAlg is a continuous functor. By Theorem 3.5, every topological group with a pro-Lie component has a pro-Lie algebra. What remains to show is that proLieComGr is closed in TopGr under the passing to products and closed subgroups. Since the identity component of a product of topological groups is the product of their identity components, and since the category of projective limits of Lie groups is closed under the formation of products by Theorem 3.3, the category proLieComGr is closed under the formation of products in TopGr. Now let G be a topological group such that G0 is a projective limit of Lie groups. Let H be a closed subgroup of G. Then H0 ⊆ G0 , and H0 is closed in G0 . By Proposition 3.2 it follows that H0 is a projective limit of Lie groups. This completes the proof of the theorem. The full subcategory of connected topological groups in TopGr was denoted ConnGr in Chapter 1, just before Theorem 1.11. It is relevant in our present context that the assignment G  → G0 : TopGr → ConnGr is a functor with good properties. We recall from Theorem 1.11 (v) that the full category ConnGr is complete while it is not closed under passing to limits in TopGr. Proposition 3.16. The functor G  → G0 : TopGr → ConnGr is right adjoint to the inclusion functor and is therefore continuous. Proof. Let |•| : ConnGr → TopGr denote the inclusion functor and let εG : |G0 | → G be the inclusion morphism. If C is a connected topological group and |C| the underlying topological group, then for every morphism f : |C| → G of topological groups, since f (C) is connected and contains 1, we have f (C) ⊆ G0 and thus there is a unique corestriction f  : C → G0 such that f = εG  |f  |. This shows that (•)0 : TopGr → ConnGr is right adjoint to | • |. All right adjoints preserve limits. (See e.g. [102, Theorem A3.52.].) Because of the equation L(G) = L(G0 ), the functor L : proLieComGr → proLieAlg factors in the form (•)

L

proLieComGr −−−→ proLieComGr ∩ ConnGr −−−→ proLieAlg.

Weakly Complete Topological Vector Spaces and Lie Algebras

143

Weakly Complete Topological Vector Spaces and Lie Algebras Information on weakly complete topological vector spaces is collected in Appendix 2. One way of explaining a weakly complete topological vector space is declaring a topological vector space as weakly complete if there is an isomorphism of topological vector spaces to some product vector space RX (see Appendix 2, Corollary A2.9). Lemma 3.17. (a) Let f : V → W be a morphism of weakly complete topological vector spaces. Then f (V ) is a closed vector subspace of W , and the natural bijection V / ker f → f (V ) is an isomorphism of topological vector spaces. (b) If V and W are closed vector subspaces of a weakly complete topological vector space U , then V + W is closed, and the function f : V /(V ∩ W ) → (V + W )/W , f (v + (V ∩ W )) = v + W is an isomorphism of topological vector spaces. Proof. See Appendix 2, Theorem A2.12. Lemma 3.18. Let g = limk∈J gk be a projective limit of a projective system {γj k : gk → gj | j ≤ k, (j, k) ∈ J × J } of finite-dimensional real vector spaces in the category of weakly complete topological vector spaces. Let γj : g → gj denote the limit maps. Then for each j ∈ J there is an index kj ≥ j such that γj kj (gk ) ⊆ γj (g). Proof. See Appendix 2, Corollary A2.10. We shall have to deal with topological groups G for which we make some standard assumptions: Notation 3.19 (General Assumptions). For a projective limit G of Lie groups there is a projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of finite-dimensional Lie groups Gj such that G = limj ∈J Gj . The limit maps are denoted fj : G → Gj , the kernels ker fj of the limit maps will be abbreviated by Kj . The finite-dimensional Lie algebras L(Gj ) will be written gj . Let us write def

def

fj k = L(fj k ) and fj = L(fj ). By Theorem 2.25 (ii), the functor L : LieAlgGr → LieAlg from the category of all topological groups having a Lie algebra and continuous group morphisms between them to the category of topological Lie algebras is continuous and thus, in particular preserves projective limits. Hence L(G) ∼ = limj ∈J L(Gj ), and we may identify the two profinite-dimensional Lie algebras. Thus {fj k : gk → gj | (j, k) ∈ J × J, j ≤ k}

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3 Pro-Lie Groups

is a projective system of finite-dimensional real Lie algebras and Lie algebra morphisms such that L(G) = lim gj j ∈J

and that the continuous Lie algebra morphisms fj : L(G) → gj are the limit morphisms. We set aj = fj (L(G)) ⊆ gj for each j ∈ J , and let αj k : ak → aj be the morphism of finite-dimensional Lie algebras induced by fj k for j ≤ k. Lemma 3.20. The system L = {αj k : ak → aj | (j, k) ∈ J × J, j ≤ k} def

is a projective system of finite-dimensional Lie algebras and surjective bonding maps. Then L(G) = lim aj . j ∈J

The limit maps αj : L(G) → aj are quotient morphisms. Proof. We apply the Fundamental Theorem on Projective Limits 2.1 (iv) to the system L and conclude that lim L = lim L. The limit maps αj : L(G) → aj are surjective and thus are quotient maps (Appendix 2, Theorem A2.8 (b)). It also follows that the bonding maps αj k : ak → aj are surjective. The following diagram illustrates the situation: a⏐j ⏐ inclj  gj

αj k

←−− a⏐k ⏐ inclk  ←−− gk fj k

··· ··· ···

←− L(G) = limj ∈J aj ⏐ ⏐id  L(G) ←− L(G) = limj ∈J gj .

Now for each j ∈ J , the subalgebra aj of the finite-dimensional Lie algebra gi dedef

termines an analytic subgroup Aj = expGj aj  of Gj such that L(Aj ) = aj . (For linear Lie groups a reference is [102, Theorem 5.52]. The proof there does not depend on the assumption that G is a linear Lie group.) Lemma 3.21. Under our general assumptions 3.19 for G = limj ∈J Gj we have (∀j ∈ J )(∃kj ≥ j, kj ∈ J ) fj kj ((Gkj )0 ) ⊆ Aj . Proof. From 3.18 we have (∀j ∈ J )(∃kj ≥ j, kj ∈ J ) fj kj (gkj ) ⊆ aj . The assertion now follows from the fact that as a finite-dimensional connected Lie group, (Gkj )0 is algebraically generated by exp gkj and Aj is algebraically generated by aj . Thus fj kj ((Gkj )0 ) = fj kj (exp gkj ) = exp L(fj kj )(gkj ) ⊆ exp aj  = Aj .

Weakly Complete Topological Vector Spaces and Lie Algebras

145

def

The morphisms fj k : Gk → Gj induce morphisms ψj k = fj k |Ak : Ak → Aj with L(ψj k ) = αj k and fj0k : (Gk )0 → (Gj )0 . Then {ψj k : Ak → Aj | (j, k) ∈ J × J, j ≤ k} def

is a projective system of analytic groups; let A = limj ∈J Aj be its limit. Each analytic group carries a topology which is in general finer than the induced topology, making the subgroup Aj into a connected Lie group Hj such that L(Hj ) = L(Aj ) = aj and that the morphisms ψj k : Aj → Ak induce morphisms of Lie groups ϕj k : Hk → Hj such that L(ϕj k ) = αj k . We have injective morphisms incl(Gj )0

inclAj

εj

Hj −−−−−→ Aj −−−−−→ (Gj )0 −−−−−→ Gj where εj is the bijective morphism of topological groups given by εj (h) = h and incl denotes the respective inclusion morphisms. We consider the projective system def

H = {ϕj k : Hk → Hj | (j, k) ∈ J × J, j ≤ k} of finite-dimensional Lie groups and let H = limj ∈J Hj denote its limit; we note that due to the continuity of the functor L we have L(H ) = lim L(Hj ) = lim aj = L(G). j ∈J

j ∈J

(L)

It is not at all clear at this time that the limit of a projective system of connected Lie groups is connected if all bonding maps are surjective and all groups in the system are connected. However, we observe the following lemma which we shall presently apply to H = limj ∈J Hj . Lemma 3.22. Assume that H is a projective limit limj ∈J Hj of finite-dimensional Lie groups satisfying the following two hypotheses: (i) For all j ∈ J the Lie group Hj is connected, and (ii) the limit maps ϕj : H → Hj , j ∈ J induce surjective morphisms L(ϕj ) : L(H ) → L(Hj ). Then H is connected. Proof. Let h be an arbitrary element of H . We shall show that arbitrarily close to h there are elements from the arc component of the identity of H ; thus the arc component of the identity is dense in H and so H is indeed connected. For a proof let U be any identity neighborhood of H . By 1.27 (i) we may assume that U = ϕj−1 (V ) for some identity neighborhood V of Hj . Since Hj is connected by hypothesis (i), and since any connected finite-dimensional Lie group is algebraically generated by the image of its exponential function, there are elements X1 , . . . , Xn ∈ L(Hj ) such that

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3 Pro-Lie Groups

ϕj (h) = exp X1 . . . exp Xn . By hypothesis (ii) the morphism L(ϕj ) : L(H ) → L(Hj ) is surjective, and thus we find elements Ym ∈ L(H ), such that Xm = L(ϕj )(Ym ) for all m = 1, . . . , n. Accordingly, expHj Xm = expHj L(ϕj )(Ym ) = ϕj (expH Ym ) in Hj for all m and so ϕj (h) = ϕj (expH Y1 ) . . . ϕj (expH Yn ) = ϕj (expH Y1 . . . expH Yn ). Let α : [0, 1] → H denote the arc in H given by α(t) = expH (t · Y1 ) . . . expH (t · Yn ). Then α(0) = 1 and α(1) = expH Y1 . . . expH Yn ∈ ϕj−1 (ϕj (h)) = h ker ϕj ⊆ hU . This proves our claim and thus finishes the proof of the lemma. Lemma 3.23. The system def

H = {ϕj k : Hk → Hj | (j, k) ∈ J × J, j ≤ k} is a projective system of quotient morphisms between finite-dimensional connected Lie groups and its limit H = limj ∈J Hj is a connected projective limit of Lie groups. The limit maps ϕj : H → Hj are quotient morphisms. Proof. Since all L(ϕj k ) = αj k are surjective, the morphisms ϕj k are surjective, and since Hk as a connected finite-dimensional Lie group is σ -compact and locally compact and Hj is locally compact, by the Open Mapping Theorem (see e.g. [102, Appendix 1, Exercise EA1.21 preceding Remark A1.60]) the morphisms ϕj k are quotient morphisms. Therefore, the limit maps ϕj : H → Hj are quotient morphisms by Theorem 1.27 (iii). The preceding Lemma 3.22 applies to show that H is connected. We illustrate the situation in the following diagram showing the limits of the various projective systems we consider: H⏐j εj ⏐  A⏐j inclAj ⏐  (G⏐j )0 incl(Gj )0 ⏐  Gj

ϕj k

←−−−− fj k

H⏐k ⏐

εk 

←−−−−

A⏐k ⏐ inclAk  fj0k ←−−−− (G⏐k )0 ⏐ incl(Gk )0  fj k ←−−−− Gk

··· ··· ··· ··· ··· ··· ···

H = lim ⏐ j ∈J Hj ⏐ε  A = lim ⏐j ∈J Aj ⏐incl  A G0 = (lim⏐j ∈J (Gj )0 )0 ⏐incl  G0 G = limj ∈J Gj .

(1)

The universal property of the limit G gives us the morphisms ε : H → A and the various inclusion morphisms incl filling in diagram (1). Notice that L(H ) = L(A) = L(B) = L(G) and we may identify L(ε) and the various maps L(incl) with idL(G) . By the concrete construction of the limits we have    Gj | (∀j ≤ k in J ) fj k (gk ) = gj , G = (gj )j ∈J ∈ 

A = (aj )j ∈J ∈ 

H = (hj )j ∈J ∈

j ∈J



j ∈J



j ∈J

 Aj | (∀j ≤ k in J ) fj k (ak ) = aj ,  Hj | (∀j ≤ k in J ) fj k (hk ) = hj .

Weakly Complete Topological Vector Spaces and Lie Algebras

147

Thus A is a subgroup of G and we may identify H with A except that its topology may be finer than the topology induced from G on A. The situation is further illustrated by the following diagram: =

=

εj

inclAj

−−−−−→ L(G) L(H ⏐ ) −−−−−→ L(A) ⏐ ⏐ ⏐ ⏐exp expH ⏐ exp A   G H − − − − − → A − − − − − → G ⏐ ⏐ ⏐ ε inclA ⏐ ⏐f ϕj ⏐ ψ j    j Hj −−−−−→ Aj −−−−−→ Gj , where ε and all εj are bijective and all incl are embeddings. For a given projective limit of Lie groups G = limj ∈J Gj , a very special type of connected projective limit of Lie groups H with quotient morphisms as limit maps emerged almost out of nowhere and it is mapped under the bijective morphism ε onto the subgroup A of G. Clearly we must identify this subgroup of G0 . Lemma 3.24. H = G0 . Proof. By Lemma 3.21, (∀j ∈ J )(∃j ≤ kj ∈ J )

fj kj ((Gkj )0 ) ⊆ Aj .

Now we notice that (Gkj )0 is locally arcwise connected and Hj is Aj equipped with the arc component topology (see [102, Theorem 5.52 (iv) and Lemma A4.1ff.]). Hence the restriction and corestriction fj kj |(Gjk )0 : (Gjk )0 → Aj factors through εj : Hj → Aj for a morphism fj kj : (Gkj )0 → Hj such that def

fj0kj = inclAj  εj  fj kj : (Gkj )0 → (Gj )0 . Temporarily, set def

G0 = lim (Gj )0 ⊇ G0

(∗)

j ∈J

in the category of topological groups and continuous morphisms. Thus for each j ∈ J there is a kj ≥ j and a commutative diagram (Gj )0 o

fj0k

j

(Gkj )0 w w w ww ww w ww ww fj kj {ww Hkj Hj o ϕ j kj

fj0k

j

 (Gkj )0

G0 β

···

inclkj εkj

inclj εj

 (Gj )0 o

···

 H inclkj εkj

···

 G0 .

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3 Pro-Lie Groups def

It follows that there is a morphism βj = fj kj |(Gkj )0  fkj |G0 : G0 → Hj which is independent of the choice of kj in as much as it agrees with fj kj  fkj k  fk |G0 for k ≥ kj . We notice that for j ≤ j  we get βj = ϕjj  βj : G0 → Hj . Thus the universal property of H = limj ∈J Hk implies the existence of a unique morphism β : G0 → H such that βj = ϕj  β. From inclA  ε  fj kj = fj0kj we conclude that inclA  ε  β = idG0 . Thus inclA  ε : H → G0 is a retraction, and since it is injective, it is an isomorphism. As it is also an inclusion map (except for continuity), we now see that it is an isomorphism. This shows G0 = H . Thus G0 is connected and so H = G0 ⊆ G0 .

(∗∗)

Now (∗) and (∗∗) imply H = G0

Pro-Lie Groups Among the projective limits of Lie groups there is a class of groups for which a projective limit representation is particularly nice. We discuss these now. Firstly, we introduce some terminology. Let G be a topological group and N (G) the set of all normal subgroups such that G/N is a finite-dimensional Lie group. Note that a normal subgroup N of a topological group is closed if and only if the quotient group G/N is Hausdorff – which is certainly the case if G/N is a Lie group. We shall occasionally call a subgroup N of a topological group G a co-Lie subgroup if it is normal and G/N is a Lie group. Thus N (G) is the set of all co-Lie subgroups; certainly G ∈ N (G). We shall say that a topological group G has arbitrarily small co-Lie subgroups if every identity neighborhood of G contains a member of N (G). We claim that in a topological group with arbitrarily small co-Lie subgroups the set N (G) is closed under finite intersections and thus is a filter basis (Glöckner’s Lemma). For a proof, let M, N ∈ N (G) and let δ : G → G/M × G/N be defined by δ(g) = (gM, gN ). Since G/M and G/N are Lie groups, they contain identity neighborhoods VM and VN , respectively, which contain no subgroups except the singleton def

ones. Then U = δ −1 (VM × VN ) is an identity neighborhood of G such that any subgroup of G contained in U is mapped to {(1, 1)} and thus is contained in ker δ = M ∩N . But G has arbitrarily small co-Lie subgroups. Thus there is a co-Lie subgroup D of G contained in U such that G/D is a Lie group. By what we just saw, D ⊆ M ∩ N . Since G/D ∼ = (G/D)/((M ∩ N )/D), the factor group G/(M ∩ N ) is a quotient group

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149

of G/D and is therefore a Lie group. Hence M ∩ N ∈ N (G), and this completes the proof. In any topological group with arbitrarily small co-Lie subgroups it makes sense to talk about the filter basis N (G), and the condition that G has small co-Lie subgroups means exactly that this filter basis converges to the identity. The Definition of Proto-Lie Groups and Pro-Lie Groups Definition 3.25. A topological group G is called a proto-Lie group if it has arbitrarily small co-Lie subgroups, that is, if (i) every identity neighborhood of G contains a normal subgroup such that G/N is a Lie group, and it is called a pro-Lie group if, in addition, (ii) G is complete. The group G is called protodiscrete, respectively, prodiscrete if the filter of identity neighborhoods has a basis of open normal subgroups, respectively, G is complete and protodiscrete. The category of all pro-Lie groups and all continuous group homomorphisms between them will be called proLieGr. By the remarks preceding the definition, condition (i) is equivalent to saying (i ) N (G) is a filter basis converging to 1. Warning. Many authors call a topological group a pro-Lie group if the set N (G) of closed normal subgroups N such that G/N is a finite-dimensional Lie group is a filter basis converging to 1 and contains compact members. Such groups are locally compact by Proposition 1.34. Pro-Lie groups in the sense of this book comprise a much larger class of topological groups; indeed we shall see presently that they form a complete subcategory of TopGr so that for instance all powers RX are pro-Lie groups whether X is finite or not. We alert the reader to the fact that in our paper [107] with M. Stroppel we used the more narrow terminology which causes all pro-Lie groups to be locally compact. At the end of the chapter we shall survey the most relevant statements which are equivalent to the definition of a pro-Lie group and thus may equally well serve as definition. In that context we shall exhibit an example of a complete abelian group G for which N (G) is not a filter basis; indeed N (G) will contain two members M and N such that M ∩ N = {1} while G is not a Lie group. Proposition 3.26. Let G be a topological group such that N (G) is a filter basis. Then the following conclusions hold:

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3 Pro-Lie Groups

(i) The map γG : G → GN (G) =

lim

N ∈N (G)

G/N

of Theorem 1.29 is an embedding morphism if and only if G is a proto-Lie group. (ii) The map γG : G → GN (G) is an isomorphism if and only if G is a pro-Lie group. (iii) For any proto-Lie group G, the set UN (G) of open identity neighborhoods U for which there is an N ∈ N (G) such that U N = N U = U is a basis of the filter of identity neighborhoods of G. Proof. Assertion (i) is an immediate consequence of the definition of a proto-Lie group (Definition 3.25), of Theorem 1.29, and of Theorem 1.30. Statement (ii) then follows from Theorem 1.33 and the fact that a pro-Lie group is always complete. Claim (iii) is then a consequence of the Fundamental Theorem of Projective Limits 1.27 (i). Proposition 3.26, in particular, shows that every proto-Lie group has a completion, and that this completion is a pro-Lie group. The example of the group of rationals in its natural topology shows that a dense subgroup of a Lie group need not be a proto-Lie group. We recall that for any family {Aj : j ∈ J } of groups the restricted product is the subgroup of j ∈J Aj consisting of all (aj )j ∈J with aj = 1 for all but a finite number of indices j ∈ J . The restricted product of a constant family {Aj : j ∈ J }, Aj = A for all j ∈ J , is denoted A(J ) . Exercise E3.6. Verify the following examples of proto-Lie groups. Example. (a) The additive group of any dense vector subspace of a weakly complete topological vector space is a proto-Lie group. (b) In particular, for any set X, the restricted product (in this case the direct sum) R(X) is a proto-Lie group when it is equipped with the topology induced from the product RX . (c) The additive group of any hyperplane in a weakly complete topological vector space (closed or not) is a proto-Lie group. (d) The  restricted product of any given family {Lj | j ∈ J } of Lie groups in the product j ∈J Lj is a proto-Lie group. [Hint. (a) Let V be a dense vector subspace of a weakly complete topological vector space W . If N is a closed vector subspace such that dim W/N < ∞, then V /(V ∩N ) ∼ = (V + N)/N is a dense vector subspace of W/N and thus agrees with W/N . Therefore limN V /(V ∩ N ) = limN W/N = W . (b) and the nonclosed case of (c) follow from (a) since a nonclosed hyperplane is dense. The closed case of (c) follows from 3.35. (d) is straightforward from the fact that the cofinite partial products are co-Lie subgroups.] Before we look at further examples, we provide an elementary, but useful tool.

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151

Proposition 3.27. For a topological group G, the following conditions are equivalent: (i) G is a proto-Lie group. (ii) There is a filter basis M of closed normal subgroups converging to 1 such that G/M is a finite-dimensional Lie group for each M ∈ M. If these conditions hold, then M is cofinal in N (G). Moreover, if G is complete, then these conditions are equivalent to (iii) G is a pro-Lie group. If (iii) holds then G ∼ = limM∈M G/M. Proof. Since (i) ⇒ (ii) is trivial by Definition 3.25 of a proto-Lie group, we prove (ii) ⇒ (i). Clearly, M ⊆ N (G). We claim that (∀N ∈ N (G))(∃M ∈ M)

N ⊇ M.

(2)

Let us begin by assuming that condition (2) is satisfied. Then we claim firstly that N (G) is closed under finite intersections and hence is a filter basis: Let N1 , N2 ∈ N (G), then by (2) there are subgroups M1 , M2 ∈ M with Nj ⊇ Mj for j = 1, 2. Since M is a filter basis, there is an M ∈ M such that M1 ∩ M2 ⊇ M. Hence N1 ∩ N2 ⊇ M. Therefore G/(N1 ∩N2 ) is a quotient group of the finite-dimensional Lie group G/M and is therefore itself a finite-dimensional Lie group. Hence N1 ∩ N2 ∈ N (G). Secondly, since M ⊆ N (G), and since M converges to 1, the filter basis N (G) converges to 1 as well. And finally, by (2), M is cofinal in N (G), whence G ∼ = limM∈M G/M = limN∈N (G) G/N by the Cofinality Lemma 1.21. Thus it remains to prove (2). So let N ∈ N (G) be given. Let U = U N be an open identity neighborhood of G such that U N/N is an identity neighborhood of the finitedimensional Lie group G/N which contains no subgroups other than the singleton one. If p : G → G/N is the quotient map, then the image filter basis p(M) converges to the identity in G/N. Hence there is an M such that p(M) ⊆ U N/N. Then the subgroup p(M) is singleton, that is M ⊆ N , which is what we had to show. If G is complete, then (i) shows that G is a pro-Lie group and by Theorem 1.33 we then know that G ∼ = limM∈M G/M. The argument used to prove (2) occurred in a similar fashion in the proof of part (vi) of Theorem 1.29. We draw attention to the fact that in the proof of the preceding result we have used the information that the quotient of a finite-dimensional Lie group is a Lie group. This is not true for Lie groups in general; there are examples of additive groups of separable Banach spaces having closed arcwise connected subgroups with zero Lie algebra (see [102, Exercise E7.17 following Example 7.39]); the respective quotients fail to be Lie groups. Examples 3.28. All classes of topological groups listed in 3.4 are in fact classes of pro-Lie groups. This is an easy exercise from Definition 3.25 of a pro-Lie group and the discussion of 3.4 in Exercise E3.2.

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Using the terminology we have available now, the results we have proved in this section may be summarised as follows Lemma 3.29 (The First Fundamental Lemma on Pro-Lie Groups). Let G be a projective limit limj ∈J Gj of Lie groups. Then the following conclusions hold: (i) The identity component G0 is a pro-Lie group and thus G0 ∼ =

lim

M∈N (G0 )

G0 /M.

def

(ii) Set M = {G0 ∩ Kj : j ∈ J } for the kernels Kj of the limit morphisms fj : G → Gj . Then M is a cofinal subset of N (G0 ); that is, for each M ∈ N (G0 ) there is a j ∈ J such that G0 ∩ Kj ⊆ M. (iii) For each j ∈ J , the natural map G0 /(G0 ∩ Kj ) → (G0 Kj )/Kj is an isomorphism, the group G0 Kj /Kj is a Lie group and a closed subgroup of G/Kj , and G0 = lim (G0 Kj )/Kj . j ∈J

(iv) For each M ∈ M, there is an injective morphism of topological groups from G/M into a Lie group. Proof. (i) Since G0 is a pro-Lie group by Proposition 3.27 and Lemmas 3.23 and 3.24, we have G0 ∼ = limM∈N (G0 ) G0 /M. (ii) Since lim N (G0 ) = 1 and fj : G → Gj is continuous for each j ∈ J , we have lim fj (N (G0 )) = 1. But Gj is a Lie group and thus has no small subgroups. Hence there is an M ∈ N (G0 ) such that fj (M) = {1}, that is, M ⊆ Kj . Thus we have a quotient morphism G0 /M → G0 /(G0 ∩ Kj ). Since quotients of finite-dimensional Lie groups are Lie groups, G0 /(G0 ∩ Kj ) is a Lie group whence G0 ∩ Kj ∈ N (G0 ) by Definition 3.25. Hence M ⊆ N (G0 ). By Theorem 1.27 (i) we know that limj ∈J Kj = 1. Then limj ∈J G0 ∩ Kj = 1. Let M ∈ N (G0 ). Then G0 /M is a Lie group, and thus there is an open identity neighborhood U of G0 such that U M = U and U/M has no nonsingleton subgroups. Then there is a j ∈ J such that G0 ∩ Kj ⊆ U . Since (G0 ∩ Kj )M/M is a subgroup of G0 /M contained in U/M we have G0 ∩ Kj ⊆ M. def

(iii) By (ii) above, G0 /(G0 ∩ Kj ) is a finite-dimensional Lie group. We set N = {Kj : j ∈ J }. By 1.27 we know that lim N = 1. So we can apply the Closed Subgroup Theorem for Projective Limits 1.34 with H = G0 . In particular, 1.34 (iv) yields the assertions of (iii). (iv) According to our conventions 3.19 we have G = limj ∈J Gj for a projective system of finite-dimensional Lie groups Gj . Each M ∈ M is of the form M = ker fj for some j ∈ J for the limit morphism fj : G → Gj . This morphism induces an injective morphism of topological groups Fj : G/Kj → Gj such that fj = Fj  qj for the quotient map qj : G → G/Kj = G/M.

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153

Note that we have shown, in particular, that every connected projective limit of Lie groups is a pro-Lie group. Proposition 3.30. (a) For a projective limit of Lie groups G, the following statements are equivalent: (i) (ii) (iii) (iv)

G is prodiscrete. G is zero-dimensional. G is totally disconnected. L(G) = {0}.

(b) A quotient of a protodiscrete group is protodiscrete. Proof. First we prove (a). (i) ⇒ (ii): By (i) G is a closed subgroup of a product of discrete groups and therefore the filter of its identity neighborhoods has a basis of open subgroups. (ii) ⇒ (iii) ⇒ (iv): This is clear. (iv) ⇒ (i): Let G = limj ∈J Gj for a projective system as in 3.19. By 1.27 (iv) we may and will assume that the limit maps fj : G → Gj have dense images. Let Dj be the discrete group Gj /(Gj )0 and let D = {Fj k : Dk → Dj | j ≤ k, (j, k) ∈ J × J } be the projective system induced by P and let D = limj ∈J Dj . Since Gj = fj (G) and the groups Dj are discrete, the limit maps Fj are surjective. Then each quotient D/ ker Fj for the limit maps Fj : D → Dj is discrete, and D = limj ∈J D/ ker Fj by Theorem 1.27 (ii). Hence D is a prodiscrete group. Now by hypothesis (iv) we have {0} = L(G) = limj ∈J L(Gj ). Then by Lemma 3.21, for each j ∈ J , there is a kj ≥ j such that fj kj ((Gkj )0 ) = {1}. Thus fj kj factors through a morphism Fj kj : Dj kj → Gj . We have a diagram Gj o

fj kj

qj

Gkj qk

 Dkj }} }} } }} }} Fj k } j } } ~} Gkj Gj o  Dj o

···

Fj kj

fj kj

G q

···

 D p

···

 G.

By an argument analogous to that used in the proof of 3.10 we conclude the existence def

of a morphism πj = Fj kj  Fkj : D → Gj which is independent of the choice of kj

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3 Pro-Lie Groups

in as much as it agrees with Fj kj  Fkj k  Fk for k ≥ kj . We notice that for j ≤ j  we get πj = fjj   πj : D → Gj . Thus the universal property of D = limj ∈J Dk implies the existence of a unique morphism p : D → G such that πj = fj  p. Hence G is a retract of D. But retracts of prodiscrete groups are easily seen to be prodiscrete. This completes the proof. Proof of (b): If G is protodiscrete, N (G) is a filter basis of open normal subgroups which converges to 1. Now let N be a closed normal subgroup of G. Define U = {N U/U : U ∈ N (G)}. Each NU is an open and hence closed normal subgroup of G and thus the N U/U are open-closed subgroups of G/N , and we claim that U converges to the identity of G/N. Let W be an open identity neighborhood of G/N and V its full inverse image in G. Then V is an open identity neighborhood of G such that NV = V . Since N (G) converges to 1, there is a U ∈ N (G) such that U ⊆ V . Then NU ⊆ NV = V and thus NU/N ⊆ W . This proves the claim and establishes (b) in view of 3.27. Lemma 3.31 (The Second Fundamental Lemma on Pro-Lie Groups). For any G that is a projective limit of Lie groups, the component factor group G/G0 is protodiscrete; if it is complete, then it is prodiscrete. Proof. We retain the notation of the proof of Proposition 3.30 and consider the commutative diagram (G⏐j )0 ⏐ incl G⏐j ⏐ quot Dj

fj0k

←−−−− (G⏐k )0 ⏐ incl fj k ←−−−− G⏐k ⏐ quot Fj k ←−−−− Dk

···

G⏐0 = limj ∈J (Gj )0 ⏐ incl G = limj ∈J Gj ⏐ ⏐q  D = limj ∈J Dj .

··· ···

The morphism q : G → D is the fill-in map given by the universal property of the limit in the last row. (In passing we note that the first row is a limit diagram in the category of connected topological groups in view of 3.16.) Since the composition incl

quot

(Gj )0 −−−→ Gj −−−→ Dj is constant, so is the composition incl

q

G0 −−−→ G −−−→ D. Hence we have a unique morphism p : G/G0 → D, p(gG0 ) = q(g). Assume that g = (gj )j ∈J ∈ G is such that p(gG0 ) = 1, i.e., (gj (Gj )0 )j ∈J = q(g) = 1 in  limj ∈J Dj ; thus g ∈ j ∈J fj−1 ((Gj )0 ) = limj ∈J (Gj )0 = G0 . This shows that p is injective. The sets Fj−1 (1) are basic identity neighborhoods of D by 1.27 (i). As p−1 Fj−1 (1) = fj−1 ((Gj )0 )/G0 and this is an open-closed subgroup we see that p is

Pro-Lie Groups

155

def

an embedding. Therefore G/G0 may be identified with the subgroup S = im q = {(gj (Gj )0 )j ∈J : (gj )j ∈J ∈ G} of D. Let Nj = Fj−1 (1). Then Nj is an open-closed normal subgroup of D and S∩Nj is an open-closed normal subgroup of S. Since limj ∈J Nj = 1 we have limj ∈J S ∩ Nj = 1.  Hence G/G0 ∼ = S is a protodiscrete group and S = j ∈J SNj is prodiscrete. If G/G0 is complete, then G/G0 ∼ = S and G/G0 is prodiscrete. Before we continue, we record an independent elementary lemma. Lemma 3.32. (i) Let f : A → B be a quotient morphism of topological groups with discrete kernel. Then there is an open symmetric identity neighborhood V of A and an open symmetric identity neighborhood W of B such that f |V : V → W is a homeomorphism, and for every subgroup K of B contained in W there is a subgroup S of A contained in V such that f (S) = K. (ii) If A is connected and B is a pro-Lie group, then A is a pro-Lie group. Proof. (i) Let U be a symmetric open identity neighborhood of A such that U 2 ∩ ker f = {1}. Then f (U ) is an open symmetric identity neighborhood of B. So f |U : U → f (U ) is continuous, open and surjective; if u1 , u2 ∈ U and f (u1 ) = f (u2 ), then u1 u−1 ∈ (ker f ) ∩ U 2 . Thus f |U is a homeomorphism. Now let 2 V be an open symmetric identity neighborhood in A such that V 2 ⊆ U , and set def

W = f (V ). Then f |V : V → W is a homeomorphism onto an open identity neighborhood of B. Define ϕ : W → V to be its inverse and take w1 , w2 ∈ W such that w1 w2 ∈ W . Set vj = ϕ(wj ), j = 1, 2 and v = ϕ(w1 w2 ). Then (f |U )(v) = (f |V )ϕ(w1 w2 ) = w1 w2 . Further v1 v2 ∈ V 2 ⊆ U . Then (f |U )(v1 v2 ) = f (v1 )f (v2 ) = (f |V )ϕ(w1 )(f |V )ϕ(w2 ) = w1 w2 = (f |U )(v). Since (f |U ) is injective, we conclude v = v1 v2 , that is, ϕ : W → V is a homeomorphism such that (∀w1 , w2 ∈ W ) (w1 w2 ∈ W ) ⇒ (ϕ(w1 w2 ) = ϕ(w1 )ϕ(w2 ).

(∗)

In particular, if w ∈ W then w−1 ∈ W and ww −1 = 1 ∈ W and thus ϕ(w)ϕ(w −1 ) = ϕ(1) = 1; therefore ϕ(w−1 ) = ϕ(w)−1 . Now let K be a subgroup of B contained in W . Let g1 , g2 ∈ ϕ(K). Then there are elements w1 , w2 ∈ K ⊆ W such that gj = ϕ(wj ), j = 1, 2 and w1 w2−1 ∈ K ⊆ W . Hence g1 g2−1 = ϕ(w1 )ϕ(w2 )−1 = ϕ(w1 )ϕ(w2−1 ) = ϕ(w1 w2−1 ) ∈ ϕ(K). It follows that ϕ(K) ⊆ V is a subgroup of A. (ii) Now let N be a normal subgroup of B contained in W ; set M = ϕ(N ). Since N is normal in B, in particular wNw−1 = N for all w ∈ W . Applying ϕ we obtain vMv −1 = M for all v ∈ V . Since A is connected, V generates A algebraically and thus M is normal in A. As f : A → B is an open morphism, it induces an open morphism P : A/M → B/N via P (aM) = f (a)N, whose kernel consists of all aM such that f (a) ∈ N . Let m ∈ V ⊆ A be the unique element such that f (m) = f (a). Then am−1 ∈ ker f . We write D = ker f shortly for the discrete kernel of f . Then ker P = DM/M. We claim that ker P is discrete. For this purpose we note that DV M/M is an open identity neighborhood of A/M. An element dM ∈ ker P is in

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3 Pro-Lie Groups

DV M/M iff dM = δvN for some elements δ ∈ D and v ∈ V , that is, iff d = δvm for δ and v as before and m ∈ M. Then δ −1 d = vm ∈ D ∩ V M ⊆ D ∩ V 2 = {1}. Thus v = m−1 ∈ M and d ∈ M ⊆ V ; and since D ∩ V = {1} this entails d = 1. We have shown that DV M/M ∩ ker P = {M}, and this proves the claim that ker P is discrete. Hence P implements a local isomorphism, and so A/M and B/M are locally isomorphic. Therefore, if B is a pro-Lie group, then the filter basis N (B) converges to 1 whence all sufficiently small members N ∈ N (B) are contained in W . Then for all sufficiently small N ∈ N (B), the normal subgroups ϕ(N ) of A form a filter basis converging to 1 in A, and are contained in N (A). If M ∈ N (A), then A/M is a Lie group which has no small subgroups, and so A has an identity neighborhood UM such that all members ϕ(N) ∈ N (A) contained in UM are contained in M. Thus {ϕ(N ) : N ∈ N (B), N ⊆ W } is cofinal in N (A). So A is a proto-Lie group. Now A/D ∼ = f (A) is an open subgroup of B and thus is a pro-Lie group; moreover D, as a discrete group, is complete. Therefore A is complete (see [176], p. 225, 12.3). Hence A is a pro-Lie group as asserted. Lemma 3.33 (The Third Fundamental Lemma on Pro-Lie Groups). Let G be a topological group such that G0 is a finite-dimensional Lie group and assume that there is an injection f : G → L into a finite-dimensional Lie group L. If G/G0 is a protodiscrete group, then G is a finite-dimensional Lie group. Proof. We must show that G0 is open in G. First we make some reductions. Since f (G) is a Lie group as a closed subgroup of a Lie group, we may and will assume that L = f (G). Next, since f −1 (L0 ) is open in G there is no loss in assuming that L = L0 , i.e. that L is connected. Let M = f (G0 ). Then M is a closed normal subgroup of L and f induces an injective map G/f −1 (M) → L/M. Now G/f −1 (M), being a quotient of the protodiscrete group G/G0 is protodiscrete by 4.3 (b) and is, at the same time, without small subgroups. Hence it is discrete, that is, f −1 (M) is open. We may therefore assume that G = f −1 (M), i.e. that M = L. Thus we may assume that f (G0 ) is dense in L.  → L and form the pullback Now we consider the universal covering q : L F

P ⏐ −−−−−→ ⏐ Q G −−−−−→ f

 L ⏐ ⏐q  L.

 : f (g) = q()}. ˜ ∈ G× L ˜ If p = (g, ) ˜ ∈P In terms of elements, we have P = {(g, ) ˜ = 1 and thus g = 1 as f is injective. and F (p) = 1, then ˜ = 1, whence f (g) = q() Thus p = (1, 1) and this shows that F is injective. ˜ ∈ ker Q. Next F maps ker Q isomorphically onto ker q. Indeed let p = (g, ) ˜ Then 1 = Q(p) = g and then qF (p) = q() = f (g) = 1, that is, F (p) ∈ ker q. ˜ = f (1), whence p def Conversely, if ˜ ∈ ker q, then 1 = q() = (1, q) ∈ P , and Q(p) = 1, i.e. p ∈ ker Q and F (p) = q. Now let V be an identity neighborhood of

Pro-Lie Groups

157

 such that V ∩ ker q = {1} and assume that p = (g, ) ˜ ∈ ker Q ∩ (G × V ); then g = L ˜ thus ˜ ∈ V ∩ ker q = {1}. Thus p = 1 and ker Q is Q(p) = 1 and 1 = f (g) = q(); discrete in P . If (U ×V )∩P is an identity neighborhood and p = (u, v) ∈ (U ×V )∩P , then Q(p) = u and f (u) = q(v), whence Q((U × V ) ∩ P ) = U ∩ f −1 q(V ), and this set is an identity neighborhood. Thus the morphism Q is open and thus, since its kernel is discrete, it implements a local isomorphism. Therefore G is a Lie group if and only if P is a Lie group. Thus we must show that P is a Lie group, that is, that P0 is open.  and normal analytic subgroups in Now F (P0 ) is a normal analytic subgroup of L,  is F (P0 ) ker q, simply connected Lie groups are closed. The full inverse of f (G0 ) in L  (P0 ). Since ker q and thus this group is dense, and F (P0 ) ker q/F (P0 ) is dense in L/F  the group L/F  (P0 ) is abelian and simply connected, hence is isomorphic is central in L, to a vector group Rn . Thus F induces an injective morphism of P /P0 into the vector  (P0 ) and so P /P0 has no small subgroups. The quotient morphism group L/F quotG

Q

P −−−−−→ G −−−−−→ G/G0 vanishes on P0 and therefore factors through P /P0 : Q∗

quotP

quotG Q = (P −−−−−→ P /P0 −−−−−→ G/G0 ). We have ker Q∗ = P1 /P0 with P1 = Q−1 G0 . The following is a diagram of abelian topological groups (ι an injective morphism): P /P0 ⏐ ⏐ ∗ Q

ι

−−−→ Rn

G/G0 . def

The morphism Q = Q|P1 : P1 → G0 is a covering morphism of the Lie group G0 with kernel ker Q ∼ = ker q and thus is a Lie group containing the closed normal subgroup P0 = (P1 )0 . Then ker Q∗ = P1 /P0 is a totally disconnected Lie group and is therefore discrete. Since G/G0 has arbitrarily small open subgroups by the hypothesis of protodiscreteness, Lemma 4.5 applies to Q∗ and shows that P /P0 has arbitrarily small open subgroups (that is, P /P0 is a protodiscrete group). But ι injects P /P0 into Rn , and thus P /P0 has an identity neighborhood in which the singleton group {P0 } is the only subgroup; this subgroup, therefore, is open and so P0 is open which is what we had to show. Now we are ready for the principal result of this entire chapter. The Pro-Lie Group Theorem Theorem 3.34. Every projective limit of Lie groups is a pro-Lie group.

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Proof. By the First Fundamental Lemma 3.29, a group G that is a projective limit of Lie groups has a filter basis M of closed normal subgroups M converging to 1 such that G0 M/M is a connected Lie subgroup of G/M, and that there is an injective morphism of G/M into a finite-dimensional Lie group. By the Second Fundamental Lemma 3.31, G/G0 has a basis O of open normal subgroups converging to the identity. It follows that for each M ∈ M, the factor group G/M has a filter basis of open normal subgroups U/M such that every open set V containing G0 M/M contains one of the U/M, U ∈ O. Thus every G/M, M ∈ M, satisfies the hypotheses of the Third Fundamental Lemma 3.33. As a consequence of 3.33, G/M is a finite-dimensional Lie group. Then by Proposition 3.27 it follows that G is a pro-Lie group. The Closed Subgroup Theorem for Pro-Lie Groups Theorem 3.35. A closed subgroup of a pro-Lie group is a pro-Lie group. Proof. The proof is immediate from Theorem 3.34 and Proposition 3.2. The Completeness Theorem for Projective Limits of Lie Groups 3.3 may now be reformulated as saying that the full subcategory of pro-Lie groups in the category of topological groups is closed under the formation of all limits and the passage to closed subgroups. We may record this as follows. Completeness Theorem for Pro-Lie Groups Theorem 3.36. (i) The full subcategory proLieGr of pro-Lie groups in the category TopGr of topological groups is closed under the formation of all limits and is therefore complete. (ii) Every limit of pro-Lie groups, in particular, every limit of finite-dimensional Lie groups is a pro-Lie group. (iii) There is a functor F : TopGr → proLieGr such that given any topological group G, there is a pro-Lie group F G and a morphism ηG : G → F G with dense image such that for every morphism f : G → H into a pro-Lie group H there is a unique morphism f  : F G → H such that f = f   ηG . TopGr

G ⏐ ⏐ ∀f  H

proLieGr

ηG

−−−−−→ F⏐G ⏐  f −−−−−→ idH

H

F⏐G ⏐  ∃!f H

(iv) Free pro-Lie groups exist. Specifically: there is a functor FproLieGr : Top0 → proLieGr from the category of pointed topological spaces into the category of pro-Lie groups such that there is a natural morphism ϕX : X → |FproLieGr (X)| of pointed topological spaces mapping the base point X0 of X to the identity of the topological group

Pro-Lie Groups

159

FproLieGr (X) such for every base point preserving continuous function F : X → H into a pro-Lie group H there is a unique morphism of pro-Lie groups f  : FproLieGr (X) → H such that f = |f  |  ϕX . Top0

X ⏐ ⏐ ∀f 

proLieGr

ϕX

−−−−−→ |FproLieGr ⏐ (X)| ⏐  f

|H | −−−−−→ idH

|H |

FproLieGr ⏐ (X) ⏐  ∃!f H

The subgroup generated in FproLieGr (X) generated algebraically by ϕX (X) is free. If X is completely regular, then ϕX is a topological embedding. Proof. (i) and (ii) are immediate consequences of the preceding results. See also Theorem 1.11 (i), (ii). Condition (iii) is an immediate consequence of (i) and the Retraction Theorem for Full Closed Subcategories of TopGr 1.41. (iv) The category TopGr of topological groups has a free functor FTopGr : Top0 → TopGr (see [71]) whose universal property is encapsulated in the following diagram. Top0

X ⏐ ⏐ ∀f 

TopGr

ψX

−−−−−→ |FTopGr ⏐ (X)| ⏐  f

|H | −−−−−→ idH

|H |

FTopGr ⏐ (X) ⏐  ∃!f H

We recall the retraction functor F : TopGr → proLieGr of (iii) above. Then the composition FproLieGr = F  FTopGr satisfies the requirement. Now let X be completely regular. Then X is topologically embedded into the free compact group via a natural map of pointed spaces ρX : X → |FCompGr (X)| (see for instance [102, Lemma 11.3]). Since the free compact group is a pro-Lie group, the universal property provides us with a morphism ρ  : FproLieGr (X) → FCompGr such that ρX : X = ρ   ϕX , and if ρX is an embedding, ϕX is an embedding. Since the group FTopGr (X) is algebraically free over the image of X, the subgroup ϕX (X) of FproLieGr (X) generated by the image of X under ϕX is free. Theorem 3.36 (iii) should be compared with the more familiar situation of the left adjoint to the inclusion functor CompGr → TopGr, where the Bohr compactification of G takes over the role of F G. Indeed the inclusion CompGr → proLieGr gives rise to a left adjoint mapping each pro-Lie group to its Bohr compactification.

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For results concerning free compact and free compact abelian groups see [94], [95], [96], [97], [100], and [101]; much, but not all, of the content of these sources is summarized in [102].

Some Examples After the Completeness Theorem 3.36 and the Closed Subgroup Theorem 3.35 it is not hard to realize that the class ofpro-Lie groups is quite large, because every closed subgroup of an arbitrary product j ∈J Lj of finite-dimensional Lie groups Lj is a pro-Lie group. The subclass of abelian pro-Lie groups alone is rather big, and we shall devote the entire Chapter 5 to them. After having discussed much of the general foundations of pro-Lie groups rather extensively, the reader might wish to be reminded of some simple examples which fail to be pro-Lie groups while being, say, locally compact. Both of the groups listed in the following two examples are totally disconnected but fail to have arbitrarily small open normal subgroups Example 3.37. Let L be a compact Lie group, for instance L = Z/2Z or L = R/Z. The classical example of a semidirect product LZ σ Z with the shift action of Z on the product is a locally compact group which is not a pro-Lie group because LZ × {0} is the unique smallest normal open subgroup. Example 3.38. The group Sl(2, Qp ) is a p-adic Lie group and thus is a totally disconnected locally compact group which is not a pro-Lie group (in the sense of this book) because it has no nontrivial normal subgroups except {±1}

An Overview of the Definitions of a Pro-Lie Group The mathematical content of this chapter is highly technical, but it is crucial for any deeper understanding of the definition of a pro-Lie group. It therefore serves a useful purpose if we summarize the possible definitions and comment on their relative merits. We recall that with any topological group G we associate canonically the set N (G) of all normal subgroups N such that G/N is a Lie group; occasionally we call the members of N (G) the co-Lie subgroups. We say that a topological group G has arbitrarily small co-Lie subgroups if every identity neighborhood of G contains a member of N (G). The First Definition of Pro-Lie Group Definition A. A topological group is a pro-Lie group if it is complete and has arbitrarily small co-Lie subgroups. It was this definition that we chose in the present chapter to define pro-Lie groups in Definition 3.6.

An Overview of the Definitions of a Pro-Lie Group

161

The Second Definition of Pro-Lie Group Definition B. A topological group G is a pro-Lie group if there is a projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of morphisms between Lie groups such that G is isomorphic to its projective limit limj ∈J Gj . In this chapter we presented this definition in Definition 3.1 in order to define projective limit of Lie groups. The Third Definition of Pro-Lie Group Definition C. A topological group G is a pro-Lie group if it is isomorphic (algebraically and topologically) to a closed subgroup of a product of Lie groups. The major outcome of this chapter is The Equivalence Theorem of the Definitions of Pro-Lie Groups Theorem 3.39. The First, Second, and Third Definitions of Pro-Lie Groups are equivalent. Proof. A ⇒ B: In the circumstances of the first definition, by Glöckner’s Lemma preceding Definition 3.25, the set N (G) is a filter basis. Then Proposition 3.26 (ii) shows that G is isomorphic to the projective limit of Lie groups limN ∈N (G) G/N . B ⇒ A: This was established in the Pro-Lie Group Theorem 3.34. B ⇒ C: This is a consequence of Proposition 1.18 which made the general definition of a projective limit concrete in the category of topological groups.  C ⇒ A: If P = j ∈J Gj for a family of Lie groups Gj , then P is a pro-Lie group in the senseof Definition A, because P is complete and has arbitrarily small co-Lie subgroups j ∈J \F Gj , F ⊆ J finite. Let G be a closed subgroup of P . Then the Closed Subgroup Theorem 3.35 shows that G is a pro-Lie group in the sense of Definition A. In this chapter we chose Definition A for defining the concept of a pro-Lie group. This definition is the group theoretician’s choice because it provides the most immediate access to the information provided to us by the theory of finite-dimensional Lie groups. How powerful this approach is we shall see in the next chapter when we study the behavior of the Lie algebra functor on the quotient morphism G → G/N . However, using merely the information given in Definition A we find it impossible to conclude directly that the category of pro-Lie groups is complete. A proof of the completeness of the category of pro-Lie groups is easy if it is based on Definition B. On the other hand it may be very hard to draw any conclusion on the structure of a topological group G given as a projective limit G = limj ∈J Gj .

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Finally, Definition C formulates the concept of a pro-Lie group very much in the spirit of universal algebra and the theory of varieties. It looks rather simple, but experience shows that knowing a topological group (or any topological algebra, for that matter) to be a subgroup (or subalgebra) of a large product in general says very little about its structure. But Definition C may motivate one to look for product structures when we unfold the structure theory of pro-Lie groups. The two Definitions B and C are rather close – closer in fact that one might surmise at first glance; the Closed Subgroup Theorem for Projective Limits 1.34 is a tool that appears to be needed in one form or another. It is clear from the definition of a pro-Lie group that at the center is an important invariant attached to every topological group G, namely, the set N (G) of all normal subgroups N such that G/N is a Lie group. This set contains G and indeed all open normal subgroups if there are any. The set is partially ordered with respect to the relation ⊆. If S is any nonempty subset 0 1 of N (G), we may set N = S which is the smallest closed subgroup containing all members of S. This subgroup is normal, and since S = Ø we find an S ∈ S such that G/S is a Lie group and S ⊆ N. Then G/N ∼ = (G/S)/(N/S) is a quotient group of a Lie group and is therefore a Lie group, that is, N ∈ N (G). Thus every nonempty subset of N (G) has a least upper bound in N (G). This looks suspiciously as though N (G) should be a complete lattice, that is a partially ordered set in which every subset has a least upper bound and thus, as a consequence, every subset has also a greatest lower bound. But this is not the case, as the following example due to H. Glöckner (see [65, Appendix p. 452]) will show. Recall first that we say that two subgroups M and N of a group are said to be disjoint if M ∩ N = {1}. Example 3.40 (H. Glöckner). There is a complete topological abelian group G with two disjoint closed subgroups Nk , k = 1, 2 such that both quotient groups G/Nk are Lie / N (G). groups, while G itself fails to be a Lie group. So N1 , N2 ∈ N (G) but N1 ∩N2 ∈ Proof. We use the fact, that the additive group of rational numbers carries a grouptopology τ such that it is a nondiscrete complete topological group Qτ with respect to this topology (see [141]). It is nondiscrete and countable and thus cannot be locally compact because of the Baire Category Theorem; in particular, it is not a Lie group. Let  = {(r, r) : r ∈ R} ⊆ R2 and set G = (Qτ ×{0})+. We let b : Qτ × R → G be the bijective map b(q, r) = (q + r, r) and let j : G → R2 be the inclusion morphism. We give G that group topology which makes b an isomorphism of topological groups. Then G is not a Lie group since G/ ∼ = Qτ is not a Lie group. We use the projections def

pr k : R2 → R, k = 1, 2, to get morphisms pk = pr k j |G : G → R which are continuous, surjective and open; that is, they are quotient morphisms. We set Nk = ker pk . Then (q + r, r) ∈ N1 iff q + r = 0, that is N1 = {0} × Q, and (q + r, r) ∈ N2 iff r = 0 and thus N2 = Qτ × {0}. As kernels of morphisms of topological groups, both Nj are closed and G/Nk ∼ = R since both pk are quotient morphisms. Thus N1 , N2 are in N (G).

An Overview of the Definitions of a Pro-Lie Group

163

The example has some interesting aspects which illustrate various tricky features of this set-up: (i) The inclusion map j : G → R2 may be identified with the injective morphism g  → (g + N1 , g + N2 ) : G → G/N1 × G/N2 . Even though each of the quotient maps G → G/Nj is open, this inclusion map is not open onto its image G∗ = Q × R, and G does not inherit the induced topology from this map. The group G∗ is not a Lie group. (ii) The sum N1 + N2 = Qτ × Q is direct, and it has a nondiscrete topology. The quotient group (N1 + N2 )/N1 = (Qτ × Q)/({0} × Q) has basic zero neighborhoods of the form    & 1 1 ' ((U ∩ Qτ × Q) + ({0} × Q))/({0} × Q) = − n , n ∩ Q × Q /({0} × Q) # where U is a basic zero neighborhood of G of the form (U  × {0}) + (r, r) : − n1 < & $ ' r < n1 with a basic zero neighborhood of Qτ such that U  ⊆ − n1 , n1 . Thus (N1 + N2 )/N1 is isomorphic to Q with the topology induced from R. So the standard bijective morphism N2 → (N1 +N2 )/N1 is not open and thus is not an isomorphism of topological groups. This is an interesting failure of the second isomorphism theorem, because N2 ∼ = Qτ is countable, hence a σ -compact group. (iii) We note that N1 + N2 N1 + N2 G G + ⊆ G∗ ⊆ × , N1 N2 N1 N2 which we may read as Q2 ⊆ G∗ ⊆ R2 with the topology on Q2 that is induced from R2 . The group G∗ is dense, but not equal to R2 . In order to help the reader to develop a feeling for the position of the category proLieGr of pro-Lie groups within the category TopGr of all topological groups we recall that by the Completeness Theorem for Pro-Lie Groups 3.36, the category proLieGr is closed in TopGr under the formation of all limits and that (as a consequence) for each topological group G there is a unique pro-Lie group F G and a morphism ηG : G → F G with dense image such that every morphism f : G → H into a pro-Lie group factors uniquely through ηG . This formalism is completely analogous to that of the Bohr compactification of a topological group. What is the Bohr compactification AG there is the “pro-Lie-ification” F G here. Since the Bohr compactification AG as a compact group is a pro-Lie group, the Bohr compactification morphism G → AG factors through a canonical morphism F G → AG. The pro-Lie-ification also allows us to verify easily, that the category of pro-Lie groups has free objects over the category of pointed topological spaces.

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3 Pro-Lie Groups

Postscript In Definition 3.25 of this chapter we introduce a notion of pro-Lie group which is both more general and more natural than that common in the literature, and we summarize equivalent formulations of this definition towards the end of the chapter. Indeed let G be a topological group and N (G) the set of all closed normal subgroups such that G/N is a finite-dimensional Lie group. We call a topological group a pro-Lie group if G is complete and has arbitrarily small normal subgroups N such that G/N is a Lie group. We have seen in Chapter 1 that in these circumstances, G is naturally isomorphic to the projective limit limN ∈N (G) G/N and that this limit is locally compact if and only if there is a compact element N in the filter basis N (G). It is this more special situation that the literature generally refers to when it speaks of pro-Lie groups. The entire structure theory we develop in this book concentrates on pro-Lie groups and therefore on the category proLieGr of all pro-Lie groups and all continuous homomorphisms between them. So proLieGr is a full subcategory of the category TopGr of all topological groups. We have noticed that every pro-Lie group gives rise to a projective system of finitedimensional Lie groups {pMN : G/N → G/M | (M, N ) ∈ N (G) × N (G), M ⊇ N} whose limit is naturally isomorphic to G in such a way that the limit morphisms are the quotient morphisms pN : G → G/N. This is indeed a very special kind of projective limit. If an arbitrary projective system {fj k : Lk → Lj | (j, k) ∈ J × J, j ≤ k} of morphisms of topological groups between finite-dimensional Lie groups is given for def

some directed set J , then we say that G = limj ∈J Lj is a projective limit of Lie groups – implicitly assuming that we are talking about finite-dimensional real Lie groups. With this terminology at hand we can say that every pro-Lie group is a projective limit of Lie groups; in our summary of possible definitions, this is the implication A ⇒ B. The converse emerges here as a difficult question, but in the fundamental Pro-Lie Group Theorem 3.34 we show that all projective limits of Lie groups are pro-Lie groups; in the summary B ⇒ A. Thus the words “projective limit of finite-dimensional Lie groups” and “pro-Lie group” are synonyms. This is a highly nontrivial fact as we have seen in this chapter. While we proved this fact here from scratch, using the basic theory of finite-dimensional Lie groups, A. A. George Michael was able to give a very short proof that is based on the solution of Hilbert’s Fifth Problem [61]. The primary Definition A (= Definition 3.25) of pro-Lie group uses the set N (G) which is canonically attached to any topological group. Caution is needed because N (G) is not always a filter base, as Glöckner’s Example 3.40 shows, while it is in any pro-Lie group, as was shown in a paragraph preceding 3.25.

Postscript

165

The category proLieGr contains all compact groups and all locally compact abelian groups. That proLieGr is a rich environment is evident from the fact that it also contains, firstly, all finite-dimensional Lie groups and arbitrary products of these, and, secondly, the additive groups of all Lie algebras of compact groups (see [102, Proposition 9.45]) and of locally compact abelian groups (loc. cit., p. 355, Theorem 7.66 (i)); furthermore, the exponential function from the Lie algebra of a locally compact abelian group to the group itself is a morphism in proLieGr. We show that all pro-Lie groups have Lie algebras whose additive topological groups are pro-Lie groups. It is a relatively easy matter to establish that the full subcategory of TopGr of projective limits of Lie groups is closed under the formation of all limits in TopGr and is therefore a complete category in its own right. This is done in 3.8. Thus the category proLieGr of pro-Lie groups is complete. It would be a challenging task to prove this directly from the definition of pro-Lie groups; in fact we do not know how to proceed in this fashion. It is somewhat curious that a proof of the fact that every closed subgroup of a pro-Lie group is a pro-Lie group is harder than the completeness result itself. We use here the “Closed Subgroup Theorem for Projective Limits” 1.34 which we provided in Chapter 1. In the process of establishing all of this we have proved that a pro-Lie group is totally disconnected iff L(G) = {0} iff it is prodiscrete; these equivalences are plausible, but by no means obvious. Yamabe [206], [207] has shown that every locally compact group contains an open subgroup which is a pro-Lie group (see also [145], [129]). It is that portion of a locally compact group which we shall recognize as amenable to further structural analysis. The bottom line: We have arrived at the very substantial and hopefully manageable category proLieGr of topological groups with profinite-dimensional Lie algebras. It is suitable in all respects in which any category of locally compact groups is defective: – it is closed under all limits and passing to closed subgroups, and it contains all finite-dimensional Lie groups; – it permits a Bohr compactification-like reflection of the category of all topological groups into the category of pro-Lie groups proLieGr and it has a free functor Top → proLieGr; – it has an excellent – albeit in general infinite-dimensional – Lie theory; – it is closed under passing to the additive groups of the Lie algebras. And, in addition it still has the following property: – it includes all compact groups, all locally compact abelian groups, and all connected locally compact groups and thus is the true background theory for any Lie theory of locally compact groups. It is therefore feasible (at least temporarily) to introduce categories which appear to be particularly suitable envelopes of proLieGr, namely, LieAlgGenGr, the full subcategory of all topological groups having a generating Lie algebra. In 4.22 (i) below we shall prove that every pro-Lie group has a generating Lie algebra. We show that

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3 Pro-Lie Groups

proLieGr it is also contained in the category proLieAlgGr of topological groups which have pro-Lie algebras. But proLieAlgGr is complete; it does in fact contain arbitrary

products of finite-dimensional Lie groups. The following diagram gives an overview of the principal categories that we have considered.

TopGr

kkkk kkkk LieAlgGenGr

LieAlgGr

SSSS SSSS proLieAlgGr

SSSS SSS

kkk kkk proLieComGr

kkkk kkkk proLieGr

kkk kkk k k kk

CompGr

LCGr??

LieGr

SSSS SSSS

SSS SSS SSS S

proLieGr

LCAbGr

The morphisms of all categories in this diagram are continuous homomorphisms between topological groups. We need tabulate only their objects: Name of the category TopGr LieAlgGr LieAlgGenGr proLieAlgGr proLieComGr proLieGr LCGr CompGr LieGr LCAbGr

Objects of the category topological groups topological groups having a Lie algebra topological groups having a generating Lie algebra topological groups having a profinite-dimensional Lie algebra topological groups having a pro-Lie group identity component pro-Lie groups locally compact groups compact groups (finite-dimensional) Lie groups locally compact abelian groups

Complete? yes yes ??? yes yes yes no yes no no

Postscript

167

The questions marks in “LCGr??” are a warning that in this book we do not reprove the fact that locally compact groups have pro-Lie identity components and thus also have profinite-dimensional Lie algebras; see [129], [145], [206], [207]. We do know that the categories CompGr, LCAbGr and LieGr do have profinite-dimensional Lie algebras and thus are indeed subcategories of proLieAlgGr. The question marks in the line for LieAlgGenGr are there because we do not know whether the category of topological groups having generating Lie algebras is complete; it is closed under the formation of products, but we did not show that it is closed under the passage to closed subgroups. Another unsettled question that we leave behind is whether for a pro-Lie group G the component factor group G/G0 is complete, that is, is a prodiscrete group.

Chapter 4

Quotients of Pro-Lie Groups

We have proved that every projective limit of finite-dimensional Lie groups is a pro-Lie group, that the category of pro-Lie groups is complete and that every closed subgroup of a pro-Lie group is a pro-Lie group. We now address the topic of quotient groups of pro-Lie groups. Quotients arise naturally in the canonical decomposition of any morphism f : G → H between topological groups: If K is the kernel of f then there is a unique bijective morphism f  : G/K → f (G) defined by f  (gK) = f (g) so that the standard diagram commutes G ⏐ ⏐ quot G/K

f

−−−−−→

H  ⏐ ⏐incl −−−−−→ f (G), f

where quot : G → G/K is the quotient morphism and incl : f (G) → H is the inclusion embedding. Thus the first thing in understanding the structure of morphisms is understanding quotients and quotient morphisms. It belongs to the basic theory of locally compact groups to establish that quotients of locally compact groups are locally compact. The fact that quotients of finite-dimensional Lie groups are Lie groups belongs to the less elementary pieces of information of the basic theory. It is inevitable that we address the question of quotients of pro-Lie groups. Indeed, we see shortly that a quotient group of a pro-Lie group need not be a pro-Lie group. We then ask whether the category of pro-Lie groups “has quotients”. But for this question to make sense we would first have to make precise what we mean by a quotient in a category. Alas, from a category theoretical point of view, quotients are hard to put into an entirely satisfactory framework. We review this observation early in this chapter. We then show that the failure of pro-Lie groups to have complete quotients occurs already for the abelian pro-Lie groups RX as soon as X has at least continuum cardinality, and we notice that this phenomenon occurs in the supposedly familiar context of connected compact abelian groups for whose theory we refer e.g. to [102, Chapters 7 and 8]. On the positive side, we show that the Lie algebra functor on proLieGr preserves quotient morphisms; and we eventually prove a result which implies that all quotient groups of connected pro-Lie groups modulo connected closed normal subgroups are complete and therefore are pro-Lie groups. Prerequisites. No new prerequisites are demanded in this chapter.

Quotient Objects in Categories

169

Quotient Objects in Categories Quotient objects in a category arise when morphisms ϕ : A → B are decomposed in the form ϕ = με where ε : A → C is an epic and μ : C → B is some morphism such that whenever ϕ is factored in the form ϕ = ba with a : A → E and a monic b : E → B, then a has a factorisation a = a  ε. We mention in passing that in many categories where such a factorisation ϕ = μ  ε exists, μ is in fact itself a monic; these categories are said to have epic-monic factorisation. Examples are the categories of abelian groups, the category of groups, the category of rings, and their topological-algebraic counterparts. If categories have additional structure, then canonical constructions of factorisations ϕ = μ  ε are possible. Let us briefly describe how, and let us not aspire to the utmost possible generality but work on a level of generality that is still perfectly adequate for the categories we are dealing with in this book. In fact, categories of groups typically have singleton objects; in category theoretical terms they fit the following description: A null object in a category is an object 0 such that for each object A in the category there are unique morphisms 0 → A and A → 0. It is seen immediately that null objects are unique up to isomorphism. If null objects exist, then for every pair (A, B) of objects there is a morphism A → 0 → B, called a null morphism or a constant morphism, easily seen to be unique and denoted by 0BA : A → B. Categories with null objects are called pointed categories, see e.g. [102, Definition A3.6]. All categories whose objects are groups of one form or another are clearly pointed, and a constant morphism is one mapping everything to the identity element of the range group. Assume that ϕ : A → B is a morphism in a pointed category. If the equalizer (respectively, coequalizer) of ϕ and 0BA exists, then it is called the kernel κϕ : ker ϕ → A of ϕ (respectively, the cokernel qϕ : B → coker ϕ of ϕ). If a pointed category has kernels and cokernels, then for any morphism ϕ : A → B we can form the cokernel of κϕ : ker ϕ → A and denote it qϕ : A → A/ ker ϕ. Now ϕ  κϕ = 0B ker ϕ = ϕ  0A ker ϕ , and by the definition of qϕ there is a unique μϕ : A/ ker ϕ → B such that ϕ = μϕ  qϕ . If ϕ = ba is any factorisation with a : A → E, then 0B ker ϕ = ϕκϕ = baκϕ and the universal property of the kernel κb : ker b → E imply that there is a unique morphism k : ker ϕ → ker b such that κb k = aκϕ . If b is monic then κb is the null morphism and aκϕ = κb k = 0; thus, by the definition of the cokernel, there is an a  : A/ ker ϕ → E such that a = a  qf . Thus the factorisation ϕ = μϕ  qϕ has the required universal factorisation property. In such categories as the category of abelian groups, modules, groups, topological groups, etc., the inclusion morphism of the standard kernel ker ϕ into A is the categorical kernel and the standard quotient map A → A/ ker f is the categorical quotient qϕ . In these categories, the factor μϕ of a factorisation ϕ = μϕ  qϕ is automatically a monic. In general however, as we shall see in the category of pro-Lie groups, in the factorisation ϕ = μϕ qϕ itself, the morphism μϕ need not be monic. Nevertheless, the best category theoretical definition of a quotient morphism in a pointed category with kernels and cokernels is this: A morphism q : A → Q is called a quotient morphism if there is a morphism ϕ : A → B such that Q = A/ ker ϕ and q = qϕ .

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4 Quotients of Pro-Lie Groups

We should note that this formalism does have a category theoretical dual which associates with ϕ : A → B the kernel jϕ : C(ϕ) → B of the cokernel B → coker ϕ which one should duly call the coquotient of ϕ, giving rise to the canonical decomposition A ⏐ qf ⏐  A/ ker ϕ

ϕ

−−−−−→ −−−−−→ ϕ

B  ⏐j ⏐ϕ C(ϕ).

But we shall not pursue this further at this point, in particular since cokernels and coquotients in all but categories such as that of abelian groups and its variants are not what they seem at first sight. In the category proLieGr, however, the next theorem tells us what we can expect in terms of quotients and canonical decompositions of morphisms, and subsequently we shall inspect examples which show that we cannot expect to do much better. Indeed, as a consequence of the next theorem, the category proLieGr of pro-Lie groups is closed in TopGr under the formation of (group theoretical) quotients that happen to be complete. For any morphism f : G → H of pro-Lie groups with kernel K = ker f , the morphism qf defined as the composition of the (group theoretical) quotient morphism G → G/K with the embedding of G/K into its completion (G/K)N (G/K) is the cokernel of inclK : K → G and thus is the category theoretical quotient in proLieGr if H is a pro-Lie group. But we shall also see shortly, that the factorisation f = μf  qf is not epic-monic in general.

Quotient Groups of Pro-Lie Groups The Quotient Theorem for Pro-Lie Groups Theorem 4.1. (i) A quotient group of a proto-Lie group is a proto-Lie group and is isomorphic as a topological group to a dense subgroup of a pro-Lie group; in particular, every proto-Lie group has a completion. If the quotient group is complete, then it is a pro-Lie group. (ii) Let f : G → H be a morphism of topological groups from a pro-Lie group into def a complete group, and set K = ker f . Then there is a canonical decomposition: G ⏐ ⏐ q

f

−−−−−→

G/K ⏐ γG/K ⏐  (G/K)N (G/K)

H  ⏐ ⏐incl f (G)  ⏐ ⏐id

−−−−−→ f

f (G).

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171

In this diagram all groups are complete with the possible exception of G/K; further, G and (G/K)N (G/K) are pro-Lie groups. In particular, there is a factorisation f = μf  qf , qf = γG/K  q and μf = incl f  with an epic qf such that for any factorisation f = me of morphisms between complete groups for a monic m and some morphism e there is a factorisation of this morphism e in the form of e = e  qf . (iii) The morphism qf : G → (G/K)N (G/K) is the cokernel of inclK : K → G in the category proLieGr. (iv) (The Weak Pro-Open Mapping Theorem for Pro-Lie groups) Assume that G is a proto-Lie group such that G/N has countably many components for all N ∈ N (G). Let U be the set of open identity neighborhoods of G. If f : G → H is a surjective morphism onto a proto-Lie group, then (∀U ∈ U, M ∈ N (H )) Mf (U ) is open in H. If G has countably many components, then G/N has countably many components for all N ∈ N (G). Proof. (i) Let G be a pro-Lie group and K a closed normal subgroup. Define f : G → def

H = G/K to be the open quotient morphism. For N ∈ N (G), the set N K is a closed subgroup of G containing K, and since f is a quotient map and N K is K-saturated, the def

set N ∗ = f (N K) ⊆ H is closed and agrees with f (N ). Then N ∗ is a closed normal subgroup of H , and since f is open, we have that H /N ∗ ∼ = G/f −1 (N ∗ ) = G/N K ∼ = (G/N)/(NK/N ) is a finite-dimensional Lie group as a quotient of a finite-dimensional Lie group. Let M = {N ∗ | N ∈ N (G)}. Then M is a filter basis of closed normal subgroups of H such that all factor groups H /M, M ∈ M are Lie groups. Since N (G) converges to 1 as G is a pro-Lie group, from the continuity of f we conclude that f (N (G)) = {f (N) | N ∈ N (G)} converges to 1 in H . But since H is regular, i.e. the filter of identity neighborhoods has a basis of closed sets, M converges to 1 in H . Thus H is a proto-Lie group and by Theorems 1.29 and 1.30 we have a natural dense embedding morphism γH : H → HM into a pro-Lie group HM . By Definition 3.1, if H is complete, then, being a proto-Lie group, H is a pro-Lie group. (ii) Since H is complete by hypothesis, f (G) is the completion of f (G). There is no loss of generality in assuming H = f (G). By the standard canonical decomposition of a morphism, we have an injective morphism of topological groups F : G/K → H , F (gK) = f (g), such that f = F  q and F (G/K) = f (G). The pro-Lie group (G/K)N (G/K) is the completion of the proto-Lie group G/K according to (i). Hence F extends uniquely to a morphism f  : (G/K)N (G/K) → H , and thus f = f   γG/K  q = F  q. Finally set qf = γG/K  q and assume that f = m  e for a morphism e : G → C between complete groups and for a monomorphism m : C → H . Then K = ker f = (me)−1 (1) = e−1 (ker m) = e−1 ({1}) = ker e. Now we apply to e in place of f what we just derived for f ; hence there is a unique morphism e : (G/K)N (G/K) → C such that e = e qf . (iii) Assume that ϕ : G → X is a morphism in proLieGr such that ϕ  inclK = 0, that is, ϕ(K) = {1}. Then there is a unique morphism ψ : G/K → X given by

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ψ(gK) = ϕ(g). Since X is complete as a pro-Lie group, ψ extends uniquely to the completion of G/K, i.e. there is a unique morphism ϕ  : (G/K)N (G/K) → X such that ϕ = ϕ  γG/K q = ϕ  qf . Since the category proLieGr is full in TopGr, this is what we had to show. def (iv) Let K = ker f . Then G/K is a proto- Lie group by (i) above, and f is open iff the induced bijective morphism F : G/K → H is an isomorphism. We claim that (G/K)/(N/K) ∼ = G/N has countably many components for any N/K ∈ N (G/K). Since N/K ∈ N (G/K) we know that N ∈ N (G), and thus G/N has only countably many components. It is therefore no loss of generality if we assume that f : G → H is bijective, as we shall do from now on. We note that if G has countably many components, then G/N has countably many components for each N ∈ N (G). Indeed, let ν : N → G/G0 be any surjective function. The quotient map G → (G/N )/(G/N )0 factors through G/G0 yielding a surjective function μ : G/G0 → (G/K)/(G/K)0 . Then μν : N → (G/N )/(G/N )0 is surjective, proving the claim that G/N has countably many components. def

It follows as in the proof of (i) that C = {f (N ) : N ∈ N (G)} is a filter basis of closed normal subgroups of H converging to 1. Let M ∈ N (H ) and U an open identity neighborhood of H satisfying MU = U M = U such that U/M contains no subgroups of the Lie group H /M other than the singleton one. Then we find an N ∈ N (G) such that f (N) ⊆ U ; thus f (N ) ⊆ M. Then fMN : G/N → H /M, fMN (gN) = fMN (g)M, is a surjective morphism between two locally compact groups. A connected locally compact group is algebraically generated by any compact identity neighborhood (see [102, Proposition A4.25 (iii)]); having countably many components, G/N is therefore σ -compact. Then the Open Mapping Theorem for Locally Compact Groups (see for instance [102, EA1.25]) shows that fMN is open. If pN : G → G/N and qM : H → H /M are the quotient morphisms, then qM  f = fMN  pN is open. Hence for each M ∈ N (H ) and each open identity neighborhood U of G the subset Mf (U ) is an M-invariant open identity neighborhood of H . Of course, Theorem 4.1 says in particular that any quotient of a pro-Lie group is a proto-Lie group. This may be perceived as an unsatisfactory weakness of proLie groups. We shall see shortly that we have intriguing examples showing that, for instance, the pro-Lie group RR has incomplete quotients. This will show that, sadly, this weakness cannot be remedied. We shall see in Chapter 5, Proposition 5.2, that the free abelian group Z(N) has a nondiscrete nonmetrizable prodiscrete and thus pro-Lie topology τ . If δ is the discrete topology on Z(N) , then the identity morphism f : (Z(N) , δ) → (Z(N) , τ ) is a nonopen bijective morphism between two countable pro-Lie groups. Thus a better Open Mapping Theorem than 4.1 (iv), alas, cannot be proved either. In later chapters, however, we shall see that connectivity makes all the difference, and we shall prove an Open Mapping Theorem saying that surjective morphisms between connected pro-Lie groups are open.

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In the proof of Theorem 4.1 we gave a self-contained proof of the factorisation property in (ii) even though it follows from (iii) via the category theoretical arguments we outlined in the paragraphs preceding 4.1. In [176, p. 196, Remark (b)] it is observed that a product of abelian complete metrizable topological groups may have incomplete quotient groups. Moreover, it is proved that every topological abelian group is a quotient of a complete topological abelian group ([176, 11.1]). Indeed, O. V. Sipacheva has proved that every topological group is a quotient of a complete topological group. (See [41, p. 100], [184, corollary to Theorem 4.1.6.].) We shall now set about showing that a product of 2ℵ0 copies of R has incomplete quotients; such examples occur in the immediate vicinity of compact connected abelian groups. Thus examples that illustrate the limitations of our Theorem 4.1 arise even in the theory of this fairly well documented class of topological groups.

The Exponential Function of Compact Abelian Groups and Quotient Morphisms This requires a detour into the theory of compact abelian groups. Namely, we shall investigate when the exponential function exp : L(G) → G of a compact connected abelian group G is open onto its image. For a detailed exposition of the exponential function of compact abelian groups we refer to [102], notably Chapters 7 and 8. We recall that L(G) = Hom(R, G) is the topological vector space of all one parameter subgroups, and that the exponential function is given by evaluation via exp X = X(1). We note that L(G) is isomorphic to a product of copies of R (see [102, Theorem 7.66 (i) and Theorem 7.30 (ii)]). For any compact abelian group G, the exponential function expG : L(G) → G (see [102, Theorem 7.66]) is a morphism of abelian topological groups. Let Ga denote the arc component of the zero element 0. Then the corestriction expG : L(G) → Ga of the exponential function to its image is a surjective morphism of topological groups. Since we deal here exclusively with abelian groups, we shall write them additively. The neutral element is zero, written 0. Since G is a compact abelian  is a discrete abelian group, and since G is connected, G  is group, its character group G torsion-free. We observe that the image of the exponential function expG : L(G) → G is Ga and that Ga = exp L(G) is dense in G. (For a proof, see [102, Theorem 8.30].) Separable abelian groups. A subgroup G of an abelian group A is said to split if there is a subgroup H of A such that A = G ⊕ H . Likewise we say that a short exact sequence j

0→G→A→B→0

(E)

is split if j (G) is a split subgroup of A. This is tantamount to saying that the equivalence class of (E) in Ext(B, G) is zero. (See e.g. [102, Lemma A1.49ff.]) A subgroup G of

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an abelian group A splits automatically if G is divisible or A/G is free. (See e.g. [102, Corollary A1.36 and Proposition A1.15.]) A subgroup P of an abelian group A is called a pure subgroup if the following condition is satisfied. (i) (∀p ∈ P , a ∈ A, n ∈ N) n · a = p ⇒ (∃x ∈ P ) n · x = p. One observes directly (see also [102, PropositionA1.24]) that for a torsion-free group A, this condition is equivalent to either of the following conditions. (ii) The factor group A/G is torsion-free. (iii) (∀n ∈ N, a ∈ A) n · a ∈ G ⇒ a ∈ G. In a torsion-free abelian group A every subgroup G is contained in a unique pure subgroup [G] = {a ∈ A : (∃n ∈ N) n · a ∈ G}. (See [102, Proposition A1.25].) Moreover, the following conditions are equivalent (see [102, Proposition A1.64]): (a) Every finite rank pure subgroup of A is free. (b) Every countable subgroup of A is free. An abelian group satisfying these conditions is said to be ℵ1 -free. We are interested in a subclass of the class of ℵ1 -free groups. Proposition 4.2. Let A be a torsion-free abelian group. Then the following conditions are equivalent: (i) (∀a ∈ A) [Z · a] is free and splits. (ii) Every rank one pure subgroup is free and splits. (iii) Every finite rank pure subgroup is free and splits. Proof. Clearly, (iii) ⇒ (ii) ⇔ (i). We must prove (i) ⇒ (iii). Step 1. We first note that if A satisfies (i) then every pure subgroup B of A satisfies (i): Indeed let b ∈ B; since B is pure, the pure subgroup [Z · b] generated by b in B is pure in A, and as a subgroup of the free pure subgroup generated by b in A it is free. Since A satisfies (i), we have A = [Z · b] ⊕ K with a suitable subgroup K. By the modular law, B = [Z · b] ⊕ (K ∩ B). Setting H = K ∩ B we obtain B = [Z · b] ⊕ H as claimed. Step 2. We claim that every finite rank pure subgroup is free; in other words we show that A is ℵ1 -free. Let G be a finite rank pure subgroup of A and assume that G is a counterexample of minimal rank n. From (i) we know n > 1. Let 0 = g ∈ G. Since G satisfies (i), G = [Z · g] ⊕ H and [Z · g] is free. Since the subgroup H has smaller rank than G, it is not a counterexample and is therefore free. Hence G is free and thus cannot be a counterexample. Step 3. Now we prove that every finite rank pure subgroup G splits. By Step 2, N G = m=1 Z · em , where N ∈ N. By (i) there is a subgroup H1 of A such that A = Z · e1 ⊕ H1 and G = Z · e1 ⊕ (H1 ∩ G). Assume that H1 ⊇ H2 ⊇ · · · ⊇ Hn , n < N has been constructed in such a fashion that A = Z · e1 ⊕ · · · ⊕ Z · em ⊕ Hm

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and G = Z · e1 ⊕ · · · ⊕ Z · em ⊕ (Hm ∩ G), m = 1, 2, . . . , n. By the first step we may apply (i) to the pure subgroup Hn and find a subgroup Hn+1 of Hn such that Hn = Z · en+1 ⊕ Hn+1 and Hn ∩ G = Z · en+1 ∩ (Hn+1 ∩ G). This yields a descending family of subgroups Hn such that A = nm=1 Z · em ⊕ Hn and em ∈ Hn for m > n. We set H = HN . Then A = N m=1 Z · em ⊕ HN = G ⊕ H , as was to be shown. Definition 4.3. We say that an abelian group A is an S-group if it satisfies the equivalent conditions of Proposition 4.2. The S-groups have been called separable [49] which is not an advisable terminology here because we will deal with topological abelian groups for which the adjective separable refers to groups having a dense countable subset, and this is entirely different. One might have called S-groups strongly ℵ1 -free; our terminology reflects the “strongly” as well. Every free group is an S-group, since every subgroup of a free group is free, and the quotient of a free group modulo a pure subgroup is free (see e.g. [102, Proposition A1.24 (ii)]); thus every pure subgroup splits. A Whitehead group is an abelian group A such that Ext(A, Z) = {0}, that is, every extension 0→Z→G→A→0 splits. Example 4.4. The group A = ZN has the following properties: (i) A is an S-group. (ii) A is not a Whitehead group. (iii) The subgroup Z(N) of A is a countable free pure subgroup which does not split. Proof. (i), (ii) The group ZN is an ℵ1 -free group which is not a Whitehead group: see e.g. [102, Example A1.65]. We verify Condition 4.2 (ii) for A = ZN . Let P be a rank one pure subgroup of A and let k = (kn )n∈N be an element of P such that [Z · k] = P . Then the greatest common divisor of {kn : n ∈ N} is 1. The decreasing sequence gcd{k1 , k2 , . . . , kn } is eventually constant; that is there is a natural number N such that k1 , . . . , kN have the greatest common divisor 1. Then the subgroup PN generated in the finitely generated free group ZN by (k1 , . . . , kN ) is pure. By the Elementary Divisor Theorem (see e.g. [102, Theorem A1.10]) applied to ZN , after choosing a new basis, we may assume that PN = Z × {0} × · · · × {0}. It is therefore no loss of generality to assume that (kn )n∈N = (1, 0, . . . , 0, kN +1 , kN +2 , . . . ). Set G = {0} × Z × Z × · · · . Then ZN = P ⊕ G and ZN is an S-group as asserted. (iii) If m · (kn )n∈N ∈ Z(N) for some m ∈ Z then mkn = 0 for all but a finite number of n ∈ N. Then (kn )n∈N . Thus Z(N) is a pure subgroup of Z N which is obviously countable and free. The group ZN /Z(N) is a torsion-free algebraically compact group and contains a copy Zp of the p-adic integers for each prime as a direct summand. (See e.g. [58, p. 176, 42.2 and p. 169, 40.4.]) Since Zp contains countable groups which

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are not free (e.g. q1∞ Z for any prime q different from p), and since ZN is ℵ1 -free, ZN cannot contain a subgroup isomorphic to A/Z(N) . Example 4.5. There is an abelian group B with a subgroup C ∼ = Z such that B/C ∼ = ZN and that every morphism B → Z annihilates C. The group B is an ℵ1 -free group which is not an S-group. Proof. In the proof of Proposition A1.66 of [102], the following lemmas are proved: Lemma A. Let E = [0 → C → B → X → 0] be any extension of C ∼ = Z by an abelian group X. Then there is a homomorphism f : B → Z whose restriction to C is nontrivial if and only if E represents an element of finite order in Ext(X, Z). Lemma B. Ext(ZN , Z) contains 2(2

ℵ0 )

elements of infinite order.

Taken together, these lemmas yield the existence of a torsion-free group B and a cyclic subgroup C such that B/C ∼ = ZN , and that every morphism B → Z vanishes on C. The subgroup C is a subgroup of rank 1 which does not split, and since B/C is torsion-free, C is a pure subgroup ' +C &of B. Thus B is not an S-group. If P is a finite rank pure subgroup of B, then P C is a finite rank pure subgroup of B/C ∼ = A and is therefore finitely generated free; its full inverse image P  in B is a finitely generated torsion-free group and is, therefore, free. Thus P as a subgroup of a free group is free. Hence B is an ℵ1 -free group. In this area of the theory of abelian groups, ZN is a universal test example. For instance, Proposition 4.2 cannot be complemented by another equivalent condition which would say: Every countable pure subgroup splits. The example shows, in particular, that the class of S-groups is properly smaller than that of ℵ1 -free groups and is not contained in the class of Whitehead groups. Every Whitehead group is an S-group (see [49, p. 226]). Thus the class of S-groups is properly bigger than the class of Whitehead groups and thus a fortiori properly bigger than the class of free groups. ℵ1 -free groups  S-groups   Whitehead  groups  free groups Strengthening local connectivity for compact abelian groups. A locally compact  = Hom(G, T). abelian group G is completely characterized by its Pontryagin dual G (See e.g. [102, Chapters 7 and 8].) A topological group G is locally connected if and only if its identity component G0 is open in G and is locally connected; a connected locally compact abelian group G contains a unique characteristic maximal compact

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subgroup C and a subgroup V ∼ = Rn such that the morphism (v, c)  → v + c : V × C → G is an isomorphism of topological groups. (See e.g. [102, Theorem 7.57].) In discussing local connectivity of a locally compact abelian group G, it is no loss of generality to assume that G is compact and connected. A locally compact abelian  is discrete and group G is compact and connected if and only if its character group G torsion-free. (See e.g. [102, Proposition 7.5 (i), and Corollary 8.5].) Local connectivity of a compact connected abelian group is characterized as follows. Proposition 4.6. For a compact connected abelian group G, the following statements are equivalent: (i) There are arbitrarily small compact connected subgroups N such that G/N is a finite-dimensional torus group.  is the directed union of pure finitely generated free sub(ii) The character group G groups.  is ℵ1 -free. (iii) G (iv) G is locally connected. Proof. For a proof see [102, Theorem 8.36]. We compare this proposition with the following Proposition 4.7. For a compact connected abelian group G, the following statements are equivalent: (i) There are arbitrarily small compact connected subgroups N for which there is a finite-dimensional torus subgroup TN of G so that (n, t)  → n+t : N ×TN → G is an isomorphism of topological groups.  is the directed union of finitely generated free split sub(ii) The character group G groups.  is an S-group. (iii) G Proof. The equivalence of (i) and (ii) follows at once from duality.  is torsion-free. Let P be a rank one pure (ii) ⇒ (iii): By (ii), the abelian group G  By (ii) there is a finitely generated  The P = [Z · a] for some a ∈ G. subgroup of G.  free split subgroup F of G containing a. Since a direct summand is a pure subgroup we have P = [Z · a] ⊆ F . As a pure subgroup of a finitely generated torsion-free  P is a split group, P is a direct summand of F , and since F is a direct summand of G,  subgroup of G.  is the directed union of all of its finite (iii) ⇒ (ii): As a torsion-free abelian group, G rank pure subgroups P ; by (iii), every such P is split and free, and thus (ii) follows. The comparison of Propositions 4.6 and 4.7 justifies the following definition. Definition 4.8. A locally compact abelian group is said to be strongly locally connected if its identity component is open and its unique maximal compact connected subgroup satisfies the equivalent conditions of Proposition 4.7.

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In particular, a compact connected abelian group is strongly locally connected if and only if its character group is an S-group. def  N . Then G is a strongly locally connected and connected Example 4.9. Let G = Z but not arcwise connected compact abelian group. There is a compact connected, locally connected, but not strongly locally connected group H of weight 2ℵ0 containing G such that H /G is a circle group. G has a metric torus group quotient which is not a homomorphic retract.

Proof. A compact connected abelian group H is arcwise connected if and only if its  is a Whitehead group. (See e.g. [102, Theorem 8.30 (iv)].) The character group H claim thus follows by duality from Examples 4.4 and 4.5. The class of connected strongly locally connected compact abelian groups is properly larger than that of torus groups and properly smaller than that of connected locally connected compact abelian groups. The exponential function of strongly locally connected groups. We shall investigate when the exponential function expG : L(G) → G of a compact abelian group G is open onto its image in the present context. For a detailed exposition of the exponential function of compact abelian groups we refer to [102], notably Chapters 7 and 8. We need to know here that L(G) = Hom(R, G) is the topological vector space of all one parameter subgroups, i.e. continuous group morphisms X : R → G, where Hom(R, G) is given the topology of uniform convergence on compact sets. The exponential function is given by evaluation via exp X = X(1). By duality, L(G) may also be viewed  R) with the topology of pointwise convergence. We as the vector space Hom(G, note that L(G) is isomorphic to a product of copies of R (see [102, Theorem 7.66 (i) and Theorem 7.30 (ii)]). For any compact abelian group G, the exponential function expG : L(G) → G (see [102, Theorem 7.66]) is a morphism of abelian topological groups. Let Ga = im expG denote the arc component of the zero element 0. (See [102, Theorem 8.30.]) We note the exact sequence: expG  Z) → 0 0 → K(G) → L(G) −−−−−→ G → Ext(G,

(exp)

where K(G) = ker expG . The corestriction expG : L(G) → Ga of the exponential function to its image is a surjective morphism of topological groups. A surjective morphism between topological groups is open if and only if it is a quotient morphism. Thus expG is open iff the induced bijective morphism of topological groups L(G)/K(G) → Ga is an isomorphism. Proposition 4.10. Let G be a connected and strongly locally connected compact abelian group. Then expG : L(G) → Ga is open.  Then P ∈ P is Proof. Let P denote the set of pure finite rank subgroups of G.  such  a finitely generated free split subgroup of G. We select a subgroup SP ⊆ G def  = P ⊕ SP . Let NP = P ⊥ denote the annihilator of P in G and TP = S ⊥ that G P

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the annihilator of SP . By duality (n, t)  → n + t : NP × TP → G is an isomorphism of topological groups, that is, G = NP ⊕ TP algebraically and topologically.  are naturally isomorphic by the Annihilator MechaThe groups TP , G/NP and P nism (see [102, Theorem 7.64]) and thus are finite-dimensional torus groups. The morphism expNP × expTP : L(NP ) × L(TP ) → NP × TP is naturally equivalent to the exponential function of NP × TP . Let UP be the set of arcwise connected open zero-neighborhoods U of L(TP ) mapped homeomorphically onto an open zero neighborhood V of TP by expTP ; such neighborhoods U and V exist as TP is a Lie group. Then (expNP × expTP )(L(NP ) × U ) = (NP )a × V = (NP × V ) ∩ (NP × TP )a . It follows that expG (L(NP ) ⊕ U ) is an identity neighborhood of Ga in the topology induced from that of G. We claim that {L(NP ) ⊕ U : P ∈ P , U ∈ UP } is a basis for the open zero-neighborhoods of L(G), where L(NP ) is naturally considered as a cofinite-dimensional vector subspace of L(G). Once this claim is established, expG : L(G) → Ga is open and the proof of the proposition is complete.   = Since G P = colimP ∈P P by duality, we have G = limP ∈P G/NP . The functor L = Hom(R, −) preserves projective limits (see e.g. [102, Proposition 7.38 (iv)]; in fact L preserves all limits). Furthermore, L preserves quotients (see [102, Theorem 7.66 (iv)]). Hence L(G) = limP ∈P L(G)/L(NP ). Let rP : L(G) → L(G)/L(NP ) denote the quotient morphism, qP : L(G)/L(NP ) → L(TP ) the natural isomorphism, and pP = qP ◦ rP : L(G) = L(NP ) ⊕ L(TP ) → L(TP ) the projection. For any zero neighborhood W of L(G), by the Fundamental Theorem on Projective Limits 1.27 (i), there is a P ∈ P and a zero neighborhood U ∈ UP in L(TP ) such that pP−1 (U ) ⊆ W . Since pP−1 (U ) = L(NP ) ⊕ U , the claim is proved. We now get the result which shows that Theorem 4.1 cannot be improved. The product RR is a pro-Lie group (see 3.4 (ii) and 3.28), and it is of the simplest kind while not being locally compact. def

Corollary 4.11. (i) The uncountable product V = RR has a closed totally disconnected algebraically free subgroup K of countable rank such that the quotient V /K is incomplete and its completion G is a compact connected and strongly locally connected abelian group of continuum weight. (ii) Let q : V → V /K denote the quotient map and γV /K : V /K → G be the completion map. There is a morphism f : V → C into a compact hence complete group whose kernel is K and which has the property that the factorisation map f  : G → C determined uniquely by f = f   γV /K  q is not injective.  N be the compact abelian group of 4.4. The rank of ZN Proof. (i): Let G = Z agrees with the cardinal of ZN and that is the cardinal 2N of the continuum. Then N L(G) = Hom(R, G) = Hom(ZN , R) ∼ = RR . Thus we take for V the ad= R2 ∼ ditive group of L(G) and K = K(G) and know that ε : V /K → Ga is an isomorphism of topological groups by Proposition 4.10. The completion of Ga is G, and w(G) = card ZN = 2ℵ0 . The kernel K(G) is algebraically isomorphic to Hom(ZN , Z) (see [102, Theorem 7.66 (ii)]). But Hom(ZN , Z) ∼ = Z(N) (see [49, p. 61, Corollary 2.5]). Thus K is free of countable rank.

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(ii): In [102, Example A1.65] one finds the construction of a character χ : ZN → T of order 2 (i.e., 2 · χ = 0 in additive notation) which does not factor in the form ϕ

p

ZN −−→ R −−→ T,

p(r) = r + Z.

Then, as an element of G = Hom(ZN , T), the character χ is not in the image Ga of Hom(ZN , p) = expG : Hom(ZN , R) = L(G) → G. Set Z = {0, χ }. Then Z is a closed subgroup of G such that Ga ∩ Z = {0}. Put C = G/Z and let f  : G → C be the quotient morphism whose kernel is Z. The restriction F : Ga → C is injective. Let f : V → C be defined by f (X) = F (expG X). By (i) the corestriction q : V = L(G) → Ga of the exponential function is a quotient morphism, and F = f  q is the canonical epic-monic factorisation of F . Since G is isomorphic to the completion G of Ga and the inclusion Ga → G is the completion morphism γGa : Ga → G, the assertion follows. The significance of 4.11 (ii) is as follows: In the category of all complete topological abelian groups, the completion of a quotient plays the role of a quotient in the category as it has the expected universal properties; nevertheless, it will in general fail to have familiar properties as 4.11 (ii) illustrates. The possible incompleteness of quotients plays a somewhat disturbing role in the general theory of pro-Lie groups. The simplest nontrivial projective limits of finite-dimensional Lie groups are the products RX . The product RN is metrizable and complete, hence every quotient is complete. Corollary 4.11 shows that the “next largest product”, RR already has incomplete quotients, and it is remarkable that there are such quotients whose completion is compact. While we are utilizing the exponential function of compact abelian groups for the purpose of exhibiting incomplete quotients, we record additional information that is relevant in this context. Proposition 4.12. Assume that the corestriction expG : L(G) → Ga of the exponential function of a compact connected abelian group is a quotient morphism. Then (i) Ga has arbitrarily small open arcwise connected identity neighborhoods in the topology induced from that of G. (ii) G is locally connected.  is ℵ1 -free. (iii) G Proof. (i) A quotient morphism is open. But L(G) is a locally convex topological vector space and thus has arbitrarily small arcwise connected neighborhoods of zero which are mapped onto open identity neighborhoods of Ga by expG . (ii) Let W be an identity neighborhood of G. Then there is an identity neighborhood V such that V 2 ⊆ W . By (i), Ga has an open arcwise connected identity neighborhood U satisfying U ⊆ V . Then the closure U of U in G is contained in V ⊆ V V ⊆ W . There is an open set UG in G such that U = UG ∩ Ga , and since Ga is dense in G we have U = UG . Also, since U is arcwise connected, U is connected

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181

and is an identity neighborhood in G. Thus U is a connected identity neighborhood in G which is contained in W . Thus G is locally connected. (iii) This follows from Proposition 4.6 and (ii) above. Thus the exponential function of a compact connected abelian group can be open onto its image only if the group is locally connected. Proposition 4.13. If the arc component Ga of the zero element of a compact connected abelian group G is locally arcwise connected, then the corestriction expG : L(G) → Ga of the exponential function is open. Proof. On the group G there is a filter basis N (G) converging to 1 and consisting of closed compact subgroups N such that G/N is a finite-dimensional torus. By [102, Theorem 7.66 (iv)], there are arbitrarily small identity neighborhoods of G of the form N ⊕ V where V = expG U for an open n-cell neighborhood U in a finite-dimensional vector subspace F of L(G) such that (n, X)  → n + expG X : N × U → N ⊕ V is a homeomorphism. Now Na ⊕ V = (N ⊕ V )a is the arc component of 0 in Ga ∩ (N ⊕ V ). Since Ga is locally arcwise connected, arc components of open sets of Ga are open in Ga , and thus Na ⊕ V is open in Ga . Therefore Ga has arbitrarily small identity neighborhoods of the form Na ⊕ V . However, these are of the form Na ⊕ V = expN L(N ) ⊕ expG U = expG (L(N ) ⊕ U ) = expG (L(N ) ⊕ U ). Now we know that G = limN ∈N (G) G/N and just as in the proof of 3.1 we conclude that therefore L(G) = limN ∈N (G) L(G)/L(N ) and that L(G) has arbitrarily small neighborhoods of the form L(N )⊕U . This proves that expG is an open morphism. In fact one can show, that strong local connectedness is both necessary and sufficient for the exponential function of a compact abelian group to be surjective onto its image. Indeed the following theorem holds. Theorem 4.14 (Characterisation Theorem for Strong Local Connectivity of Compact Connected Abelian Groups). For a compact connected abelian group G and its zero arc-component Ga , the following conditions are equivalent: (i) (ii) (iii) (iv)

G is strongly locally connected. The exponential function expG : L(G) → G is open onto its image. Ga is locally arcwise connected.  is an S-group, that is, every finite rank pure subgroup of G  is free and is a G direct summand.

Proof. See [106, Theorem 5.1]. We have presented the implications (iv) ⇔ (i) ⇒ (ii) ⇒ (iii) in Propositions 4.7, 4.10, and 4.12. The implication (ii) ⇒ (i) is harder to prove.

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Remarks 4.15. (i) If a topological group G contains a closed central subgroup N such that N ∼ = RR , then G has an incomplete quotient group. (ii) Let {Aj : j ∈ J } be a family of abelian topological groups such that def  card J ≥ 2ℵ0 and each Aj contains a copy of R. Then G = j ∈J Aj has an incomplete quotient group. (iii) Let {Ej : j ∈ J } be a family of Hausdorff topological vector spaces such that def  card J ≥ 2ℵ0 . The E = j ∈J Ej has an incomplete quotient group. (v) If a topological group has a quotient group which has an incomplete quotient group, then it itself has an incomplete quotient group. (vi) Let {Gj : j ∈ J } be a family of topological groups such that card J ≥ 2ℵ0 and each Gj contains closed normal subgroups Mj ⊆ Nj such that Nj /Mj ∼ = R and [G, Nj ] ⊆ Mj (where [A, B] denotes the subgroup generated by the commutators def  aba −1 b−1 , a ∈ A, b ∈ B). Then G = j ∈J Gj has an incomplete quotient group. Exercise E4.1. Verify the assertions in Remarks 4.15 [Hint. (i) From the example in Corollary 4.11 above we find a closed, totally disconnected subgroup D of N such that N/D is incomplete. Since N is central, D is normal in G and then G/D contains N/D, an incomplete group, as a closed subgroup. The remaining assertions are straightforward.] The example in Corollary 4.11 (i) and the examples in Exercise E4.1 show very clearly that Theorem 4.1 (i) cannot be improved to read that quotients of pro-Lie groups are pro-Lie groups; most of our examples are in fact connected. In [34] it is shown that the class of abelian topological groups which are isomorphic to a product C × RX × ZY where C is a compact group and X and Y are countable sets is closed under the passage to closed subgroups and to quotient groups. This points out the fact that quotient groups of RX are complete if X is countable. In this sense the example of RR is minimal in the class of weakly complete topological vector spaces, if one momentarily accepts the Continuum Hypothesis. In [116] it is proved that a closed connected subgroup of RX is a closed vector subspace; consequently it is a direct summand algebraically and topologically (see [102, Theorem 7.30 (iv)]; in particular the quotient is a weakly complete topological vector space. More generally, we shall see shortly sufficient conditions on a pro-Lie group G and a closed normal subgroup N for G/N to be a pro-Lie group. (See Theorem 4.28 below).

The One Parameter Subgroup Lifting Theorem The lifting of one parameter subgroups deals with the following situation: Assume that f : G → H is a quotient morphism and Y ∈ L(H ); under which circumstances is there an X ∈ L(G) such that L(f )(X) = Y ?

The One Parameter Subgroup Lifting Theorem

Lemma 4.16. Assume that

183

ϕ

P ⏐ −−→ R ⏐ ⏐ ⏐ π Y G −−→ H

(1)

f

def

is a pullback of topological groups. Set K = ker ϕ. Then the following conditions are equivalent: (i) (ii) (ii ) (iii) (iv)

K is a semidirect factor and ϕ is surjective. ϕ is a retraction. ϕ|P0 : P0 → R is a retraction, where P0 is the identity component of P . There is an X ∈ L(G) such that L(f )(X) = Y . There is a subgroup R of P such that KR = P and K ∩ R = {1}, and further that ϕ|R : R → R is open.

These conditions imply: (v) There is a closed subgroup R of P such that KR = P and K ∩ R = {1}. Proof. (i) ⇔ (ii): The equivalence of (i) and (ii) is a standard exercise in topological group theory (see e.g. E1.5). (ii) ⇒ (ii ): If a morphism σ : R → P satisfies ϕ  σ = idR , then σ (R) ⊆ P0 as R is connected, and thus its corestriction σ : R → P0 satisfies ϕ  σ = idR . (ii ) ⇒ (ii): Conversely, if σ : R → P0 satisfies ϕ  σ = idR , then its coextension σ : R → P satisfies ϕ  σ = idR . (ii) ⇒ (iii): If X : R → P is a one parameter subgroup satisfying ϕ  X  = idR def

then X = π  X : R → G is a one parameter subgroup of G such that L(f )(X) = f  X = f  π  X = Y  ϕ  X  = Y  idR = Y . (iii) ⇒ (ii): Assume Y = L(f )(X) = f X. Then for all r ∈ R we have f (X(r)) = Y (r). Now the explicit form of the pullback is P = {(g, r) ∈ G × R | f (g) = Y (r)} and ϕ(g, r) = r (see e.g. Theorem 1.5). Hence (X(r), r) ∈ P for all r ∈ R and if we set X (r) = (X(r), r), then X  : R → P is a morphism satisfying ϕ(X  (r)) = r for all r. (i) ⇒ (iv) ⇒ (v) is trivial. (iv) ⇒ (ii): The morphism ϕ|R : R → R is continuous and open. Thus ϕ(R) is an open subgroup of R and therefore equals R. So ϕ|R is surjective, and since K ∩ R = {0} it is also injective. Hence it is an isomorphism of topological groups and thus is invertible; the coextension σ : R → P of (ϕ|R)−1 : R → R satisfies ϕ  σ = idR . Lemma 4.17. If f in the pullback (1) is surjective, then ϕ is surjective. If f is open, then ϕ is open. If f is a quotient morphism so is ϕ. Proof. Surjectivity: if r ∈ R then, since f is surjective, there is a g ∈ G such that f (g) = Y (r).

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4 Quotients of Pro-Lie Groups

Openness: The filter of identity neighborhoods of P has a basis of open sets of the form W = P ∩ (U × I ), where U is an open identity neighborhood of G and I an open interval around 0 in R. Then ϕ(W ) = {r ∈ I | (∃g ∈ U ) f (g) = Y (r)} = I ∩ Y −1 (f (U )). Since f is an open map, f (U ) is an open subset of H and thus by the continuity of Y , the set ϕ(W ) is open. Quotients: This assertion follows from the combination of the preceding two. Lemma 4.18. In the pullback (1), assume that G is a pro-Lie group. Then P is a pro-Lie group. Proof. By Theorem 1.5 (a) we have P = {(g, r) ∈ G × R : f (g) = Y (r)} in the category TopGr of topological groups. Since G and R are pro-Lie groups, the product G× R is a pro-Lie group since by Theorem 3.3 the category proLieGr is closed under the formation of products. Since the groups G, R, and H are all Hausdorff, the subgroup P is closed. Hence by Theorem 3.35 it follows that P is a pro-Lie group. We now are ready for a proof of the lifting of one parameter subgroups. This is not easy because in the absence of countability assumptions, this requires the Axiom of Choice, and the absence of compactness in the present situation forces us to rely on completeness and the convergence of Cauchy filters. The proof will require from the reader a certain facility handling “multivalued morphisms” as a special type of binary relations; but most of what is required will be self-explanatory in the proof. Lemma 4.19 (The One Parameter Subgroup Lifting Lemma). Let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Then every one parameter subgroup Y : R → H lifts to one of G, that is, there is a one parameter subgroup σ of G such that Y = f  σ . Proof. By Lemmas 4.16, 4.17, and 4.18, we may assume that H = R and we have to show that f is a retraction. Let K = ker f . Since {N ∗ = f (N) | N ∈ N (G)} converges to 0 in R, and since there are no subgroups in ]−1, 1[ other than {0} there is an N ∈ N (G) such that f (N) = N ∗ = {0}, and thus N ⊆ K. Then for all N ∈ N (G), N ⊇ N , the morphism f induces a quotient morphism fN : G/N → R, fN (gN ) = f (g), and fN (gN ) = 0 iff f (g) = 0 iff g ∈ K, that is, ker fN = K/N . If we let pN : K → K/N and qN : G → G/N denote the quotient morphisms, then we have a commutative diagram 1 →

N ⏐ ⏐ incl 1 → K ⏐ ⏐ pN  1 → K/N

idN

−−−→

N ⏐ ⏐ incl f incl −−−→ G −−→ R ⏐ ⏐ → 0 ⏐q ⏐id N  R fN incl −−−→ G/N −−→ R → 0

(2)

The One Parameter Subgroup Lifting Theorem

185

with exact rows and columns. Due to the fact that the exponential map of a Lie group is a local homeomorphism at 0, an open morphism ψ : L1 → L2 between Lie groups induces an open and therefore surjective morphism L(ψ) between their Lie algebras: L(ψ)

L(L ⏐ 1 ) −−−→ L(L ⏐ 2) ⏐expL expL1 ⏐   2 L1 −−−→ L2 . ψ

Hence there is a morphism σN : R → G/N such that fN  σN = idR . The binary −1  σN : R → G satisfies the following conditions: relation  = qN def

(i) (0) = N and every (r) ⊆ G is a coset modulo N. (ii) The graph of  is a closed subgroup of R × G. (iii) We have a commutative diagram of binary relations of which all but  are functions: idR R −−−→ R ⏐ ⏐ ⏐ ⏐id   R f G −−−→ R (3) ⏐ ⏐ ⏐ ⏐id qN   R G/N −−→ R. fN

A binary relation  : R → G satisfying (i), (ii) and (iii) will be called a multivalued morphism associated with N . The set S of all multivalued morphisms  : R → G associated with some N ∈ N (G) is partially ordered under containment ⊆. By Zorn’s Lemma we find a maximal filter F ⊆ S. It is our goal to show def

that M = {(0) |  ∈ F } is cofinal in N (G). Assuming that this is proved, we note that for each r ∈ R and  ∈ F the subset (r) is a coset N x = xN with N = (0) ∈ N (G), and thus (r)(r)−1 = N x(N x)−1 = N; since M converges to 1, we conclude that {(r) |  ∈ F } is a Cauchy filter basis. Since G is complete, it converges to an element σ (r) ∈ G, giving us a function σ : R → G. As each (r), being a coset modulo N = (0) ∈ M, is closed, we have σ (r) ∈ (r) for all  ∈ F . Consequently, since (3) is commutative for each  ∈ F for N = (0) we have the following commutative diagram for all N ∈ M: R ⏐ ⏐ σ G ⏐ ⏐ qN  G/N

idR

−−−→ f

−−−→ −−→ fN

R ⏐ ⏐id  R R ⏐ ⏐id  R R.

(4)

The upper rectangle shows that f  σ = idR , and the fact that each qN   : R → G is a morphism of topological groups shows that qN  σ : R → G/N is continuous.

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4 Quotients of Pro-Lie Groups

Theorem 2.1 (i) shows that G has arbitrarily small open identity neighborhoods U satisfying U N = U for some N ∈ M. Then if V is a zero neighborhood of R such that −1 (U/N ) = U . This shows that σ is continuous. qN (σ (V )) ⊆ U/N , then σ (V ) ∈ qN Hence σ is the required coretraction for f . Thus the remainder of the proof will show that M is cofinal in N (G). Suppose that this is not the case. Then there exists an N ∈ N (G), N ⊇ N such that M ⊆ N for all M ∈ M ⊆ N (G). Let us temporarily fix M; then M ∩ N ∈ N (G), and thus def

G† = G/(M ∩ N) is a Lie group: G

MNG GG ww GG w GG ww w w GG w w N M GG ww GG w GG ww GG ww G ww M ∩N

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

G† .

This shows that for fixed M everything takes place in the Lie group G† in which def

def

M † = M/(M ∩ N) and N † = N/(M ∩ N) are closed normal Lie subgroups with M † ∩ N † = {1}. Thus μ : M † → G/N, μ(m(M ∩ N )) = mN is a morphism of Lie groups mapping M † bijectively onto MN/N and inducing an isomorphism of Lie algebras L(M † ) → L(MN/N ) ⊆ L(G/N ). Now M † is a closed normal subgroup of the Lie group G† and thus M † /M0† is a discrete normal subgroup of the Lie group G† /M0† . We let M ‡ be the open subgroup of M † containing (M † )0 and being such that M ‡ /M0† = (M † /M0† ) ∩ (G† )0 /M0† . Hence M ‡ /M0† is a discrete normal subgroup of a connected Lie group. Consequently it is finitely generated and thus countable. Therefore M ‡ has countably many components and so μ(M0† ) is an analytic subgroup Man ⊆ G/N agreeing with (MN/N )0 and having Lie algebra L(Man ) = L(MN/N ) = L((MN/N )0 ). (See [102, Theorem 5.52ff.]) Accordingly, {L(MN/N ) | M ∈ M} is a filter basis of finite-dimensional vector subspaces of L(G/N ). Hence there is a smallest element m = L(M# N/N ) in it such that for all M ≤ M# in M we have L(MN/N) = m. Let us abbreviate q(M# ∩N ) : G → G/(M# ∩ N ) by q # : G → G# , further f(M# ∩N) : G# → R by f # , and M# /(M# ∩ N ) by M # . Since L(μ) : L(M # ) → L(M # N/N ) = m is an isomorphism we have ,, q#

M(M # ∩ N) M# ∩ N

- = q # (M0# ) for M ⊆ M# in M. 0

(5)

The One Parameter Subgroup Lifting Theorem

187

There is a  # ∈ F such that M # =  # (0). Then for all  ∈ F contained in  # , the subgroup q # ((0)) of the Lie group G# is contained in q # ( # (0)), satisfies q # ((0)) = M0# , and (f #  q #  )(R) = R. Thus for all r ∈ R we have q # ( # (r)) = q # ((r)) since the right side is contained in the left and both are cosets modulo M # . In the Lie group G# we have the configuration G

M# NG GG v v GG vv v GG vv GG v v M# H N HH ww w HH ww HH ww HH ww M# ∩ N

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

G# .

−1 # Let σ # = σM# ∩N : G# → R be defined by σ = qM #   # . Then  # = qM #  σ and we have a commutative diagram of binary relations def

idR

−−−→

R ⏐ ⏐ #  G ⏐ qM # ⏐ 

f

−−−→

G/M #

−−−→ fM #

R ⏐ ⏐id  R R ⏐ ⏐id  R

(3# )

R.

def

We conclude that S = q # ( # (R)) =  # (R)/(M # ∩ N ) is a closed subgroup of G# whose Lie algebra L(S) cannot be contained in K # = K/(M # ∩ N ) = ker f # . From dim G# /K # = 1 we conclude L(G# ) = L(K # ) + L(S)

and L(S) = L(S) ∩ L(K # ) + R · X

for a suitable element X ∈ L(G# ) satisfying f # (expG# X) = 1. Setting τ : R → S, τ (r) = expG# r · X we obtain a coretraction for f # : G# → R. The binary relation def

 = (q # )−1  τ : R → G is a member of S. Moreover, for all  ∈ F we have q # ()(r) ⊇ τ (r) for all r ∈ R. Hence  ∩  is a member of S. Now the maximality of F shows that  ∈ F . But this implies that M # ∩ N = (0) ∈ M and that is a contradiction to our supposition allowing us a choice of an N such that M ∩ N = M for all M ∈ M. This contradiction finally completes the proof. There are some subtleties here which we should point out. Following Theorem 5.52 in [102] we have seen that the additive group h of a Banach space mapped surjectively onto an abelian Lie group G (which itself is quotient of a Banach space modulo a

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4 Quotients of Pro-Lie Groups

discrete subgroup) such that G has a one parameter subgroup which does not lift to h. This cannot happen if the domain is separable, but it does happen in the category of not necessarily finite-dimensional Lie groups. While being surjective, the morphism in question is not open and the Open Mapping Theorem fails. On the other hand, let G be a free abelian topological group in the sense of Graev (see [71]) generated by the space |R| underlying the topological abelian (additive) group R of real numbers. (For details on free topological groups see for instance [150]; see also [130].) Since the generating space |R| is arcwise connected and contains the identity, G itself is arcwise connected. Further, G is a kω group; hence it is a complete topological group ([150, pp. 379, 380]). Accordingly, G is not metrizable, because if it were it would be a σ -compact Baire space group and thus would be locally compact – which it is not. (See also [151].) Now we use the universal property of G to extend the identity map |R| → R to a morphism of topological groups q : G → R. If we consider |R| as embedded into G (which we may) then the restriction ρ of q to |R| is the identity map. Set K = ker q and define m : |R| × K → G by m(r, k) = rk; then m has an inverse given by m−1 (g) = (ρ −1 q(g), (ρ −1 q(g))−1 g). So m is a homeomorphism such that q m = ρ pr 1 . Thus, topologically, q is equivalent to a projection of a product onto one of the factors and is therefore a quotient map. However, algebraically G is a free group generated by the elements of |R| and thus does not contain any divisible elements, and so L(G) = {0}. Hence there is no morphism X : R → G such that q  X = idR . The quotient morphism q does not lift and L(q) : L(G) → L(R) = R is not surjective. We have seen that the functor L preserves all limits and thus, in particular, all kernels (since ker f for a morphism f of topological groups is nothing but the equalizer of f and the constant morphism). We shall say that a functor F : A → B between categories of topological groups is strictly exact if it preserves kernels and quotients. A morphism e : N → G of topological groups is said to be a strict morphism if its corestriction N → im e is an open morphism, that is, the bijective morphism N/ ker e → im e induced by it is an isomorphism of topological groups. Also recall from Theorem 4.1, that a quotient H of a pro-Lie group G is a proto-Lie group and that it has a completion H which is a pro-Lie group. Now, as a corollary of the One Parameter Subgroup Lifting Lemma we obtain the following theorem. The Strict Exactness Theorem for L Theorem 4.20. Let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Then the following conclusions hold. (i0 ) H is a proto-Lie group and the canonical embedding γH : H → HN (H ) =

lim

N ∈N (H )

H /N

according to 1.29 and 1.30 induces an isomorphism L(γH ) : L(H ) → L(HN (H ) )

The One Parameter Subgroup Lifting Theorem

189

of pro-Lie algebras. In particular L(H ) is a pro-Lie algebra. (i) The pro-Lie algebra morphism L(f ) : L(G) → L(H ) is a quotient morphism. In particular, the functor L : proLieGr → proLieAlg is strictly exact. (ii) If f

e

N −−→ G −−→ H is an exact sequence of morphisms of pro-Lie groups with a strict morphism e, then L(f )

L(e)

L(N ) −−−−→ L(G) −−−−→ L(H ) is an exact sequence of pro-Lie algebras. Proof. (i0 ) From the Quotient Theorem of Pro-Lie Groups 4.1 we recall that HN (H ) = limN∈N (H ) H /N is the completion H of the proto-Lie group H . Let νN : H → H /N denote the limit morphism and qN : H → H /N be the quotient morphism for N ∈ N (H ). From Theorem 1.29 we recall that νN  γH = qN for all N ∈ N (H ). The functor L preserves limits, and so for each N ∈ N (H ) we have a commutative diagram L(γH )

L(H −−−−→ L(H ⏐ ) ⏐ ) ⏐ ⏐ qN  L(ν) L(H /N ) −−−−→ L(H /N ),

(∗)

idL(H /N)

where the L(νN ) are the limit maps. The composition f

qN

qN  f : G −−→ H −−→ H /N is a quotient morphism of pro-Lie groups. The One Parameter Subgroup Lifting Lemma 4.19 tells us that it induces a surjective morphism L(qN  f ) : L(G) → L(H /N). Surjective morphisms of weakly complete topological vector spaces are quotient maps (see Appendix 2, Theorem A2.12 (b)). Since L is a functor, we have L(qN  f ) = L(qN )  L(f ); since L(qN )  L(f ) is surjective, L(qN ) : L(H ) → L(H /N) is surjective. Thus in the diagram (∗), the maps L(νN )  L(γH ) are all quotient maps, and this implies by Theorem 1.20 (i) that L(γH ) has a dense image. Also, L(γH ) has a zero kernel, since L preserves kernels and γH is an embedding. From Theorem 1.30 it follows that L(γH ) is an embedding. Therefore the composition L(f )

L(γH )

L(γH )  L(f ) : L(G) −−−−→ L(H ) −−−−→ L(H ) is a morphism of pro-Lie algebras with a dense image. From Appendix 2, Theorem A2.12 (a) we know that every morphism of weakly complete topological vector spaces with dense image splits and thus is surjective. Since the underlying vector spaces of L(G) and L(H ) are weakly complete by 3.8 and 3.12, the morphism L(γH )L(f ) is

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4 Quotients of Pro-Lie Groups

surjective. Therefore, the embedding L(γH ) is surjective and thus is an isomorphism. This is what we had to show. (i) We observed that L preserves kernels because kernels are limits. In (i0 ) we saw that L(H ) is a pro-Lie algebra, and the One Parameter Subgroup Lifting Lemma 4.19 shows that L(f ) is surjective, hence a quotient since surjective morphisms of weakly complete topological vector spaces are quotients by Theorem A2.12(b). Therefore, L preserves quotients, and so L is strictly exact. This completes the proof of (i). (ii) Let e

f

N −−→ G −−→ H be morphisms of pro-Lie groups such that im e = ker f and e is open onto its image. Set K = ker f , and decompose e as κ  ε where ε : N → K is the corestriction of e to its image and κ : K → G is the inclusion morphism. Then L(e) = L(κ)  L(ε) and L(κ) : L(K) → L(G) is the kernel of L(f ) since L preserves kernels, and L(ε) : L(N) → L(K) is surjective since L preserves quotients. Thus im L(e) = L(K) = ker L(f ).

It is remarkable that we do not require that H is a pro-Lie group; yet we obtain that L(H ) is a pro-Lie algebra. Such a situation can indeed arise as is shown by 4.9 and 4.10 where we have a quotient morphism from RR onto an incomplete group with a totally disconnected kernel inducing on the Lie algebra level the identity map of RR . The identity morphism Rd → R from the discrete group of real numbers to the group of real numbers with its natural topology is a bijective morphism of Lie groups inducing the zero morphism L(Rd ) = {0} → R ∼ = L(R). Thus one cannot drop the assumption in the theorem that e be a strict morphism. Sometimes one has an Open Mapping Theorem available which gives us this assumption for free – for instance if ker f is a Baire space and N is locally compact and σ -compact. In fact, we shall prove an Open Mapping Theorem for almost connected pro-Lie groups later in the book. In that context we shall show elementarily the existence of a connected proto-Lie group 2 ℵ0 H whose Lie algebra is algebraically isomorphic to Rℵ0 , while L(H ) ∼ = R2 . Thus there is no chance of proving (i0 ) of Theorem 4.20 without the assumption that H is in fact a quotient of a pro-Lie group. Corollary 4.21. (i) If N is a closed normal subgroup of a pro-Lie group G, then the quotient morphism q : G → G/N induces a map L(q) : L(G) → L(G/N ) which is a quotient morphism with kernel L(N ). Accordingly there is a natural isomorphism X + L(N) → L(f )(X) : L(G)/L(N ) → L(G/N ). (ii) Let G be a pro-Lie group. Then {L(N ) | N ∈ N (G)} converges to zero and is cofinal in the filter  (L(G)) of all ideals i such that L(G)/i is finite-dimensional. Furthermore, L(G) is the projective limit limN ∈N (G) L(G)/L(N ) of a projective system of bonding morphisms and limit maps all of which are quotient morphisms, and

The One Parameter Subgroup Lifting Theorem

191

there is a commutative diagram L(γG ) G ∼ L(G) −−−−→ L(GN (G) ) = L(limN ∈N ⏐ ⏐ (G) N ) = limN ∈N (G) ⏐ ⏐ expG  L(limN∈N (G) expG/N ) G −−−−→ GN (G) = limN ∈N (G) G/N.

L(G) L(N )

γG

Proof. Assertion (i) is an immediate consequence of the Strict Exactness Theorem 4.20. (ii) We know that L preserves limits. Thus L(γG ) : L(G) → L(GN (G) ) is an isomorphism. By (i) above, L(G/N ) ∼ = L(G)/L(N ) and thus L(G) ∼ =

lim

N ∈N (G)

L(G)/L(N ).

Thus by 1.27 (ii), the filter basis {L(N ) | N ∈ N (G)} of the kernels of the limit maps converges to 0 and the projective system of the L(G)/L(N ) has the natural quotient morphisms as bonding maps; by 1.27 (ii) it follows that the limit maps are quotient morphisms as well. It then follows that this filter basis is cofinal in  (L(G)). (Compare 1.40.) Recall from Definition 2.21 that E(G) denotes the subgroup expG L(G) algebraically generated by expG L(G). Corollary 4.22. (i) For a pro-Lie group G, the subgroup E(G) = expG L(G) is dense in G0 , i.e. E(G) = expG L(G) equals G0 . In particular, a connected nonsingleton pro-Lie group has nontrivial one parameter subgroups. Moreover, if h is a closed proper subalgebra of L(G), then expG h = E(G). (ii) A morphism f : G → H of pro-Lie groups induces a surjective (hence quotient) morphism L(f ) : L(G) → L(H ) in each of the following cases: (a) f is open. (b) f is surjective and G is almost connected. (iii) Assume that a morphism f : G → H of pro-Lie groups induces a surjective morphism L(f ) : L(G) → L(H ); by (ii) above this is the case, in particular, if f is open or if f is surjective and G is almost connected. Then the induced morphism E(f ) : E(G) → E(H ) is surjective, that is E(H ) = f (E(G)). As a consequence, H0 = f (G0 ). If H is a Lie group, then H0 ⊆ f (G). If H is in addition connected, then f is surjective. (iv) A morphism f : G → H of pro-Lie groups into a finite-dimensional Lie group inducing a surjective morphism L(f ) is locally trivial, that is, there is an open identity neighborhood V of H and a continuous map s : V → G such that f (s(h)) = h for all h ∈ V . In particular, for N = ker f , ϕ : f −1 (V ) → N × V ,

ϕ(g) = (gs(f (g))−1 , f (g))

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4 Quotients of Pro-Lie Groups

is a homeomorphism with inverse ψ given by ψ(n, h) = ns(h). (vi) Let G be a pro-Lie group and assume that for all N from a basis of N (G) the quotient G/N is connected. Then G is connected. Proof. (i) First we show that a nonsingleton connected pro-Lie group has nontrivial one parameter subgroups. Let G be a nonsingleton connected pro-Lie group. There is a g ∈ G, g = 1. Since lim N (G) = 1 there is an N ∈ N (G) such that g ∈ / N. Then G/N is a nonsingleton connected Lie group. Thus L(G/N ) = {0}. Then L(G) = {0} by 4.21 (i). Next we let G be an arbitrary pro-Lie group. The closed subgroup E(G) = expG L(G) is fully characteristic, hence normal. By the One Parameter Subgroup Lifting Lemma 4.19, every one parameter subgroup of G/E(G) lifts to one in G which is contained in E(G) by the definition of E(G). Hence L(G/E(G)) = {0}. Thus G/E(G) is totally disconnected by what we just proved, and thus G0 ⊆ E(G) ⊆ G0 . def

For a proof of the last assertion of (i), let us write g = L(G) and let h be closed subalgebra of g such that h = g. Then there is a 0-neighborhood U of g such that h + U = g. By Corollary 4.21 (ii) we find a closed normal subgroup N of G such that def

n = L(N) ⊆ U . Thus h + n = g. Since dim g/n < ∞ and all vector subspaces of a finite-dimensional real vector space are closed we know that h+n is a closed subalgebra of g. If we can prove that expG (h + n) = E(G) then certainly expG h = E(G). Thus by replacing h by h + n and renaming the subalgebra we shall assume without loss of generality that n ⊆ h. By Corollary 4.21 (i) we have a commutative diagram {0} →

n ⏐ ⏐ id {0} → ⏐ n expN ⏐  {1} → N

−−−→

incl

quot

j

q

h −−−→ h/n → {0} ⏐ ⏐ ⏐ ⏐ incl  L(j ) L(q) −−−→ ⏐ g −−−→ L(G/N ⏐ ) → {0} ⏐exp expG ⏐   G/N −−−→ G −−−→ G/N → {1},

and we may naturally identify L(G/N ) with g/n; under this identification, expG/N (h/n) gets identified with q(expG (h)) = (expG (h))N/N. Now expG hN/N = (expG (h))N/N = expG/N (h/n). By the same token, E(G)N/N = expG gN/N = expG/N (g/n) = (G/N )0 since G/N is a Lie group. Since h/n = g/n the analytic subgroup expG/N (h/n) of (G/N )0 is proper. Thus expG hN = E(G)N and since expG h ⊆ E(G), this must mean expG h = E(G). (ii) (a) Let f : G → H be an open morphism of topological groups. Then f (G) is an open, hence closed, subgroup of H (see e.g. [102, Proposition A4.25 (ii)]) and thus a pro-Lie group by Corollary 4.8. The open and surjective corestriction G → f (G) (inducing an isomorphism of topological groups G/ ker f → f (G)) is a quotient morphism between pro-Lie groups and thus induces a quotient morphism

The One Parameter Subgroup Lifting Theorem

193

L(f ) : L(G) → L(f (G)) by the Strict Exactness Theorem 4.20. Since f (G) is open in H , the inclusion j : f (G) → H induces an isomorphism L(j ) : L(f (G)) → L(H ) of topological Lie algebras. Thus L(f ) : L(G) → L(H ) is a quotient morphism. (b) By Theorem A2.12 (b) of Appendix 2, a surjective morphism of weakly complete topological vector spaces is a quotient morphism; so we have to show that L(f ) is surjective, that is, that every one parameter subgroup of H can be lifted to one of G. By Lemmas 4.17 and 4.18, it is no restriction to assume that H = R = L(H ). Assume that X ∈ L(G) is such that L(f )(X) = 0. Then L(f )(R · X) = R = L(H ). It therefore suffices to show the existence of an X ∈ L(G) which is not annihilated by L(f ). However, suppose that L(f )(L(G)) = {0}. Then f (E(G)) = {1H }. By the continuity of f it follows that f (E(G)) = {1H }. By (i) above, E(G) = G0 . Since G is almost connected, G/G0 is compact. Since f (G0 ) = {1H }, there is a morphism F : G/G0 → H = R, F (gG0 ) = f (g). Then f (G) = F (G/G0 ) is a compact subgroup of H = R and therefore is {1H }, and this contradicts the surjectivity of f . (iii) By assumption, L(H ) = L(f )(L(G)), and thus expH L(H ) = expH L(f )(L(G)) = f (expG L(G)). Consequently E(H ) = expH L(H ) = f (expG L(G)) = f expG L(G) = f (E(G)).

Thus H0 = E(H ) = f (E(G)) ⊆ f (G0 ) ⊆ H0 = H0 , and this shows f (G0 ) = H0 . Now assume that H is a connected Lie group. Then expH L(H ) is a neighborhood of 1 and thus E(H ) is an open and hence also closed subgroup of H and thus H0 ⊆ E(H ). Since E(H ) is connected, E(H ) = H0 . Now H = E(H ) = f (E(H )) ⊆ f (G), and thus H0 ⊆ f (G). If H = H0 , then f is surjective. (iv) The morphism L(f ) : L(G) → L(H ) is a surjective morphism of weakly complete topological vector spaces and therefore splits by Theorem A2.12 (a), that is, there is a morphism of weakly complete topological vector spaces σ : L(H ) → L(G) such that L(f )  σ = idL(H ) . Since H is a finite-dimensional Lie group, there is an open zero neighborhood U of L(H ) and an open identity neighborhood V of H such that expH |U : U → V is a homeomorphism. Define s : V → G by s = expG  σ  (expH |U )−1 . Let h ∈ V . Then f (s(v)) = (f  expG σ )((expH |U )−1 (h)) = (expH L(f )  σ )((expH |U )−1 (h)) = expH ((expH |U )−1 (h)) = h, as asserted. The remainder is an easy exercise by verifying that ψ  ϕ and ϕ  ψ are the respective identity functions. (v) Let qN : G → G/N denote the quotient morphism. By (ii) we have G/N = E(G/N) = qN (E(G)). Thus G = E(G)N for all N ∈ N (G) and so G0 = E(G) = E(G) = G. There is a connected commutative infinite-dimensional Banach Lie group G whose exponential function maps a proper closed hyperplane h in its Lie algebra g surjectively onto the group (see [102, Chapter 5, paragraph following Theorem 5.52]); this illustrates that 4.22 (i) is not exactly a trivial matter.

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4 Quotients of Pro-Lie Groups

Let H = Tp be the p-adic solenoid (see Example 1.20 (ii)), let G = L(H ) = R, and let f : G → H be the exponential function expH : L(H ) → H . Then L(f ) : L(G) → L(H ),

L(G) = R

is an isomorphism, but f is far from surjective. In fact, im f = Ha is the identity  arc component of H and as abelian groups, π0 (H ) = H /Ha and Ext p1∞ · Z, Z are isomorphic groups of continuum cardinality. (See Chapter 8 in [102].) The relation H0 = f (G0 ) for a quotient morphism f such as in 4.22 (iii) cannot be improved as the example of the following quotient morphism of locally compact abelian groups shows: Let G = R × Zp for the group of p-adic integers Zp , let H = G/{(n, −n) | n ∈ Z} ∼ = Tp and let f be the corresponding quotient morphism. Note that H is compact and connected. (See [102, Exercise E1.11 following Definition 1.30]. We consider Z as a subgroup of Tp as well.) Then G0 = R × {0}, and f (G0 ) = Ha = H = H0 . However, we want to point out, that the proof of 4.22 (iii) does show more than what was asserted. Observe that for any topological group G we can write E(G) = expG L(G) and E(G) = E(G). If G is a proto-Lie group, then its completion is a pro-Lie group G∗ = GN (G) and we have g ⊆ g∗ (writing G ⊆ G∗ ). If equality g = g∗ holds then expG g = expG∗ g∗  is dense in G∗ and is contained in G, and so is dense in G. Therefore: Remark 4.22 (iii)∗ . Assume that a morphism f : G → H of topological groups induces a surjective function L(f ) : L(G) → L(H ). Then the induced morphism E(f ) : E(G) → E(H ) is surjective, that is E(H ) = f (E(G)). If G0 = E(G) and H0 = E(H ), then H0 = f (G0 ). In particular, if G is a pro-Lie group and H is a proto-Lie group whose pro-Lie algebra coincides with the pro-Lie algebra of its completion, then H0 = f (G0 ). The vanishing of the Lie algebra means the absence of nontrivial one parameter subgroups. This property should be discussed here. Just for the record we repeat at this point what we showed in Proposition 3.30. In the light of the Pro-Lie Group Theorem 3.34 that result can be slightly reformulated. Corollary 4.23 = 3.30. For a pro-Lie group G the following statements are equivalent: (a) (b) (c) (d)

L(G) = {0}. G is totally disconnected. G is zero-dimensional. G is prodiscrete.



This generalizes the classical result that a totally disconnected finite-dimensional Lie group is discrete.

Sufficient Conditions for Quotients to be Complete

195

Sufficient Conditions for Quotients to be Complete In the structure theory of locally compact groups, the concept of “almost connected groups” is quite helpful, but it applies perfectly well to arbitrary topological groups.We introduced it in our preface and in Definition 1(ii) in the overview chapter, and we reiterated it many times since. For easy reference we recall: Definition 4.24 = 1(ii). A topological group G is called almost connected if G/G0 is compact. Clearly, all compact groups and all connected topological groups are almost connected. If G is almost connected and N is a closed normal subgroup of G, then G0 N /N ⊆ (G/N)0 , whence (G/N )/(G/N )0 is a quotient group of (G/N )/(G0 N/N) ∼ = G/G0 N ∼ = (G/G0 )/(G0 N /G0 ), which is in turn a quotient group of the compact group G/G0 . Thus quotients of almost connected groups are almost connected. For the next steps we observe, that the filter basis N (G) that we use to define pro-Lie groups may be “thinned out” to a cofinal filter basis M(G) if G is almost connected. We say that M and N in N (G) are close if M ∩ N is open in M and N . This relation is reflexive and symmetric. We claim that it is transitive: Indeed, if N1 ∩ N2 is open in Nj , j = 1, 2 and N2 ∩ N3 is open in Nj , j = 2, 3 then N1 ∩ N2 ∩ N3 is open in N2 and then in N1 ∩ N2 and N2 ∩ N3 ; therefore the triple intersection is also open in N1 and N3 whence N1 ∩ N3 is open in N1 and N3 . Therefore closeness is an equivalence relation, and we write M ∼ N if M and N are close. Now we define def

M(G) = {N ∈ N (G) | (M ⊆ N and M ∼ N ) ⇒ |N/M| < ∞}.

(6)

Lemma 4.25. (i) If G is an almost connected pro-Lie group, then M(G) as defined in (6) is cofinal in N (G). In particular, M(G) is a filter basis. (ii) If M(G) is cofinal, then γ : G → GM(G) = limM∈M(G) G/M is an isomorphism. Proof. (i) Let N ∈ N (G) and M ∼ N in N (G) such that N ⊇ M. The quotient def

G/M → G/N has the discrete kernel D = N/M and so induces a covering map of the identity components (G/M)0 → (G/N )0 whose kernel is K = (N/M) ∩ (G/M)0 = (N ∩G0 M)/M. Since G is almost connected, the Lie group G/M is almost connected, and thus (G/M)/(G/M)0 ∼ = G/G0 M is finite; consequently D/K is finite. The group K is central (see e.g. [102, A4.27]), in particular abelian, and it is finitely generated. (See [102, Proposition 5.75], applied to a maximal connected abelian subgroup H : all of these contain the center (see [86, p. 189], Theorem 1.2, or [85, p. 280, Satz III.7.11]).)

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4 Quotients of Pro-Lie Groups

Let tor K denote the finite torsion subgroup of K. D = N/M    K    tor K   {1}

 D/K finite 

K/ tor K ∼ = Zn(M)

 finite

The torsion-free rank n(M) of K/ tor K is a natural number. We claim that n(N ) is the maximal rank of free abelian subgroups of D: Indeed let F be a free abelian subgroup of D. Then F K/K ∼ = F /(F ∩K) is finite, hence F ∩K is a free abelian group such that rank(F ∩ K) = rank F (see e.g. [102, Theorem A1.10]). But rank(F ∩ K) ≤ n(M), and this establishes the claim. Let GN denote the simply connected covering group of (G/N )0 . Then for each M ∼ N, M ⊆ N there is a covering morphism GN → (G/M)0 . The set {n(M) | M ⊆ N, M ∼ N} is bounded by the torsion-free rank of π1 (GN ). Let us pick an M ⊆ N in the ∼-equivalence class of N with maximal n(M). Now let M  ⊆ M be a member in the ∼-class of M. Then M  ∼ N and M  ⊆ M, and thus n(M  ) = n(M) by the maximality of n(M). We claim that |M/M  | is finite. Suppose that this is not the case. Then in the sequence of coverings (G/M  )0 → (G/M)0 → (G/N )0 the kernel of the first one has a positive torsion-free rank n , and the kernel of the second has torsion-free rank n(M). Hence the torsion-free rank n(M  ) of the composition is n + n(M) > n(M), contrary to the maximality of n(M). This contradiction proves our claim that |M/M  | is finite. Hence M ∈ M(G) and M ⊆ N and this shows that M(G) is cofinal in N (G). (ii) By the Cofinality Lemma 1.21 we have lim

M∈M(G)

G/M =

lim

N ∈N (G)

G/N = GN (G) .

Since G is pro-Lie we have G ∼ = GN (G) . For almost connected pro-Lie groups, we shall see an alternative approach to M(G) in 9.45. We shall now introduce a topology which we call the Z-topology and which is reminiscent of the Zariski topology on algebraic varieties and, in particular, on algebraic groups. This topology will be defined on almost connected pro-Lie groups. While it is a T1 -topology (singletons are closed sets) it gives rise to one of those rare occasions where we choose to use topologies which are not Hausdorff. When the topology is restricted to a Lie group which is an algebraic group, it will in general be coarser than the Zariski topology. Recall that a subbasis for the set of closed sets C of a topology O(X) on a set X is a set B ⊆ C such that every closed set is the intersection of sets each of which is a finite

Sufficient Conditions for Quotients to be Complete

197

union of members of B. We say that the topology O(X) is generated by B. Every set of subsets of a set X can serve as a subbasis for a topology. Recall that a collection of subsets of a set is said to satisfy the finite intersection property if every finite subcollection has a nonempty intersection, that is, iff it generates a filter. Let X be a topological space. By the Alexander Subbasis Theorem (see e.g. [131, p. 139, Theorem 6]), the topology O(X) generated by a subbasis B for the set of closed sets of X is compact iff every set of sets from B satisfying the finite intersection property has a nonempty intersection. Note in passing that a proof of the Alexander Subbasis Theorem requires the Axiom of Choice. Definition 4.26 (The Z-topology on a pro-Lie group). Let G be a pro-Lie group and def

set B(G) = {gN | g ∈ G, N ∈ M(G)}. The topology Z(G) on G generated by B(G) as a subbasis for the set of closed sets is called the Z-topology. We shall often abbreviate Z(G) by Z. Note that we are not saying that G with the Z-topology is a topological group. Proposition 4.27. The Z-topology Z on a pro-Lie group G has the following properties: (i) All left and right translations are homeomorphisms in the Z-topology. (ii) Inversion x  → x −1 : G → G is a homeomorphism in the Z-topology.  (iii) If M(G) = {1} then Z is a T1 -topology. This is the case if G is almost connected. (iv) If G is almost connected, then Z is a compact T1 -topology. Proof. (i), (ii) It suffices to observe that the subbasis B(G) is bijectively mapped onto itself by left and right translations (since x(gN ) = (xg)N respectively, (gN )x = (gx)N) and by inversion (since (gN )−1 = N −1 g −1 = g −1 N ). (iii) By (i) it suffices to show that {1} is closed, i.e., that for each g = 1 there is a closed set containing 1 but not g. Thus let g = 1. Since M(G) = {1} there is an N ∈ M(G) such that g ∈ / N . Then N is the required closed set. (iv) We assume that G is an almost connected pro-Lie group and show that the Z-topology is compact. We proceed in steps. Step (a). Assume that G is a Lie group, in which case {1} ∈ M(G). Now let F = {gj Nj | j ∈ J } be a filter basis of subbasic sets with suitable Nj ∈ M(G). We like to show that it has a nonempty intersection and then apply the Alexander Subbasis Theorem to prove that G is compact for the Z-topology. Define F0 = {gj (Nj )0 | j ∈ J }. Since the collection of finite dimensional vector spaces {L(Nj ) | j ∈ J } satisfies the descending chain condition, there is a j0 ∈ J such that for all j ≥ j0 we have L(Nj ) = L(Nj0 ) and hence (Nj )0 = (Nj0 )0 . Thus there is no harm in assuming that the identity components of the members of F all agree with some M ∈ M(G). But then we pass to the quotient G/M. To simplify notation we then assume that M is a singleton and that all members of F are discrete normal subgroups of the Lie group G. Now we use our hypothesis that G is almost connected in its group topology. This property is preserved by the reductions we made. Hence G/G0 is finite since in a Lie

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4 Quotients of Pro-Lie Groups

group the identity component is open. Then there is a g ∈ G such that gG0 ∩ F = Ø for all F ∈ F . We may consider the filter basis g −1F instead  and assume in the end that {G0 ∩ F : F ∈ F } is a filter basis F  . Since F  ⊆ F . Therefore there is no harm in assuming F  = F . Thus we are dealing with a connected Lie group G and a filter basis F of cosets modulo discrete normal subgroups D ∈ M(G). Since {1} ∈ M(G) and {1} is open in each D ∈ M(G), D ∼ {1} for all D. By the definition of M(G), this means that D is finite. Thus the filter basis F consists of finite sets and so has a nonempty intersection. An application of the Alexander Subbasis Theorem completes the proof of the compactness of G in this case. Step (b). Assume that G is a strict projective limit of its Lie group quotients G/N , def  N ∈ M(G) and assume that H = N ∈M(G) G/M is compact in its Z-topology. We claim that then G is compact in its Z-topology. For a proof consider M, N ∈ M(G), M ⊇ N, let SMN denote the subgroup of G/M×G/N of all pairs (gM, hN ) with hN ⊆ gM. Then hM = hN. We claim that (G/M ×G/N )/SMN = (G/M ×{N })SMN /SMN is isomorphic to G/M. Indeed G/M × G/N is an almost connected Lie group, and thus is σ -compact and local compact, and G/M is a locally compact. Then by the Open Mapping Theorem for Locally Compact Groups (see for instance [79, p. 42, Theorem 5.29]) applies and  establishes the claim. Hence SMN ∈ M(G/M × G/N). Now let PMN ⊆ H = P ∈M(G) G/P be  the subgroup of all (gP P )P ∈N (G) such that gN ∈ gM M. Then PMN ∼ = SMN × P ∈M(G)\{M,N } G/P and thus H /PMN ∼ = G/M and therefore PMN ∈ M(H ). In particular PMN is Z-closed in H . Hence limP ∈M(G) G/P = M,N ∈N (G),N ⊆M PMN is a Z-closed subgroup of H . Since H is Z-compact by assumption, it follows that G ∼ = limN ∈M(G) G/N is Z-compact. After Steps (a) and (b) the proof of Claim (iv) will be finished if we prove the next and final step:  Step (c). G is a product j ∈J Gj of almost connected Lie groups Gj . We claim that G is compact in the Z-topology. Observation (i). If M ∈ M(G), then there is a unique largest cofinite subset EM of J such that j ∈EM Gj ⊆ M, where we have identified the partial product in an unmistakable fashion with a subgroup of G. Proof of Observation (i). Since G/M is a Lie group and thus has no small subgroups, we find an open neighborhood U of M such that U M = U and every subgroup of G contained in U is contained in M. Let V be a basic identity neighborhood for the product topology of G such that V ⊆ U . Then there is a finite  subset EM ⊆ J and identity neighborhoods Wj in Gj for j ∈ EM such that V = j ∈J Vj for " Wj if j ∈ EM , Vj = Gj if j ∈ J \ EM .    are two Then j ∈J \EM Gj ⊆ V ⊆ U and therefore j ∈EM Gj ⊆ M. If EM and EM such sets, then       Gj = Gj Gj ⊆ M.  j ∈J \EM ∩EM

j ∈J \EM

 j ∈J \EM

Sufficient Conditions for Quotients to be Complete

199 def

Thus  we may assume that EM was selected to be the smallest set such that P (M) = j ∈J \EM Gj , {1} × P (M) ⊆ M.  Now let N(M) be the projection of M onto j ∈EM Gj along P (M) ⊆ M. Then M = N(M) × P (M) where N(M) is a closed normal and almost connected normal subgroup of G (incidentally not containing any of the factorsGj , j ∈ EM ) such that M = N(M) × P (M). For a subset I ⊆ J let us write GI = j ∈I Gj , and let Fin(J ) denote the set of all finite subsets of J . Then we have Observation (ii). The set M(G) is the set of all M = N × GJ \E ,

E ∈ Fin(J ),

N ∈ M(GE ).

In particular, each M ∈ M(G) contains a member of M(G) of the form {1}×GJ \E , E ∈ Fin(J ).  Observation (iii). In the Z-topology, G = j ∈J Gj is compact for any infinite family of almost connected Lie groups Gj . In order to show that G is compact in the Ztopology, by the Alexander Subbasis Theorem it is sufficient to show that a filter basis B of subbasic closed sets, that is, cosets B = g(B)M(B) ∈ B,

g(B) ∈ G, M(B) ∈ M(G),

has a nonempty intersection. Notice that M(B) = B −1 B is uniquely determined by B and B ⊇ C implies M(B) ⊇ M(C), while the representative g(B) is unique only modulo M(B). If g ∈ B∩C for B, C ∈ B, then B = gM(B) and C = gM(C), whence B ∩ C = g(M(B) ∩ M(C)); in general M(G) is not closed under finite intersections. As we have seen in Observation (ii), for each B ∈ B, the normal subgroup M(B) of G is of the form M(B)EB × GJ \EB for a suitable minimal finite set EB ∈ Fin(J ) and a suitable subgroup M(B)EB ∈ M(GEB ). We write g(B) as g(B)EB × gJ \EB ∈ GEB × GJ \EB . Then we have B = g(B)EB M(B)EB × GJ \EB .

(∗)

The set of all filter bases B  containing B and having cosets g(B  )M(B  )B  as members is inductive with respect to containment ⊆. Hence it contains maximal ones by Zorn’s  Lemma. The intersection of any of these maximal filter bases is contained in B. It is therefore no restriction of generality if, in order to simplify notation, we assume that B itself is maximal. Clearly B is infinite. Let E ∈ Fin(J ); we consider the projection fE : G → GE and inspect the filterbasis fE (B) of cosets modulo normal subgroups in the almost connected Lie group GE . Equation (∗) permits us the comparison B = g(B)EB M(B)EB × GE\EB × GJ \(E∪EB ) , ker fE = {1E∩EB } × GEB \E × {1E\EB } × GJ \(E∪EB ) ,

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4 Quotients of Pro-Lie Groups

with obvious identifications of the partial products of the Gj . Accordingly we see that fE (B) = fE(E∪EB ) (g(B)EB M(B)EB × GE\EB ). Note that g(B)EB M(B)EB × GE\EB = (g(B)EB × {1E\EB })·(M(B)EB × GE\EB ) is a coset modulo the closed normal subgroup M(B)EB × GE\EB . However, M(B)EB ∈ M(GEB ), and thus the component (M(B)EB )0 is open in M(B)EB and therefore M(B)EB ∼ (M(B)EB )0 ; hence M(B)EB has finitely many components and therefore M(B)EB × GE\EB has finitely many components. The image fE(E∪EB ) ((M(B)EB × GE\EB )0 ) is an analytic subgroup of the Lie group GE and thus fE (M(B)) = fE(E∪EB ) (M(B)EB × GE\EB ) is a finite extension of an analytic subgroup of GE . In view of the fact that the set of finite extensions of analytic subgroups in an almost connected Lie group satisfies the finite descending chain condition, we have seen that (†) for each E ∈ Fin(J ) the filterbasis fE (B) contains a smallest element AE , that is, there is a C ∈ B such that for all D ∈ B AE = fE (C) ⊆ fE (D). We now claim that (‡) for each B ∈ B there is a unique element hB ∈ GEB such that def

B = {hB } × GJ \EB ∈ B and B ⊆ B. For a proof of this claim let B ∈ B. Then by (∗) we have B = g(B)EB M(B)EB × GJ \EB . By (†) the filter basis fEB (B) has a minimal element AEB ⊆ g(B)EB M(B)EB . Let a ∈ AE . Then g(B)EB M(B)EB = aM(B)EB and a ∈ fEB (C) for all C ∈ B. Hence, if for C ∈ B we set Ca = fE−1 (a) ∩ C, then B def

(∀C ∈ B) Ca = Ø,

and Ca−1 Ca = GJ \EB ∩ M(C) ⊇ GJ \(EB ∪EC ) ∈ M(G).

In particular, since fEB |GEB is (up to natural identification) the identity of GEB , (a) ∩ B = {a} × GJ \EB . If C ∈ B, then Ba ∩ C = fE−1 (a) ∩ B ∩ C. we have Ba = fE−1 B B Since B is a filterbasis, there is a D ∈ B such that D ⊆ B ∩ C. Then Ø = Da = fE−1 (a) ∩ D ⊆ fE−1 (a) ∩ B ∩ C = Ba ∩ C. Thus {Ba } ∪ B is contained in a filter basis B B  B of cosets modulo closed normal subgroups of G which are members of M(G). By the maximality of B we conclude B  = B. This means B ⊇ Ba ∈ B, and if we set

Sufficient Conditions for Quotients to be Complete

201

hB = a and B = Ba , then this is the assertion (‡). For a proof of the uniqueness of hB , assume that B contains also {hB } × GJ \EB ∈ B. Since B is a filterbasis, {hB } × GJ \EB ∩ {hB } × GJ \EB = Ø. This entails hB = hB and thus secures uniqueness. From (∗) we obtain for each pair C ⊆ B in B the comparison B = g(B)EB M(B)EB × GEC \EB × GJ \EC , × GJ \EC , C = g(C)EC M(C)EC

(∗∗) (∗∗∗)

Accordingly, we have the comparison B = {hB } × GEC \EB × GJ \EC , × GJ \EC , C = {hC }

(++) (+++)

From (++) and (+++) we have (∀B, C ∈ B) B ⊇ C ⇒ fEB EC (hC ) = hB .

(#)

Next we show that the updirected family {EB : B ∈ B} is cofinal in Fin(J ), that is, for each E ∈ Fin(J ) there is a B ∈ B such that E ⊆ EB . Since E is finite and {EB : B ∈ B} is up-directed, this is equivalent to saying that each j ∈ J is contained  in some EB , that is B∈B EB = J . By way of contradiction, suppose that this is not the case. Then there is a j ∈ J such that j ∈ / EB for all B. Then (∗) takes the following form: B = gEB MEB × Gj × GJ \(EB ∪{j }) . Then the filter basis of all sets B  = gEB MEB × {1j } × GJ \(EB ∪{j }) can be enlarged to contain B, and that contradicts the maximality of B. Thus {EB : B ∈ B} is cofinal in Fin(J ) as asserted.  Then condition (#) implies that there is a family  h = (hj )j ∈J ∈ G = j ∈J Gj such that hEB = (h j )j ∈EB ∈ G EB and thus h ∈ B∈B B. Since B ⊇ B for all B ∈ B we conclude h ∈ B. Thus B = Ø which is what we had to show to complete Observation (iii) and thereby Step c. In Step (c) we have shown that arbitrary products of families of almost connected pro-Lie groups are Z-compact. In Step (b) we have shown that if this is the case, then for any almost connected pro-Lie group G, the arbitrary strict projective limits G∼ = GM(G) = limN ∈M(G) G/M is Z-compact. This completes the proof of Claim (iv) of the proposition. The proof of assertion (iv) is delicate in so far as we have to keep clearly in focus that we want to prove compactness for the Z-topology. While it is true that an arbitrary

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product of almost  connected pro-Lie groups Gj is compact in the Tychonoff product topology of  j (Gj , Z(Gj )) this contributes nothing to the fact, proven in (iv), that  ( j Gj , Z( j Gj )) is compact, as this topology of this space in general is finer than the product topology. Exercise E4.2. Verify the details of the following examples. (i) On the groups G = Z and G = R the Z-topology is the cofinite topology, that is the topology whose closed sets are all finite sets and the set G. But Z fails to be compact in the topology having as a subbasis for its closed sets the collection {gN : g ∈ G, N ∈ N (Z)}. $ # [Hint. First observe that M(G) = {0}, Z . Regarding the second assertion, the filter basis {1 + p + p2 + · · · + p n−1 + pn Z | n ∈ N} consists of cosets of groups in N (Z) and has empty intersection in Z.] (ii) Let Rc be R with the cofinite topology. The product topology on R2c has a basis of open sets of the form R2 \ L where L is a finite union of straight lines each of which is either horizontal or vertical. Thus the diagonal of R2 is dense in R2c . It is, however, closed in R2 for the Z-topology of R2 . The diagonal is the graph of the identity morphism R → R. Thus graphs of morphisms continuous in the ordinary topology are not closed in the product space for the Z-topologies on the range and the domain in general. With this preparation we can prove the first part of the following theorem giving one set of sufficient conditions for a quotient group of a pro-Lie group to be a pro-Lie group. We complement this result by alternative sufficient conditions which are easily obtained by citing relevant literature, and by one condition whose sufficiency we shall establish much later in the book in Chapter 9. The Quotient Theorem for Pro-Lie Groups Revisited Theorem 4.28. Assume that G is a pro-Lie group and K is a closed normal subgroup. Then G/K is a pro-Lie group if at least one of the following conditions is satisfied: (i) (ii) (iii) (iv)

K is almost connected and G/G0 is complete. K satisfies the First Axiom of Countability. K is locally compact. G is almost connected and K is the kernel of a morphism whose image is a pro-Lie group.

Proof. (i) We accomplish the proof in two parts. In the first part we assume that G/G0 def

is compact, that is, that G is also almost connected. By Theorem 4.1, H = G/K is a proto-Lie group. Let f : G → H be the quotient morphism and let C be a Cauchy filter of H . We have to show that C converges. Since G is almost connected, M(G) is cofinal in N (G) by 4.6. For each N ∈ M(G) let N ∗ = f (N) and let pN ∗ : H → H /N ∗ be the quotient morphism. Then the image pN ∗ (C) in the Lie group H /N ∗ is a Cauchy filter and thus has a limit hN . In

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fact, (hN )N∈M(G) ∈ H{N ∗ |N ∈M(G)} = limN ∈M(G) H /N ∗ ∼ = HN (H ) ; indeed C has to def

converge to a point in the completion of H . Now let FN = (pN ∗ f )−1 (hN ). Then {FN | N ∈ N (G)} is a filter basis consisting of cosets of closed normal subgroups KN of G where K = ker f . We claim that KN ∈ M(G). Now KN ∈ N (G). Let M ⊆ KN in N (G) such that M ∼ KN, that is, M is open in KN. Then M ∩ K is open in K. Since K is almost connected, KM/M ∼ = K/(M ∩ K) is finite. We must show that KN/M is finite; for this purpose it is now no loss of generality, after replacing M by KM if necessary, to assume that K ⊆ M. Then M is open in MN, whence MN is closed and thus KN = MN . Now MN/M ∼ = N/(M ∩ N ) is discrete, that is M ∩ N is open in N whence M ∩ N ∼ N . Now N ∈ M(G) implies that KN /M = MN/M ∼ Since = N/(M ∩ N) is finite. Thus KN ∈ M(G) as claimed.  G is compact in the Z-topology by Proposition 4.8 (iv), there is a g ∈ N ∈N (G) FN . Then pN ∗ (f (g)) = hN for all N ∈ N (G) from which we deduce that f (g) = lim C. Thus every Cauchy filter in H converges and so H is complete. By (i), H is a proto-Lie group, and thus by the definition of pro-Lie groups (3.25) we have shown that H is a pro-Lie group. In the second part of the proof we assume that G/G0 is complete. As a quotient of K/K0 , the factor group K/(K ∩ G0 ) ∼ = (K/K0 )/((K ∩ G0 )/K0 ) is compact. Moreover, in the factor group G/(K ∩ G0 ), the subgroup G0 /(K ∩ G0 ) is closed. Hence the product (G0 K/(K ∩ G0 )) · (K/(K ∩ G0 )) = G0 K/(K ∩ G0 ) is closed in G/(K ∩ G0 ) and therefore G0 K is closed in G and so is a pro-Lie group by the Closed Subgroup Theorem 3.35. The factor group G0 K/G0 is a continuous homomorphic image of the compact group K/(K ∩ G0 ) under the morphism k(K ∩ G0 )  → kG0 and consequently is a compact group. Therefore G0 K is an almost connected proLie group. Then by Part 1 of the proof, G0 K/K is a pro-Lie group. Further G/G0 is complete by assumption. Since K/(K ∩ G0 ) is compact, its homomorphic image G0 K/K is compact. Hence G/G0 K ∼ = (G/G0 )/(G0 K/G0 ) is complete by (iii) below. Now the factor group G/K has the complete normal subgroup G0 K/K such that (G/K)/(G0 K/K) ∼ = G/G0 K is complete, and it is therefore complete by [176, p. 225, 12.3]. Thus the proof of (i) is complete. (ii) and (iii): In view of [176, p. 242, Lemma 13.13], both of these are special cases of the sufficient condition given in [176, p. 206, Theorem 11.18]. (iv) will be established later in Chapter 9 in Lemma 9.57. Of course, if G itself satisfies the First Axiom of Countability, then (ii) follows. But there are simpler alternative arguments in this case. Recall that a topological group is metrizable if and only if it satisfies the First Axiom of Countability. (See for instance [102, Theorem A4.16].) Quotients of complete metrizable topological groups are complete because Cauchy sequences can be lifted. (See [26, Chap. 9, §3, no 1, Proposition 4].) The assertion that G/N is a pro-Lie group then follows from 4.1 (i). Naturally, pro-Lie groups often fail to be metrizable. Dierolf and Roelcke explain in [176] that there is a common cover for the two conditions of metrizability and local compactness; metrizable and locally compact spaces are almost metrizable, and it is true that for a complete topological group G and

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an almost metrizable closed normal subgroup N the factor group G/N is complete. For the concept of almost metrizability we refer to [176, p. 241, Definition 13.11]. We mention that it follows from the work of Wigner [201] that there is yet another sufficient condition for the conclusion of Theorem 4.28 to hold: (ii∗ ) K is isomorphic to an arbitrary product of groups each of which satisfies the First Axiom of Countability. This condition trivially implies (ii) and is implied by (ii∗∗ ) K is isomorphic to an arbitrary product of Lie groups. We shall see in Chapter 8 that a simply connected reductive Lie group is a product of simply connected simple Lie groups and copies of R (see Theorem 8.14 and Corollary 8.15). (ii∗∗ ) is implied by (ii∗∗∗ ) K is a simply connected reductive pro-Lie group. As we already noted above in comments subsequent to the example in Corollary 4.11, in general, the quotient of a complete topological group is not necessarily complete; in the case of additive groups of topological vector spaces this was observed at an early stage ([132], [75]). Grosso modo, the completeness of quotient groups of a complete topological group G is a delicate matter, and this is why, in order to find sufficient conditions appropriate for the category proLieGr, somewhat protracted arguments were needed in the proof that Condition (i) of Theorem 4.28 is sufficient for completeness. The Axiom of Choice is buried in the proof of Z-compactness, e.g. in the Alexander Subbasis Theorem. The arguments we cited from the book of Dierolf and Roelcke are all but trivial. Only the case that G is assumed to be metrizable in 4.28 (ii) is simpler, but this is also the most restrictive condition. The following question is in a sense a converse to that which is answered by the Closed Subgroup Theorem 1.34 and the Quotient Theorem 4.1: Let G be a topological group and N a closed normal subgroup of G such that both N and G/N are pro-Lie groups. Under what circumstances is G a pro-Lie group? Among the following three examples, the first two illustrate that this fails to be the case under the rather simple circumstances of semidirect products. The third example however, showing that every pro-Lie group has a tangent bundle, illustrates that often semidirect products do work well also for pro-Lie groups. Examples 4.29. (i) Let L be a nontrivial compact Lie group and let Z act automordef

phically on P = LZ by the shift. Set G = P  Z. Then N = P × {0} is a compact normal pro-Lie subgroup, and G/N is discrete, and thus a Lie group. But G is not a pro-Lie group.

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(ii) Let D be a discrete group, and A a compact nontrivial group of automorphisms of D. Then the compact group AN acts automorphically on the discrete group V = D (N) via (an )n∈N · (dn )n∈N = (an dn )n∈N . Set G = V  AN . Then N = V × {1} is a discrete normal subgroup, thus in particular a normal Lie subgroup. The factor group G/N ∼ = AN is compact and therefore a pro-Lie group. But G is not a pro-Lie group. def (iii) For each pro-Lie group G, the semidirect product T (G) = L(G) Ad G is a well-defined pro-Lie group. Proof. Exercise E4.3. Exercise E4.3. Prove the assertions made in the description of the Examples 4.29 (i) and (ii). [Hint. (i) G has arbitrarily small identity neighborhoods of the form, U = LN\F × {1} for some finite set F ⊆ Z (where we identify LA in an obvious fashion with a subgroup of ZN for any subset A of Z). Assume that F = Ø and x = (xn )n∈Z ∈ LN\F are such that xn = 1. If xn−m ∈ F then a shift of (x, 1) is outside U . Hence U does not contain any nondegenerate normal subgroups of G. (ii) G has arbitrarily small identity neighborhoods U = {1} × AN\F with a finite subset F of N. If a = (an )n∈N ∈ AN with an = 1, then there is a d ∈ D such that / {1} × A, and thus if an (d) = d. Then (d, 1)(1, an )(d −1 , 1) = (d · an (d)−1 , an ) ∈ v = (dj )j ∈N with dj = 1 for j = n, then (v, 1)(1, a)(v −1 , 1) ∈ / {1} × AN . So U does not contain any nondegenerate normal subgroup. (iii) Any identity neighborhood of T (G) contains one of the form U ×V where U is a zero neighborhood of L(G) and V is an identity neighborhood of G. In 4.21 (ii) we see that there is a closed ideal j of L(G) such that j ⊆ U and dim L(G)/j < ∞. Likewise there is a normal subgroup of G such that N ⊆ V and G/N is a Lie group. Then j×N is a normal subgroup of T (G) such that j×N ⊆ U ×V and T (G)/(j×N ) ∼ = L(G)/jG/N is a Lie group. Thus T (G) is a pro-Lie group, as asserted.] In the proof of (iii), the reader will notice very quickly that j and N may be chosen in such a way that j = L(N ) so that in fact, setting M = L(N ) × N, by what we shall see in Corollary 4.22, T (G)/M ∼ = L(G/N ) AdG/N G/N . Now T (G/N ) = L(G/N) AdG/N G/N is the tangent bundle of the Lie group G/N . Thus T (G) is approximated by the tangent bundles of the Lie group quotients G/N of G. We record some basic facts on the natural relationship between pro-Lie groups and proto-Lie groups. Exercise E4.4. The category of proto-Lie groups and morphisms of topological groups between them is denoted by protoLie. For a topological group G we write L(G) = GN (G) = limN ∈N (G) G/N. Recall that we have a morphism γG : G → L(G) (see 1.29, 1.39) Prove the following facts: (i) For any topological group G, the topological group L(G) is a pro-Lie group. (ii) γG is an embedding if and only if G is a proto-Lie group and an isomorphism if and only if G is a pro-Lie group.

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(iii) For each morphism f : G → H of proto-Lie groups into a pro-Lie group H there is a unique morphism f  : L(G) → H such that f = f   γG . protoLie γG

proLieGr

G ⏐ ⏐ ∀f 

−−→

L(G) ⏐ ⏐  f

L(G) ⏐ ⏐  ∃!f

H

−−→

H

H.

idH

(iv) The assignment G  → L(G) on proto-Lie groups extends to a functor L : protoLie → proLieGr which is left adjoint to the inclusion functor proLieGr → protoLie. (See e.g. [102, Appendix 2, Theorem A3.28ff.].) [Hint. (i) We have L(G) = limN ∈N (G) G/N where G/N is a Lie group for each N by the definition of N (G). In particular, L(G) is a complete topological group. The limit maps νN : L(G) → G/N are quotient morphisms inducing isomorphisms  | N ∈ N (G)}  → G/N (with N  = ker νN ) by 1.29 (iii). Then by 1.27 (ii) {N L(G)/N  is is a filter basis of closed normal subgroups of L(G) converging to 1 and L(G)/N a Lie group for all N ∈ N (G). Now Proposition 3.27 shows that L(G) is a pro-Lie group. (ii) This follows from 3.26. (iii) Assume now that f : G → H is a morphism of proto-Lie groups. Since H is a pro-Lie group, it is complete. Since G is a proto-Lie group, it has a completion by 3.26 and γG : G → L(G) is the completion morphism. Hence there is a unique extension of f to a morphism f  : L(G) → H . (iv) This is now a consequence of (iii) and Theorem A3.28 of [102]. On a larger scale, the inclusion functor from the category pro-Lie groups proLieGr into the category TopGr of topological groups has a left adjoint L : TopGr → proLieGr given by the construction of Exercise E4.5.; in Chapter 2 this functor was derived from more category theoretical principles in 2.25 (iii). There is a full subcategory of the category protoLie which arises naturally from the fact that a quotient group of a pro-Lie group is a proto-Lie group. The following exercise comments on this category. Exercise E4.5. Let Q denote the full subcategory of TopGr of all topological groups which are quotient groups of pro-Lie groups. Prove the following facts about this category: (i) Q is a (full) subcategory of protoLie. (ii) Q is closed in TopGr under the formation of products and retracts. (iii) Q is closed under the formation of closed subgroups. (iv) Q is closed under the formation of quotient groups. (v) The restriction of the functor P : protoLie → proLieGr to Q is left adjoint to the inclusion functor.

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[Hint. (i) is a consequence of the Quotient Theorem for Pro-Lie Groups 4.1 (ii): Show that a product of quotient morphisms is a quotient morphism and that the composition of quotient morphisms is a quotient morphism; use these facts to show that Q is closed in TopGr under the formation of products and retracts. (iii) If H /N is a closed subgroup of G/N, then H is a closed subgroup of G and if G is a pro-Lie group, then H is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Assertions (iv) and (v) are straightforward.] Exercise E4.6. Recall that we say that a topological group has no small normal subgroups if there is an identity neighborhood not containing any subgroups except the singleton one. Let G be a pro-Lie group, N a closed normal subgroup, and qN : G → G/N the associated quotient morphism. Then we have, on the level of sets, an injective function Hom(qN , H ) : Hom(G/N, H ) → Hom(G, H ), Hom(qN , H )(f ) = f  qN . If G is a pro-Lie group and N ⊆ M in N (G), giving an associated quotient morphism qMN : G/N → G/M, then we have an associated injective function Hom(qMN , H ) : Hom(G/M, H ) → Hom(G/N, H ),

M ⊆ N, M, N ∈ N (G).

This is an injective system of sets and functions, and we obtain an injective limit colimN∈N (G) Hom(G/N, H ) for which the maps Hom(qN , H ) : Hom(G/N, H ) → Hom(G, H ) provide an injective function iN (G) : colim Hom(G/N, H ) → Hom(G, H ). N ∈N (G)

We shall identify colimN ∈N (G) Hom(G/N, H ) with a subset of Hom(G, H ), which means identifying a morphism f : G/N → H with its canonical composition f qN ∈ Hom(G, H ). Prove the following statement: Proposition A. A proto-Lie group is a Lie group if and only if it has no small subgroups. [Hint. Let U be an identity neighborhood of G which contains no nondegenerate subgroups. Since G is a proto-Lie group, there is an N ∈ N (G) that is contained in U and thus satisfies N = {1}.] Proposition B. If G is a pro-Lie group, and H is a group without small subgroups, then colim Hom(G/N, H ) = Hom(G, H ). N ∈N (G)

[Hint. Let f : G → H be a member of Hom(G, H ) and let U be an identity neighborhood of G which does not contain a nondegenerate subgroup of H . Then f −1 (U ) is an identity neighborhood of G which, according to lim N (G) = 1 contains some N ∈ N (G). Then f (N) is a subgroup of H contained in U and thus is {1}, and thus f factors in the form f = f   qN . Therefore f ∈ Hom(G/N ) after our identification of f and f  .]

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Quotients and Quotient Maps between Pro-Lie Groups In Theorem 4.1 (iv) we proved the next best thing to an Open Mapping Theorem without having made a special definition. Let us do this here. Definition 4.30. (i) A morphism f : G → H of topological groups is called relatively open if the set {f (U ) | U is an identity neighborhood of G} is a basis for the filter of identity neighborhoods of H . (ii) A morphism f : G → H from a topological group to a pro-Lie group will be called pro-open, if for each M ∈ N (H ) the function g  → f (g)M : G → H /M is open. An open morphism is a relatively open morphism, and if the range is a pro-Lie group, it is pro-open. Let G be the additive group of real numbers endowed with the following topology: consider on Q the topology induced by the natural topology of R and let the open identity neighborhoods of Q form a basis for the identity neighborhoods of R. This is a metrizable group topology on R. If H = R with its natural topology, and if f : G → H is the identity map then f is a surjective morphism of topological groups which is relatively open but fails to be open. If H = Tp is the p-adic solenoid p

p

p

lim{T ←− T ←− T ←− · · · } then H is a pro-Lie group. Let Zp be the subgroup . / 1 p 1 p 1 p lim Z/Z ←− 2 Z/Z ←− 3 Z/Z ←− · · · . p p p This group and the group Zp of p-adic integers are isomorphic. Let D be the group Zp with its discrete topology and set G = D × R. Then G is an abelian finite-dimensional Lie group (hence a pro-Lie group) with uncountably many components. Define f : G → H by f ((zn + Z)n∈N , r) = (zn + p −n r + Z)n∈N . Then f is a surjective morphism of topological groups whose kernel is isomorphic to Z (see [102, Example E1.11 following Definition 1.30]). Let πn : Tp → T be the n-th projection. Then 3 4 2 ε ε (πn f )({0}×] − ε, ε[) = − n , n + Z Z ⊆ T, p p and thus f is pro-open. But it fails to be relatively open, let alone open. In a remark following the proof of Theorem 4.1 we mentioned a nondiscrete proLie group topology τ on the countable group Z(N) making the identity map from the discrete group Z(N) to (Z(N) , τ ) a bijective morphism between two countable pro-Lie groups that fails to be open.

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Lemma 4.31. (The Pro-Open Mapping Theorem for Pro-Lie Groups) (i) Let G be a pro-Lie group with arbitrarily small elements N ∈ N (G) such that the Lie group G/N has countably many components. This is the case if G has only countably many components. If f : G → H is a surjective morphism of topological groups onto a pro-Lie group H , then f is pro-open. (ii) If f : G → H is a pro-open morphism of pro-Lie groups such that f (N ) ∈ N (H ) for all N ∈ N (), then f is relatively open. (iii) If, in addition to the hypotheses of (i) above, f is locally closed (in the sense that for any identity neighborhood W of G there is an identity neighborhood U contained in W such that f (U ) is closed), then f is open. Proof. Exercise E4.7. Exercise E4.7. Verify Lemma 4.31. [Hint. (i) This was in effect proved in Theorem 4.1 (iv). (ii) We have to show that for arbitrarily small identity neighborhoods U of G the closure of the image f (U ) is an identity neighborhood of H . Since G is a pro-Lie group, by Theorem 1.27 (i), we may assume that we are given an open identity neighborhood U of G for which there is an N ∈ N (G) such that U N = N U = U . Then by (i), f (U )N ∗ /N ∗ is an identity neighborhood of H /N ∗ and thus f (U )N ∗ = f (U )f (N ) ⊆ f (U )f (N) = f (U N) = f (U ) is an identity neighborhood of H . (iii) is an immediate consequence of (i) and (ii).] This is all we shall say here in the line of the Open Mapping Theorem. Notice that a morphism is locally closed if it is a closed map; this is the case for instance if it is a proper map. Definition 4.32. A topological group P is called procountable, if the filter basis of all open subgroups N such that P /N is countable converges to the identity.  An arbitrary product P = j ∈J Dj of any family of countable discrete groups Dj is procountable. Indeed, a complete topological group is procountable iff it is isomorphic to a closed subgroup of a product of a family of countable discrete groups. Lemma 4.33. Let P be a procountable group and f : P → H a morphism of topological groups into a connected pro-Lie group G. Then f (P ) is totally disconnected. def

Proof. Let A = f (P )0 . We have to show that A is singleton. From the Closed Subgroup Theorem 3.35 we know that A is a connected abelian pro-Lie group and that f −1 (A) is a procountable subgroup of P . For the purposes of the proof we may assume that P = f −1 (A), that is, that A = f (P ). Suppose now that A is nonsingleton. Then we have a quotient morphism q : A → L where L is a nonsingleton connected Lie def

group. Let j : A → A be the inclusion morphism. Then F = q  j  f : P → L is morphism into a Lie group with a dense connected image F (P ). Now let U be a zero neighborhood in L such that {0} is the only subgroup contained in U . Since P is procountable, there is an open normal subgroup N contained in F −1 (U ) such that

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P /N is countable. Then F (P ) = q(A) is a homomorphic image of P /N and is, therefore, a dense countable connected subgroup of L. But any countable subspace of a connected metrizable space is totally disconnected (Exercise E4.9 below). Hence q(A) is singleton. So L is singleton, contrary to what we have supposed. Exercise E4.8. A countable subset of a metrizable space is totally disconnected. [Hint. Let (X, d) be a metric space and C a countable subset. We may assume that X has at least two different points. Let c ∈ C and define f : X → R+ = [0, ∞[ by f (x) = d(c, x). Then f is continuous, f (X) ⊆ R+ is connected and is therefore a nondegenerate interval Ic containing 0 = f (c). The set f (C) is countable; hence Ic \f (C) is uncountable and clusters toward 0. There is a sequence (rn )n∈N in Ic \f (C) such that rn < n1 . Then the open balls of radius rn in C form a basis of open-closed neighborhoods of c in C.] We note that, more generally, a connected space on which the continuous real valued functions separate the points, has to have a cardinality that is at least that of the continuum. There are countable connected Hausdorff spaces. (See for instance [52, pp. 352, 353].)

Postscript There are two substantial results in this chapter. The first is that the functor L preserves quotient morphisms. More precisely, let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Let Y ∈ L(H ) be a one parameter group of H . Then there is a one parameter group X ∈ L(G) such that f  X = L(f )(X) = Y , that is, the morphism L(f ) : L(G) → L(H ) induced on the Lie algebra level is surjective. In particular, if f is a quotient morphism between pro-Lie groups then L(f ) is a quotient morphism of pro-Lie algebras. Accordingly, the functor L : proLieGr → proLieAlg not only preserves all limits, but it also preserves quotient morphisms. This is a nontrivial fact which, not surprisingly, has substantial consequences. Its proof requires the Axiom of Choice. It is important to ask whether a quotient of a pro-Lie group is a pro-Lie group. Unfortunately, the answer is in the negative, because quotient groups of complete topological groups are not necessarily complete and may not even have a completion. Indeed Sipacheva [184] showed that every topological group is a quotient group of a complete topological group. However, the quotient of a pro-Lie group is always a proto-Lie group, has a completion, and this completion is a pro-Lie group. Moreover, if f : G → H is a morphism of pro-Lie groups, and if q : G → Q is the natural dense morphism from G to the pro-Lie group completion of G/ ker f , then there is a unique morphism of pro-Lie groups f  : Q → H such that f = f  q. This indeed looks like a good universal property for a quotient object Q even though q is not surjective as we expect a quotient morphism to be. Unfortunately, f  may not be injective either. Perhaps the failure of such a category as the category of pro-Lie groups to behave neatly

Postscript

211

with respect to passing to quotient objects makes it plausible that there may be no easy category theoretical approach to quotients on the level of objects and arrows. Surely the simplest connected pro-Lie groups are the products of reals RX . The quotients of these are complete if X is countable, and we show in this chapter that RX has incomplete quotients as soon as card X ≥ 2ℵ0 . Surprisingly, this all happens in a context which we think we are familiar with, to wit, the context of compact connected abelian groups and their exponential function. So, in contrast with quotients of locally compact groups which are always locally compact, and in contrast with quotients of classical Lie groups which are always Lie groups, quotients of pro-Lie groups often fail to be pro-Lie groups – and this happens already for the additive groups of weakly complete topological vector spaces. In such circumstances it is important to have sufficient conditions under which quotients of pro-Lie groups are complete. Classically, first countable complete topological groups have complete quotients. We demonstrate a result which pertains to connected and indeed more generally to almost connected pro-Lie groups. Here we have used the terminology that a topological group G is almost connected if the factor group G/G0 modulo the identity component is compact. Indeed we show that the quotient group G/N of a pro-Lie group G modulo an almost connected closed normal subgroup N is a pro-Lie group again provided that the protodiscrete factor group G/G0 is complete. (see Lemma 3.31). The proof of this fact is not superficial. It uses the Axiom of Choice as well. We do not know an example of a pro-Lie group G such that G/G0 fails to be complete. The literature on topological groups such as [176] provides us with the information that a factor group G/N of a pro-Lie group G is a pro-Lie group whenever N is locally compact. In Chapter 9 in Theorem 9.60 we shall prove a fairly good Open Mapping Theorem for pro-Lie groups which applies whenever the domain is almost connected. We mentioned this in Theorem 4.28. In the meantime we include a weaker result showing that a surjective morphism from a pro-Lie group G with countably many components onto a pro-Lie group is “pro-open” where the pro-openness of a morphism between pro-Lie groups is a substitute for openness.

Chapter 5

Abelian Pro-Lie Groups

We saw in [102] that compact abelian groups are not simply an informative, important, and interesting special case of more general compact groups, but rather an integral part of the structure of these groups. Even at the superficial level we know that the closure of any cyclic group or of any one parameter subgroup in a locally compact group is either isomorphic to Z, respectively, to R, or else is a compact abelian subgroup. A similar situation persists in the case of pro-Lie groups. We know that in any pro-Lie group the closure of any cyclic subgroup or any one parameter subgroup is an abelian pro-Lie group. Further every locally compact abelian group is an abelian pro-Lie group. A topological group is called a weakly complete vector group if it is isomorphic to the additive group of a weakly complete topological vector space, and therefore is isomorphic to RJ for some set J . (See Corollary A2.9 in Appendix 2.) We shall see that the structure of connected abelian pro-Lie groups reduces completely to that of weakly complete vector groups and compact connected abelian groups for which we have a rather exhaustive structure and Lie theory (as shown, for instance, in [102, Chapters 7 and 8]). As the structure theory of pro-Lie groups unfolds we shall have to recall results which we prove in this chapter. There are certain aspects of the duality theory of locally compact abelian groups which extend to abelian pro-Lie groups although this larger class does not have a perfect duality theory outside the domain of connected groups. We shall discuss duality theory as well. Prerequisites. We need the general theory of pro-Lie groups up to and including Chapter 4 and elements of the duality theory of topological abelian groups such as it is, for instance, presented in [102, Chapter 7]. In particular, the theory of weakly complete topological vector spaces and their duality theory is assumed to be known (see Appendix 2).

Examples of Abelian Pro-Lie Groups We begin by offering some orientation of the class of abelian pro-Lie groups by presenting a list of examples. Let us firstly recall (see for instance [199]) the following basic types of locally compact nondiscrete fields: (a) The field R of real numbers. (b) The field Qp of p-adic rationals for some prime p. (c) The field GF(p)[[X]] of Laurent series in one variable with the exponent valuation over the field with p elements.

Examples of Abelian Pro-Lie Groups

213

All other nondiscrete locally compact fields are finite extensions of these; in cases (a) and (b) the characteristic is 0 and in case (c) it is finite. Of course, every field F with the discrete topology is a locally compact field.  ∞ 1  Let Z(p∞ ) = n=1 p n · Z /Z denote the Prüfer group of all elements of p-power order in T = R/Z. We consider on the group Z of integers, the group Q of rationals, and the Prüfer group Z(p∞ ) as endowed with their discrete topologies. Examples 5.1. Let J be an arbitrary infinite set. The following examples are abelian pro-Lie groups. (i) All locally compact abelian groups. (ii) All products of locally compact abelian groups, specifically: (a) the groups RJ ; (b) the groups (Qp )J ; (c) the groups QJ for the additive group of rational numbers Q with its discrete topology; (d) the groups ZJ for the group Z of integers with its discrete topology; (e) the groups Z(p∞ )J . An infinite product of noncompact locally compact groups is not locally compact, so, for infinite J , none of the groups in (ii)(a)–(ii)(e) is locally compact, but if J is countable, they are Polish (that is, completely metrizable and second countable). A countable product of discrete infinite countable sets in the product topology is homeomorphic to the space R \ Q of the irrational numbers in the topology induced by R (see [26, Chap. IX, §6, Exercise 7]). So QN , ZN , and Z(p∞ )N are abelian prodiscrete groups on the Polish space of irrational numbers. These elementary examples show, that the category of abelian pro-Lie groups is considerably larger than that of locally compact groups. The groups in (ii)(a)–(ii)(c) are divisible and torsion-free, and the groups in (ii)(e) are divisible and have a dense torsion group. Recall from the Closed Subgroup Theorem for Pro-Lie Groups 3.35 that every closed subgroup of a pro-Lie group is a pro-Lie group, and from the Completeness Theorem for Pro-Lie Groups 3.36 that the product of any family of pro-Lie groups is a pro-Lie group. It is then clear that from even simple examples a wealth of examples can be easily constructed. There is a less obvious but very instructive example which we present separately. This example is based on the example which we discussed in Chapter 4 right after the Quotient Theorem for Pro-Lie Groups 4.1: see Proposition 4.2 and the discussion leading up to this proposition and all the information presented subsequently through the Characterisation Theorem for Strong Local Connectivity of Compact Connected Abelian Groups 4.14. This discourse is due to Poguntke and the authors as described in [106]. However, for the present example we need only what we shall specify explicitly here.

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5 Abelian Pro-Lie Groups

If J is any set and j ∈ J , then δj : J → R is defined by " 1 if x = j , δj (x) = 0 otherwise. For any subset S ⊆ RJ let S denote the subgroup algebraically generated by S as is usual. Proposition 5.2. The free abelian group Z(N) has a nondiscrete topology making it into a prodiscrete abelian group F in such a fashion that the following conditions are satisfied: (i) There is an injection j : F → W mapping F isomorphically (algebraically and topologically) onto a closed subgroup of the weakly complete vector group W of topological dimension 2ℵ0 (that is, W ∼ = RR ) such that W/F is an incomplete group whose completion is a compact connected and locally connected group, and that the R linear span span F is dense in W . def

(ii) The subset B = {δn : n ∈ N} satisfies F = B and j (B) is unbounded in W . (iii) If K ⊆ F is any compact subset, then there is a finite subset M ⊆ N such that k ∈ K ⊆ Z(N) implies that the support supp(k) = {m ∈ N : k(m) = 0} is contained in M. In particular, every compact subset of F is contained in a finite rank subgroup of F . Proof. (i) Let G be the character group of the discrete group ZN . Then G is a compact, connected and locally connected but not arcwise connected group. Let Ga denote the arc component of the identity element in G. It was proved in [106] that the corestriction of the exponential function expG : L(G) → Ga is a quotient map. Let W = L(G) ∼ = ℵ0 Hom(ZN , R) ∼ = R2 and F = ker expG ⊆ L(G) = W . The exponential function of (locally) compact abelian groups is extensively discussed in [102, Proposition 7.36ff.]; in [102, Theorem 7.66], the kernel of the exponential function is denoted by K(G). Now F as a closed subgroup of a pro-Lie group is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups. It is totally disconnected (see [102, Theorem 7.66 (ii)]), and so by Lemma 3.31 F is a prodiscrete group. By [102, Proposition 7.35 (v)(d)], the linear span span F = span K(G) is dense in L(G) = W . Since Ga does not contain any copy of a cofinite-dimensional closed vector subspace of L(G), the function expG : L(G) → Ga cannot induce a local isomorphism and thus its kernel F = K(G) is not discrete. Furthermore, from [102, Theorem 7.66 (ii)] we observe that F ∼ = Hom(ZN , Z), algebraically, and from [49, p. 53 Corollary 15 and p.560, Corollary 24], we get that  : Z(N) → Hom(ZN , Z), ((pm )m∈N )((zm )m∈N ) = m∈N pm zm is an isomorphism of abelian groups and that, accordingly, F is a free group algebraically generated by (B).

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215

(ii) We now prove that (B) ⊆ Hom(ZN , Z) ⊆ Hom(ZN , R) = W is unbounded. Note that (δn ) : ZN → Z is simply the evaluation evn : ZN → Z, evn (f ) = f (n). Since ZN is considered with the discrete topology, the topology on W ∼ = Hom(ZN , Z) N is that of pointwise convergence, that is, the topology induced from ZZ . Let s : N → Z be an arbitrary element of ZN . Then pr s : Hom(ZN , Z) → Z is given by pr s (ϕ) = ϕ(s) and thus pr s ((B)) = {evn (s) : n ∈ N} = {s(n) : n ∈ N} = im s. Therefore the projection of B into the s-component of Hom(ZN , Z) ⊆ Hom(ZN , R) ∼ = W is bounded if and only if s is bounded. Since there are unbounded elements s ∈ ZN , the set j (B) is unbounded in W . (iii) Let K ⊆ Z(N) be a compact subset and suppose that it fails to satisfy the claim. Then there is a sequence of elements kn ∈ K such that Mn = max supp(kn ) is a strictly increasing sequence. Now we define a function s : N → Z recursively as follows: Set s(1) = 0 and s(m) = 0 for m ∈ / {Mn : n ∈ N }. Assume that s(m) has been defined 5 m for 1 ≤ m ≤ Mn in such a way that σm = M j =1 km (j )s(j ) ≥ m. Now solve the inequality def

n + 1 ≤ σn+1 =

Mn 6

kn+1 (j )s(j ) + kn+1 (Mn+1 )s(Mn+1 )

j =1

#5 $ ). The s-projection pr ((K)) = k(n)s(n) : k ∈ K ⊇ {σn = for s(M n+1 s n∈ N 5 k (m)s(m) : m ∈ J } contains arbitrarily large elements σ ≥ n and thus cannot n m∈N n be bounded, in contradiction to the compactness of K. We realize that (ii) is implied by (iii); we preferred to give separate proofs for better elucidation of this remarkable example. The group F cannot be metrizable, because as a complete abelian group its underlying space would be a Baire space and thus as a countable topological group would have to be discrete. It is therefore noteworthy that there are pro-Lie groups whose underlying space is not a Baire space. As a countable group, it is trivially σ -compact, that is the countable union of compact sets. We observe in conclusion of our brief discussion of examples of abelian pro-Lie groups that we have seen prodiscrete abelian groups on the space R \ Q of irrationals and on a countable nondiscrete space.

Weil’s Lemma In the domain of locally compact groups, Weil’s Lemma says: Let g be an element of a locally compact group and g the subgroup generated by it. Then one (and only one) of the two following cases occurs (i) n  → g n : Z → g is an isomorphism of topological groups. (ii) g is compact.

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Exercise E5.1. Prove Weil’s Lemma. [Hint. We prove minutely more: Let E = R or E = Z and consider a morphism X : E → G. There is no loss in assuming that G = X(E), that is, that G is abelian and X has a dense image and so X is an epimorphism of abelian topological groups. Let K = ker X. The closed subgroup K of E is either cyclic or R. Thus E/K is singleton, a circle, or finite, or isomorphic to E. Since X factors through the quotient map E → E/K with a morphism X : E/K → im X, E ⏐ ⏐ quot E/K

X

−−→ −−→

A  ⏐ ⏐incl im X,

X

E/K and thus im X is compact in the first two cases. In these cases G is compact and we are in Case (ii) since im X is dense in G. We may and shall assume henceforth that X is injective and thus is a monic of topological abelian groups. There are elementary proofs of Weil’s Lemma in the literature, see for instance [102, Proposition 7.43]. A proof using the duality of locally compact groups and a : G →E  is a monic and epic of locally bit of Lie theory is as follows. The dual X compact abelian groups and thus is injective and has dense image. Since R has no  has no nonsingleton compact subgroups and thus, nonsingleton compact subgroups, G in particular, has no small subgroups. Hence it is of the form Rp × Tq × D for a  is an injective torsion-free discrete group D. Then p + q ≤ 1 (considering that X p+q  linear map from R → R). Case p + q = 0: Then A is discrete and so A is  maps A 0 ∼ compact. Case p = 1: Here the injective map X = R isomorphically onto ∼  E = R, and D = 0 follows. Case q = 1: Similarly, A = E = T and thus A ∼ = Z.] Weil’s Lemma for Pro-Lie Groups Theorem 5.3. Let E be either R or Z and X : E → G a morphism of topological groups into a pro-Lie group. Then one (and only one) of the two following cases occurs: (i) r  → X(r) : E → X(E) is an isomorphism of topological groups. (ii) X(E) is compact. def

Proof. The closed subgroup A = X(E) is an abelian pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Thus we may assume that G is abelian and the image of X is dense. Now let N ∈ N (G) and pN : G → G/N the quotient map. Then G/N is an abelian Lie group for a morphism pN X : E → G/N with dense image. By Weil’s Lemma for locally compact groups, either pN  X is an isomorphism of topological groups or else G/N is compact. If M ⊇ N in N (G) and pN  X is an isomorphism, then pM  X and the bonding map G/M → G/N are isomorphisms as well. Thus there are two mutually exclusive cases:

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217

(A) (∀N ∈ N (G)) pN  X is an isomorphism of topological groups and all bonding maps G/M → G/N are isomorphisms. (B) There is a cofinal subset Nc (G) ⊆ N (G) such that G/N is compact. In Case (A), the limit G ∼ = limN ∈N (G) G/N is isomorphic to E. In Case (B) let N ∈ N (G) and let ↑N = {P ∈ N (G) : P ⊆ N }. Then for all P ∈ N (G) the quotient G/P is compact. By the Cofinality Lemma 1.21 (ii) we know G∼ = limP ∈↑N G/P . Since all G/P are compact for P ∈ ↑N , the group G is compact in this case, and the theorem is proved. Definition 5.4. (i) Let G be a topological group. Then comp(L)(G) denotes the set of all X ∈ L(G) such that expG R · X is compact. A one parameter subgroup X ∈ comp(L)(G) is called a relatively compact one parameter subgroup. Furthermore, comp(G) denotes the set {x ∈ G : x is compact}. An element x ∈ comp(G) is called a relatively compact element of G. (ii) A topological abelian group G is said to be elementwise compact if G = comp(G). It is said to be compact-free if comp(G) = {0}. In any topological abelian group G, the set comp(G) of relatively compact elements is a subgroup, since g, h ∈ comp(G) implies that gh is contained in the compact subgroup gh. A discrete abelian group is elementwise compact if and only if it is a torsion group; it is compact-free if and only if it is torsion-free. If f : G → H is a morphism of abelian pro-Lie groups, then f (comp(G)) ⊆ comp(f (G)) ⊆ comp(H ); accordingly f (comp(G)) ⊆ comp(H ). The quotient morphism f : Z → Z/2Z shows that in general f (comp(G)) = comp(H ) even for quotient morphisms. Theorem 5.5. Let G be an abelian pro-Lie group. Then (i) comp(G) is a closed subgroup of G and therefore an elementwise compact proLie group in its own right. (ii) comp(G) ∼ = limN ∈N (G) comp(G/N ). (iii) For N ∈ N (G), let CN ⊆ G be the closed subgroup of G containing N for  which CN /N = comp(G/N ). Then comp(G) = N ∈N (G) CN and comp(G) = limN∈N (G) CN /N (iv) The factor group G/ comp(G) is compact-free.  Proof. (i) Since G ∼ = limN ∈N (G) G/N  ⊆ N ∈N (G) G/N we may assume that G is a closed subgroup of a product P = j ∈J Lj of abelian Lie groups Lj . Any abelian Lie group is of the form L = Rp × Tq × D for a discrete group D (see for instance [102, Corollary 7.58 (iii)]). Then comp(L) = {0}× Tq ×tor D for the torsion subgroup tor D of D. Hence comp(L) is closed. Thus pr j (comp(G)) ⊆ pr j (comp(G)) ⊆ comp(Lj ) as comp(Lj ) is closed. Now let g = (gj )j ∈J ∈ comp(G) ⊆ P . Then gj ∈ comp(Lj ) def  and thus gj  is compact and g ∈ K = j ∈J gj . As a product of compact groups, K is compact and thus g ⊆ K is compact and thus g ∈ comp(G). Hence comp(G) is closed and thus a pro-Lie group by the Closed Subgroup Theorem 3.35.

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5 Abelian Pro-Lie Groups

(ii) The assignment G → comp(G) defines a functor from the category of abelian pro-Lie groups to itself. If G is a pointwise compact abelian pro-Lie group and H is any abelian pro-Lie group, then any morphism of topological groups f : G → H factors through comp(H ), that is, is of the form f = inclcomp(H ),H f  with a unique morphism f  : H → comp(H ). That is, comp is a right adjoint to the inclusion functor of the full subcategory of pointwise compact abelian pro-Lie groups in the category AbproLieGr of abelian pro-Lie groups. See for instance [102, Proposition A3.36]. Therefore it preserves limits (see for instance [102, Theorem A3.52]). Since comp preserves limits, it preserves, in particular, projective limits. (iii) follows from the Closed Subgroup Theorem for Projective Limits 1.34 (v). (iv) Let g comp(G) be nonzero in G/ comp(G) that is, g ∈ / comp(G). Then by / comp(G/N ) and then (ii) there is an N ∈ N (G) such that g ∈ / CN . Then gN ∈ (gN) comp(G/N ) is not relatively compact in (G/N )/ comp(G/N ). Since the quotient morphism G → G/N maps comp(G) into comp(G/N ), there is an induced quotient morphism F : G/ comp(G) → (G/N )/ comp(G/N ) given by F (g comp(G)) = (g/N) comp(G/N ). As this element is not relatively compact, the element g comp(G) cannot be relatively compact in G/ comp(G). Thus G/ comp(G) is compact-free. The examples 5.1 (ii)(b) and (e) are elementwise compact abelian pro-Lie groups which are not locally compact since J is infinite. The examples 5.1 (ii)(a), (c), and (d), and the example in Proposition 5.2 are compact-free abelian pro-Lie groups. In the first part of the following definition we recall when a topological group G with identity component G0 is called almost connected (see Definitions 1 and 4.24). Definition 5.6. Let G be a topological group. (i) G is said to be almost connected if G/G0 is compact. (ii) G is said to be compactly generated if there is a compact subset K of G such that G = K. (iii) G is said to be compactly topologically generated if there is a compact subset K of G such that G = K. Lemma 5.7. Let f : G → H be a morphism of topological groups and assume, firstly, that G is almost connected and, secondly, that f (G) is dense in H . Then H is almost connected. In particular, each quotient group of an almost connected group is almost connected. Proof. Since f (G0 ) is connected and contains the identity, f (G0 ) ⊆ H0 and therefore the morphism ϕ : G/G0 → H /H0 , ϕ(gG0 ) = f (g)H0 is well-defined. By assumption G/G0 is compact. So the continuous image ϕ(G/G0 ) is compact and thus, since H /H0 is Hausdorff, is closed in H /H0 . Since f (G) is dense in H , it follows that ϕ(G/G0 ) is dense in H /H0 . Therefore, H /H0 = ϕ(G/G0 ), and so H /H0 is compact. Proposition 5.8. Assume that G is an abelian pro-Lie group satisfying at least one of the two conditions (i) G is almost connected;

Vector Group Splitting Theorems

219

(ii) G is topologically compactly generated. Then comp(G) is compact and therefore is the unique largest compact subgroup of G. Proof. We may assume that G = comp(G) and must show that G is compact. Then by Theorem 5.5 (ii) we have G = limN ∈N (G) comp(G/N ); it therefore suffices to verify that comp(G/N ) is compact for all N. Let N ∈ N (G). The Lie group G/N is isomorphic to L = Rp × Tq × D with a discrete group D (compare the proof of Theorem 5.5 (i)), and thus comp(L) = {0} × Tq × tor(D) for the torsion group tor(D) of D. Thus we have to verify that tor(D) is finite. In case (i), G/N ∼ = L is an almost connected Lie group by Lemma 5.7 and thus D is itself finite. In case (ii), G/N ∼ = L has a dense subgroup generated by a compact set, and this is then true for the discrete factor D. Then D is finitely generated and thus is isomorphic to the direct sum of a finite group and a finitely generated free group (see for instance [102, Theorem A1.11]). Thus tor(D) is finite in this case as well. Lemma 5.9. Assume that G is an abelian pro-Lie group. Then L(comp(G)) = comp(L)(G)), and there is a closed vector subspace W such that (X, Y )  → X + Y : W × comp(L)(G)) → W ⊕ comp(L)(G) = L(G) is an isomorphism of weakly complete topological vector spaces. Proof. By Definition 5.4, a one parameter subgroup X : R → G is in comp(L)(G) iff X(R) is compact iff X(R) ∈ comp(G). Thus comp(L)(G) = L(comp(G)). Since comp(G) is a closed subgroup of G by Theorem 5.5 (i), L(comp(G)) is a closed vector subspace of L(G). Then it has an algebraic and topological vector space complement, by Theorem A2.11 (i) of Appendix 2.

Vector Group Splitting Theorems Lemma 5.10. Let G be an abelian pro-Lie group, and assume that comp(G) is compact. Then the following conclusions hold. (i) There is a weakly complete vector group W and a compact abelian group C which is a product of circle groups such that G may be considered as a closed subgroup of W × C such that G ∩ ({0} × C) = comp(G). (ii) G/ comp(G) is embedded as a closed subgroup into the weakly complete vector group W . Proof. (i) For each N ∈ N (G) we have an embedding iN : G/N → W (N )×C(N ) for a finite-dimensional vector group W (N) and a finite-dimensional torus C(N ). Hence

220 G∼ =

5 Abelian Pro-Lie Groups

lim

N∈N (G)

G/N ⊆





i=

N∈N (G) iN

G/N −−−−−−−→

N ∈N (G)



W (N ) × C(N ) = W × C

N ∈N (G)

 for a  weakly complete vector group W = N ∈N (G) W (N ) and a compact group C∼ = N∈N (G) C(N). Since i is an embedding we may write G ⊆ W × C and assume that G is closed. Then comp(G) ⊆ comp(W × C) = {0} × C. But conversely, as every element in {0} × C is relatively compact, we have G ∩ ({0} × C) ⊆ comp G. (ii) By (i) above we may assume G ⊆ W × C for a weakly complete vector group W and torus C and {0} × C is the maximal compact subgroup of W × C. Hence comp(G) ⊆ {0} × C, and since C is compact, comp(G) ⊆ {0} × C. Let p : P → W be the projection of W × C onto W with kernel {0} × C. Then p is a proper and hence closed morphism of topological groups; therefore p|G : G → p(G) is a quotient morphism onto a closed subgroup of W . Since ker(p|G) = comp(G) we have G/ comp(G) ∼ = p(G) and the lemma follows. We remark in passing that an abelian Lie group, being isomorphic to a group × Tq × D for some discrete group (see for instance [102, 1 Exercise E5.18], or Corollary 7.58), is isomorphic to a subgroup of a connected abelian Lie group if and only if it is algebraically generated by a compact subset. If a topological abelian group is isomorphic to the additive topological group of a weakly complete topological vector space, it is called a weakly complete vector group. Rp

Lemma 5.11. Let G be an abelian pro-Lie group such that comp(G) = {1}. Then expG : L(G) → G0 is an isomorphism of topological groups. Proof. It is no loss of generality to assume that G = G0 , and we shall do that from now on. If X ∈ ker expG , that is X(1) = 0, then expG R.X = X(R) is a homomorphic image of R/Z and is therefore compact. Hence X(R) ⊆ comp(G) = {1} and thus X = 0. So expG is injective. Let N ∈ N (G). Since G is connected, G/N is a connected abelian Lie group, and thus the hypotheses of Lemma 5.10 are satisfied. Thus by Lemma 5.9 we may and will now assume that G is a closed subgroup of a weakly complete vector group V and we let i : G → V denote the inclusion map. We may identify V with L(V ) and expV with idV . There is a commutative diagram L(i)

L(G) −−→ ⏐ expG ⏐  G −−→ i

V ⏐ ⏐id  V V.

The morphism L(i) implements an isomorphism of L(G) onto a closed vector subspace of V . This implies that the corestriction expG : L(G) → expG L(G) is an isomorphism of topological groups, and thus expG L(G) is a closed vector subspace of V via i. By Corollary 4.22, exp L(G) is dense in G = G0 . Since expG L(G) is closed, we have G = expG L(G), and expG : L(G) → G is an isomorphism.

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Lemma 5.12 (Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups). Let G be an almost connected abelian pro-Lie group. Then there is a closed subgroup V of G such that expG |L(V ) : L(V ) → V is an isomorphism of topological groups and that (X, g)  → (expG X) + g : L(V ) × comp(G) → V ⊕ comp(G) = G is an isomorphism of topological groups. In particular, every connected abelian pro-Lie group is isomorphic to RJ × C for some set J and some compact connected abelian group C. Proof. By Theorem 5.5 (i), comp(G) is a closed subgroup. def

Let q : G → H = G/ comp(G) be the quotient morphism. By the Strict Exactness Theorem for L 4.20 we have a commutative diagram of strict exact sequences incl

L(q)

incl

q

0 → comp(L)(G) −−−→ L(G) −−−→ L(H ⏐ ⏐ ⏐ ) →0 ⏐ ⏐ ⏐exp expG | comp(L)(G) expG  H 0→ comp(G) −−−→ G −−−→ H → 0. By Lemma 5.10, the morphism L(q) splits, that is, there is a morphism of weakly complete topological vector spaces σ : L(H ) → L(G) such that L(q)  σ = idL(H ) . By 5.5 (iv) we have comp(H ) = {1}. Now we use the fact that comp(G) is compact by Proposition 5.8 to conclude that H is complete and thus is a pro-Lie group by Theorem 4.28 (iii). Since G is connected, H is connected. Then Lemma 5.11 implies that expH : L(H ) → H is an isomorphism. We define s : H → G by s = expG  σ  −1 −1 −1 exp−1 H . Then qs = qexpG  σ expH = expH  L(q) σ  expH = expH  expH = idH . Thus q splits. Now set V = s(H ) and let μ : V × comp(G) → G be defined by μ(v, g) = v + g and ν : G → V × G by ν(g) = (g − s(q(g)), s(q(g))). Then μ and ν are inverses of each other, and this completes the proof of the lemma. By applying the Vector Group Splitting Lemma 5.12 to the identity component G0 of an abelian pro-Lie group we see at once that: In any abelian pro-Lie group G, the identity component has a unique largest compact subgroup comp(G0 ) and G0 is isomorphic to a direct product of G0 / comp(G0 ) and comp(G0 ), where G0 / comp(G0 ) is isomorphic to the additive group of the weakly complete topological vector space L(G)/ comp(L)(G) and comp(G). Thus connected abelian pro-Lie groups are already reduced to weakly complete vector groups (Example 5.1 (ii)(a)) and compact connected groups. Definition 5.13. Let G be an abelian pro-Lie group. Let V be any closed subgroup of G such that (i) V is isomorphic to the additive topological group of a weakly complete topological vector space,

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(ii) (v, c)  → v +c : V ×comp(G0 ) → G0 is an isomorphism of topological groups. Then V is called a vector group complement. A vector group complement in a connected abelian pro-Lie group is not unique, but all vector group complements are isomorphic to G/ comp(G). We shall in fact want to know whether we can find a complement that is invariant under the action of a compact group of automorphisms of G. We therefore inspect the issue a bit more closely. Since all of this happens in the identity component of an abelian pro-Lie group, we shall not restrict the generality if we assume that G is in fact connected. Assume that  is a topological group and G is a pro-Lie group. Assume that (ω, g)  → ω · g :  × G → G is an automorphic action, that is, each g  → ω · g is an automorphism of topological groups and 1 · g = g, ω1 ω2 · g = ω1 · (ω2 · g) for all g ∈ G, and ω1 , ω2 ∈ . Since L is a functor, we have an action (ω, X)  → ω · X of  def on g = L(G) such that exp(ω · X) = ω · (exp X). By this definition if X : R → G is a one-parameter subgroup of G, then ω.X : R → G is defined by (ω · X)(r) = ω · X(r). If (ω, g)  → ω · g :  × G → G is a continuous function then we claim that (ω, X)  → ω · X :  × g → g is also continuous: Indeed, recall that on g = Hom(R, G) we consider the topology of uniform convergence on compact subsets and let C ≥ 0, ω ∈ , X ∈ g, and U an identity neighborhood of G. Let V be an identity neighborhood of G such that ω · V ⊆ U and consider a Y ∈ g such that (∀r ∈ [−C, C]) Then

X(r)−1 Y (r) ∈ V .

(∗)

(ω · X)(r)−1 (ω · Y )(r) = ω(X(r)−1 Y (r)) ⊆ ω · V ⊆ U

for all r ∈ [−C, C]. By the continuity of ϕ :  × R → G, ϕ(ρ, r) = (ω · X)(r)−1 (ρω · Y )(r), for each r ∈ [−C, C], there is an identity neighborhood r of  and an open neighborhood Wr of r ∈ R such that ϕ(r × Wr ) ⊆ U . Since [C, C] is compact, we find  def  r1 , . . . , rn ∈ [−C.C] such that [−C, C] ⊆ nm=1 Wrm . Let  = nm=1 rm . Then ϕ( × [−C, C]) ⊆ U . That is, for all ω ∈ ω and all Y satisfying (∗) we have (ω · X)(r)−1 (ω · Y )(r) ∈ V . And this proves our claim. We shall say that (ω, g)  → ω · g :  × G → G is a continuous action iff it is an action and a continuous function.

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If G is a topological abelian group and C a closed subgroup, then there is a complementary subgroup V of G such that G = V ⊕ C algebraically and topologically iff the morphism p : G → G, p(v + c) = c satisfies p 2 = p and C = im p; conversely, if such a function is given then ker p is a complementary subgroup for C. Proposition 5.14. (i) Let V be any vector space complement of a connected abelian pro-Lie group G and let us write G = V × C with C = comp(G). For any morphism of topological groups f : V → C, the subgroup graph(f ) = {(v, f (v)) : v ∈ V } is also a vector group complement. All vector group complements are obtained in this way. If f : V → C is given, then αf : G → G, αf (v, c) = (v, c + f (v)) is an automorphism of G mapping V ×{0} to graph(f ), while Pf : G → G, Pf (v, c) = (0, c−f (v)) is an idempotent endomorphism of G whose kernel is a vector space complement and whose image and fixed point set is C. (ii) Let  be a compact group and assume that there is a continuous automorphic action  × G → G on the connected abelian pro-Lie group G. Then there is a vector space complement in G that is invariant under . Proof. (i) Exercise E5.2 below. (ii) Since C is a characteristic subgroup, it is invariant under the action of , and it follows that L(C) is invariant in g = L(G). The vector subspace L(C) has a vector space complement in the weakly complete vector space g. (See Chapter 7, Theorem 7.7 (iv).) Thus we have an idempotent endomorphism of topological vector spaces q : g → g, im q = L(C). Since  is compact and acts continuously and linearly on the locally convex complete vector space g, the integral 7 p= ωqω−1 dω : g → g ω∈

exists and is an idempotent equivariant endomorphism of topological vector spaces, that is, it satisfies p(ω · X) = ω · p(X) and im p = L(C). (See [102, Example 3.37 (ii).]) def

Then W = ker p is an  invariant vector subspace of g such that g = W ⊕ L(C) algebraically and topologically. So, using the terminology of (i) above, writing G = V × C and identifying L(G) with L(V ) × L(C), we can now invoke (i) for L(G) and write W = {(X, Y + λ(X)) : X ∈ L(V ), Y ∈ L(C)} for a suitable linear morphism ϕ : L(V ) → L(C). The exponential function expV : L(V ) → V is an isomorphism. We define f : V → C by f (expV X) = expC ϕ(X) and get a vector space complement expG W = {(v, f (v)) : v ∈ V } ⊆ V × C, and by (i) again, this an algebraic and topological complement for C, that is, G = expG W ⊕ C. Moreover, if ω ∈ , then ω · (X, ϕ(X)) = (X  , ϕ(X )) for some X , whence ω · (expV X, f (expV X)) = ω · expG (X, ϕ(X)) = expG ω(X, ϕ(X)) = expG (X , ϕ(X )) = (expV X , f (expV X )) ∈ Vϕ . Thus expG W is -invariant.

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Exercise E5.2. Prove Remark 5.14 (i). [Hint. The function α : G0 → G0 , α(v, c) = (v, c + f (v)) is an automorphism of topological groups mapping V × {0} onto graph(f ). If W ⊆ G0 = V × C is a vector group complement, then since W is a vector group complement there is a projection pr W : G0 → W with kernel {0} × C. Let pr C : G0 → C denote the projection with kernel V × {0}. Define f : V → C by pr C (pr W (v, 0) − (v, 0)). Then f is a morphism of topological groups; if w = (v, c) ∈ W then w = pr W (v, 0) and w − (v, 0) = (0, c) whence c = f (v). The assertions on αf and Pf are straightforward.] We now work towards removing the hypothesis of almost connectivity from the Vector Group Splitting Lemma 5.12. Lemma 5.15. Let W be a weakly complete topological vector space and G a proto-Lie group. Assume that f : W → G is a bijective morphism of topological abelian groups. Then f is an isomorphism of topological groups.  denote the completion of G. Then G  is a connected abelian pro-Lie group; Proof. Let G by theVector Group Splitting Lemma 5.12, it is therefore of the form V ⊕C algebraically and topologically for a weakly complete topological vector subgroup V and a compact  → V denote the projection onto V along C. The function subgroup C. Let pr V : G def

ϕ = pr V f : W → V is a dense morphism of weakly complete vector groups and is therefore a quotient morphism of weakly complete topological vector spaces by Theorem A2.12 (a) of Appendix 2. Thus there are closed vector subspaces V1 and V2 of W such that W = V1 ⊕ V2 algebraically and topologically such that V2 = ker ϕ and ϕ  i : V1 → V , where i : V1 → W is the inclusion, is an isomorphism of weakly  σ = f i(ϕi)−1 complete topological vector spaces. Now the morphism σ : V → G, −1 −1 satisfies pr V σ (v) = pr V f  i  (ϕ  i) = ϕ  i  (ϕ  i) = idV . This means that  = σ (V ) ⊕ C, σ (V ) = f (V1 ). In order to simplify notation, after replacing V by G def

σ (V ), if necessary, we may actually assume that V = f (V1 ). Then D = f (V2 ) ⊆ C, and we have G = V × D with a dense subgroup D of C. We recall that G is a protoLie group; then D ∼ = G/V is a connected proto-Lie group and a dense subgroup of a compact group. Let N ∈ N (D); then D/N is a Lie group and a dense subgroup of C/N . A Lie group is complete, and thus D/N is closed in C/N, that is, D/N = C/N for all N ∈ N (D).  Hence G is complete and f : W → V × C Therefore D = C and thus G = G. is bijective. We can write W = W1 × W2 such that f = f1 × f2 where f1 is an isomorphism from W1 onto V and f2 : W2 → C is a bijective morphism of topological abelian groups from a weakly complete vector group W2 onto a compact group C. In particular, every point in C is on a one parameter subgroup, that is exp : L(C) → C is surjective. But C, as a bijective image of a real vector space, is torsion-free and divisible.  is divisible and torsion-free and so is a rational vector space, that is, a direct Thus C . If C = Q J for some sum of copies of Q, and therefore C is a power of copies of Q

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)J as L preserves limits, and expC may set J , then L(C) may be identified with L(Q J ∼ ) → Q  is not surjective, because the be identified with (expQ  ) . But expQ  : R = L(Q  nontrivial compact homomorphic images of R are isomorphic to R/Z ∼ = T and T ∼ =Q  . Thus J = Ø and C = {0}. This means that G = V and since  T ∼ =Q∼ = Z ∼ =Q W → V is an isomorphism of topological groups. f ∼ = Note that Lemma 5.15 is a very special example of an Open Mapping Theorem (see 9.60 and its proof). Lemma 5.16. Assume that G is an abelian topological group with closed subgroups G1 and H such that G1 is either (a) a weakly complete topological vector subgroup, or (b) a compact subgroup. Assume further that G1 + H = G, that G1 ∩ H = {0}, and that G/H is a proto-Lie group. Then μ : G1 × H → G, μ(v, h) = v + h is an isomorphism of topological groups. Proof. Clearly μ is a bijective morphism of topological groups. We must show that its inverse is continuous. The morphism β : G1 → G/H , β(v) = v + H is a continuous bijection. In Case (a), by Lemma 5.15 β is open. In Case (b) it is a homeomorphism since G1 is compact. That is, β −1 : G/G0 → G0 is continuous in both cases. Let def

q : G → G/G0 be the quotient map. Then α = β  q : G → G0 is a morphism of topological groups, and μ−1 (g) = (α(g), g − α(g)) is likewise a morphism of topological groups. Notice that by the Quotient Theorem for Pro-Lie Groups 4.1, the quotient G/H is a proto-Lie group if G is a pro-Lie group. Proposition 5.17. Assume that G is an abelian proto-Lie group and that G1 is a closed connected subgroup which is a finite-dimensional Lie group. Then there is a closed subgroup H such that the morphism (v, h)  → v + h : G1 × H → G is an isomorphism of topological groups. Proof. We claim that it is no loss of generality to assume that G is a pro-Lie group.  be the completion of G. Since the subgroup G1 is a finite-dimensionIndeed, let G al Lie group, it is locally compact. Locally compact subgroups are closed (see for instance [102, Corollary A4.24.]). Thus G1 is also a closed subgroup of the pro If we can show that there is a closed subgroup H  of G  such that Lie group G.  → G  is an isomorphism of topological groups, then (v,  h) → v +  h : G1 × H def  we get a subgroup of G such that (v, h)  → g + h : G1 × H → G setting H = G ∩ H is an isomorphism of topological groups. This proves our claim; from here on we shall assume that G is a pro-Lie group.

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The subgroup G1 , being a finite-dimensional Lie group, has no small subgroups. Hence there is a zero neighborhood U such that {0} is the only subgroup contained in G1 ∩ U . Now let N ∈ N (G) be contained in U . Then N ∩ G1 ⊆ U ∩ G1 = {0}. Recall that G/N is a Lie group and (G1 + N )/N is isomorphic to G1 /(G1 ∩ N ) ∼ = G1 by the Closed Subgroup Theorem for Projective Limits 1.34 (iv). So (G1 + N )/N is isomorphic to Rp ⊕ Tq and is a closed subgroup of G/N which is isomorphic to Rm ⊕ Tn ⊕ D for a discrete subgroup D. It is therefore a direct summand algebraically and topologically, that is, there is a closed subgroup H of G containing N such that H /N is a complementary summand for (G1 +N )/N . Thus G1 +H = G and G1 ∩H ⊆ def

G1 ∩ N = {0}. Then if C = comp(G1 ) = {0}, it follows from Lemma 5.16 (a), that (v, h) → v + h : G1 × H → G is an isomorphism of topological groups. Therefore, in the general case, G/C ∼ = G1 /C × (H + C)/C and thus there is a vector subgroup V of G such that G ∼ = V × (C + H ); it remains to be observed that C + H ∼ = C × H. But that is Lemma 5.16 (b). Next we need a lemma on weakly complete topological vector spaces. Recall that an affine subspace A of a vector space W is a subset of the form A = g + V for some vector subspace V . The affine subspace is linear iff g ∈ V . Lemma 5.18. Let W be a weakly complete topological vector space and F a filter basis of closed affine subspaces. (a) Assume that F is a closed vector subspace of W and all A ∈ F are linear. Then F+



F =



F + H.

(∗)

If, in addition, dim E/F < ∞, then   F+ F = (F + H ).

(∗∗)

H ∈F

H ∈F

(b) Assume that all A ∈ F are linear. Then the following conditions are equivalent: (i)  lim F = 0. (ii) F = {0}.  (c) F = Ø. (d) Assume that F ⊆ N (W ). Then the following conditions are equivalent: (i) lim F = 0. (ii) F is a basis of N (W ). Proof. See Theorems A2.13 and A2.14 of Appendix 2. In terms of a terminology that has been used for situations like this, (c) can be expressed in the following form: Weakly complete topological vector spaces are linearly compact.

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We shall prove next that every weakly complete vector subgroup in an abelian pro-Lie group splits. Another way of expressing this in a category theoretical fashion is this: Weakly complete vector groups and torus groups are relative injectives for the class of embeddings in the category of abelian pro-Lie groups. For a full understanding recall that for a class E of monomorphisms, an object I in a category is called a relative injective for E , if for any monic ι : A → B from E and any morphism f : A → I there is a morphism F : B → I such that f  ι = f . idI

I −−→  ⏐ f⏐ A −−→ ι

I  ⏐ ⏐F B.

(See for instance [102, paragraph following 9.77].) By definition, an object is an injective, if it is a relative injective for the class of all monomorphisms. Exercise E5.3. A pushout in a category is a pullback in the opposite category (see Definition 1.1 (iii)). Show Proposition A. Let I be an object in a category and consider the following two statements: (i) I is a relative injective for E . (ii) Any monic ι : I → B from E is a coretraction, that is there is an ε : B → I such that ε  ι = idI . Then (i) ⇒ (ii) and if the category has pushouts pushing forward E -monics, then the two conditions are equivalent. This is the case in the category AbTopGr of abelian topological groups and in the category AbproLieGr of abelian pro-Lie groups for the class E of closed embeddings. Proposition B. Any product of relative injectives is a relative injective. [Hint. Proof of A: For a proof of (i) ⇒ (ii) simply take A = I and f = idI in the definition. For a proof of the reverse implication let ι : A → B be a monomorphism from E and f : A → I any morphism into an object I satisfying (ii). Form the pushout ϕ

I −−→  ⏐ f⏐ A −−→ ι

P  ⏐β ⏐ B.

(∗)

Since pushouts push forward E -monics and ι is an E -monic, the morphism ϕ is an E -monic as well. Hence by (ii) there is an ε : P → I such that ε  ϕ = idI . Set F = εβ. Then F  ι = εβι = εϕf = idI f = f . ϕ

ε

ι

idB

I −−→ P   −−→ ⏐ ⏐β f⏐ ⏐ A −−→ B −−→

I  ⏐ ⏐F B.

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Finally, we recall from Theorem 1.5 (a), that the pushout (∗) is formed by creating first the coproduct P  of I and B and then the coequalizer e : P  → P of the two morphisms copr I f and copr B ι (assuming that coproducts and coequalizers exist). In the category AbTopGr of abelian topological groups P  = I × B and copr I (x) = def

(x, 0), copr B (b) = (0, b). The coequalizer e is the cokernel of δ = copr I f − copr B ι, that is the quotient morphism e : I × B → (I × B)/Q with Q = δ(A) = {(f (a), −a) : a ∈ A}, where we have assumed A = A ⊆ B and ι to be the inclusion map. Notice that Q is closed in I × B: If (x, b) = limj (f (aj ), −aj ) for a net (aj )j ∈J in A, then b ∈ A = A and x = limj f (aj ) = f (−b) since f is continuous. (Indeed, Q is in essence the graph of f .) Now ϕ : I → (I × B)/Q is given by ϕ(x) = (x, 0) + Q. Then x ∈ ker ϕ iff (x, 0) ∈ Q, that is, (x, 0) = (f (a), −a) for some a ∈ A. Then a = 0 and as a consequence x = f (a) = f (0) = 0. Thus ϕ maps I bijectively onto ((I × {0}) + Q)/Q = (I × A)/Q. Since Q is closed in B, the group I × A is closed in I × A. We claim that the morphism ψ : I → (I × A)/Q, ψ(x) = (x, 0) + Q is open. So let U be a symmetric identity neighborhood of I . Find a symmetric identity neighborhood V of I such that V + V ⊆ U . Now we let W = f −1 (V ); this is a symmetric identity neighborhood of A since f is continuous. Now we show that ψ(U ) contains (V × W ) + Q/Q and this will show that ψ(U ) is an identity neighborhood of (I × A)/Q and thus prove the asserted openness of ψ. Let (v, w) ∈ V × W . We claim that there are elements u ∈ U and a ∈ A such that (u, 0) + (f (a), −a) = (v, w). Indeed, we let a = −w. Then u − f (w) = v, implies u = v + f (w) ∈ V + V ∈ U and this proves the claim. In the category AbproLieGr everything remains the same except that in order to form the pushout in the category AbproLieGr we have to form the completion P of (I × B)/Q, because (I × B)/Q is a proto-Lie group but not necessarily a pro-Lie group (see Chapter 4, the Quotient Theorem for Pro-Lie Groups 4.1). Thus ϕ : I → P is the composition of the closed embedding morphism γ : I → (I × B)/Q and the dense embedding morphism η : (I × B)/Q) → P . But I is a pro-Lie group and is therefore complete, and since ϕ(I ) = η(γ (I )) is an isomorphic copy of I it is a complete and therefore closed subgroup of B. Hence ϕ is a closed embedding in this case as well. Proof of B. Let {Ij : j ∈ J } be a family of relative injectives and ι : A → B a mono from E . Assume that f : A → j ∈J Ij is any morphism. Fix j since Ij is an E injective, there is an Fj : B → Ij such that Fj  ι = pr j f . Then the universal  property of the product (see Definition 1.4 (i)) give us a morphism F : B → j ∈J Ij such that pr j F = Fj . Then pr j F ι = Fj  ι = pr j f and this implies F  ι = f by the uniqueness in the universal property of the product.] Theorem 5.19. Assume that G is an abelian pro-Lie group with a closed subgroup G1 and assume that there are sets I and J such that G1 ∼ = RI × TJ . Then G1 is a

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homomorphic retract of G, that is, a direct summand algebraically and topologically. So, G ∼ = G1 × G/G1 . Proof. By Exercise E5.3, Proposition A, applied to the category AbproLieGr of abelian pro-Lie groups, we are claiming that G1 is a relative injective for the class of closed homomorphic embeddings in AbTopGr. By Exercise E5.3, Proposition B above, it suffices to prove this for each of the cases (i) G1 = R, and (ii) G1 = T. For each of these cases Proposition 5.17 applies and proves the claim. It may be a useful exercise to present an independent and category theory free proof of the preceding theorem in the case that G1 ∼ = RI is a weakly complete topological vector space. It proceeds as follows: Let S be the set of all closed subgroups S of G satisfying the following conditions: (i) S ∩ G1 is a vector group. (ii) (∀g ∈ G) S ∩ (g + G1 ) = Ø. We claim that (S, ⊇) is an inductive poset. For a proof of the claim let T be a chain def  in S and set T = T . We  have to show that T satisfies (i) and (ii). (i) We note T ∩ G1 = S∈T S ∩ G1 , and all S ∩ G1 are closed vector groups; hence their intersection is a closed vector group. (ii) Let g ∈ G; we must show that T ∩ (g + G1 ) = Ø. Now for each S ∈ T we find an sS ∈ S ∩ (g + G1 ) by (ii). Then g + G1 = sS + G1 . We claim that (∗) S ∩ (g + G1 ) = sS + (S ∩ G1 ). Indeed if s ∈ S ∩(g+G1 ), then s ∈ g+G1 = sS +G1 and thus s −sS ∈ S ∩G1 and thus s ∈ sS +(S∩G1 ). Conversely, if s ∈ S∩G1 , then sS +s ∈ S∩(sS +G1 ) = S∩(g+G1 ). By (i) we know that S ∩ G1 is a (closed) vector subgroup VS of G1 . Thus from (∗) we obtain (∗∗) Ø = (S − g) ∩ G1 = sS − g + VS , where sG − g ∈ G1 . Now the family {sS − g + VS : S ∈ T } is a filter basis of closed affine subspaces of the vector group G1 . By Lemma 5.18 (c), there is a  t∈ (sS − g + VS ) ∈ G1 , S∈T



 and thus t + g ∈ S∈T (sS + VS ) = S∈T S ∩ (g + G1 ) = T ∩ (g + G1 ). This completes the proof that (S, ⊇) is inductive. Using Zorn’s Lemma, let H be a minimal member of S. We claim that H ∩G1 = {0}. Suppose that the claim were false. def

Then H1 = H ∩ G1 is a nonzero weakly complete vector group. Let N be a vector subgroup of H1 , such that H1 /N is a finite-dimensional vector group (for instance, one isomorphic to R). Now G/N as a quotient of a pro-Lie group, is a proto-Lie group by Theorem 4.1. Then by Proposition 5.17, there is a closed subgroup S of H containing N such that (H1 /N ) + (S/N ) = H /N, and (H1 /N ) ∩ (S/N ) = {N }. Thus

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H1 ∩ S = N is a vector subgroup and G1 + S = G1 + H = G and so the subgroup S of H /N satisfies (i) and (ii) above. The minimality of H then entails S = H and thus N = H1 ∩ S = H1 and that is a contradiction to the choice of N. This proves our claim that there is a closed subgroup H of G such that G = G1 + H and G1 ∩ H = {0}. Thus the function (v, h)  → v + h : G1 × H → G is a bijective morphism of topological groups by Lemma 5.16. Recall from the Vector Group Splitting Lemma 5.12 and Definition 5.13 that every abelian pro-Lie group has a vector group complement. Vector Group Splitting Theorem for Abelian Pro-Lie Groups Theorem 5.20. Let G be an abelian pro-Lie group and V a vector group complement. Then there is a closed subgroup H such that: (i) (v, h)  → v + h : V × H → G is an isomorphism of topological groups. (ii) H0 is compact and equals comp G0 and comp(H ) = comp(G); in particular, comp(G) ⊆ H . (iii) H /H0 ∼ = G/G0 , and this group is prodiscrete. (iv) G/ comp(G) ∼ = V × S for some prodiscrete abelian group S without nontrivial compact subgroups. (v) G has a characteristic closed subgroup G1 = G0 + comp(G) which is isomorphic to V × comp(H ) such that G/G1 is prodiscrete without nontrivial compact subgroups. (vi) The exponential function expG of G = V ⊕ H decomposes as expG = expV ⊕ expH where expV : L(V ) → V is an isomorphism of weakly complete vector groups and expH = expcomp(G0 ) : L((comp(G0 )) → comp(G0 ) is the exponential function of the unique largest compact connected subgroup; here L(comp(G0 )) = comp(L)(G) is the set of relatively compact one parameter subgroups of G. (vii) The arc component Ga of G is V ⊕Ha = V ⊕comp(G0 )a = im L(G). Moreover, if h is a closed vector subspace of L(G) such that exp h = Ga , then h = L(G). Proof. (i) By Theorem 5.19, H exists such that (i) is satisfied. (ii) Let us write G = V × H . Then G0 = V × H0 . Since V × {0} is a vector group complement, G = (V × {0}) ⊕ comp(G0 ) algebraically and topologically. Then the projection of G0 onto H0 along V maps the compact subgroup comp(G0 ) onto H0 . Thus H0 is compact. So {0} × H0 ⊆ comp(G0 ), and since G0 = V × H0 , the factor group comp(G0 )/({0} × H0 ) is isomorphic to a subgroup of V . Since V as a vector group has no nontrivial compact subgroup, comp(G) = {0} × H0 follows. (iii) Retaining the convention G = V × H after (i), we have G0 = V × H0 . Then G/G0 =

V × H ∼ (V × H )/(V × {0}) ∼ = = H /H0 . V × H0 (V × H0 )/(V × {0})

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By the Closed Subgroup Theorem 3.35, H is a pro-Lie group. Since H0 is compact, H /H0 is complete by Theorem 4.28 (iii). Thus by Lemma 3.31 H /H0 is a prodiscrete group (iv) Again we write G = V × H and have comp(G) = {0} × comp(H ). Thus G/ comp(G) =

V ×H ∼ = V × (H / comp(H )). {0} × comp(H )

By Lemma 5.5 (iii), H / comp(H ) is compact-free. By (iii) above, H /H0 is prodiscrete. As a quotient of the prodiscrete group H /H0 , the quotient H / comp(H ) ∼ = (H /H0 )/(comp(H )/H0 ) is a protodiscrete group by Proposition 3.30 (b). (v) comp(G) is a closed characteristic subgroup and by (iv) the factor group G/ comp(G) decomposes into a direct product V ×S in which V ×{0} is the connected component and thus is characteristic. The kernel G1 of the composition of the quotient morphism G → G/ comp(G) and the projection G/ comp(G) → S is a closed characteristic subgroup equal to G0 + comp(G) and G/G1 ∼ = S. Applying (i) to G1 we get G1 ∼ = V × comp(G). (vi) follows immediately from (i) and Definition 5.4 (i) in view of the fact that for any topological vector space V the exponential function expV : L(V ) → V , expG X = X(1), is an isomorphism of topological vector spaces as all one parameter subgroups are of the form X = r  → r · v for a unique vector v = vX . See also Lemma 5.15. (vii) As G = V ⊕ H is a direct product decomposition we have Ga = Va ⊕ Ha . But V , as the additive topological group of a topological vector space is arcwise connected, and Ha = (H0 )a = comp(G0 )a . By [102, Theorem 8.30 (ii)], we have comp(G0 )a = L(comp(G0 )) = L(H ). Thus from (vi) we get Ga = expG L(G). Now let h be a closed subalgebra of L(G). If h = L(G), then expG h = E(G) = Ga by Corollary 4.21 (i). In the context of conclusions (iv) and (v) we recall from 5.10 (ii) that every prodiscrete group without compact subgroup embeds into a weakly complete vector group. But Proposition 5.2 shows that such embeddings may have bizarre properties. For locally compact abelian groups, 5.20 (i) yields a core result of their structure theory; it is presented practically in every source book on locally compact abelian groups (see for instance [102, Theorem 7.57]). The examples in 5.1 (ii)(b)–(e) illustrate certain limitations of this main result. The examples in 5.1 (ii)(b) and (ii)(e) show what prodiscrete pointwise compact groups may look like; neither has a compact open subgroup and therefore both fail to be locally compact. The examples in (ii)(b) are torsion-free and divisible, the examples in (ii)(e) have a dense proper torsion subgroup and are divisible. The examples in 5.1 (ii)(c) and (d) are compact-free; those in (ii)(c) are divisible, those in (ii)(d) have no nondegenerate divisible subgroups. Thus unlike in the locally compact case, we cannot expect that inside the factor H , the subgroup comp(H ) is open, or, equivalently, that the factor group G/G0 has an open compact subgroup.

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It is easy to mix these examples. There are compact abelian groups in which the component does not split (see [102, Example 8.11]). Theorem 5.20 completely elucidates the structure of the identity component G0 , it largely clarifies the structure of G1 (although comp(G) is best understood in the locally compact case), and it reduces the more subtle problems on G to the compactfree prodiscrete factor group G/G1 . One should recall the example in Proposition 5.2 which typically might occur as a prodiscrete factor group. Corollary 5.21. Let G be an abelian pro-Lie group and V a vector group complement. Then the following statements are equivalent: (i) (ii) (iii) (iv)

G/G0 is locally compact. G/V is locally compact. comp(G/V ) is locally compact and open in G/V . There is a locally compact subgroup H of G containing comp(G) as an open subgroup such that such that (v, h)  → v + h : V × H → G is an isomorphism of topological groups.

Proof. (i) ⇔ (ii): We have G/G0 ∼ = (G/V )/(G0 /V ) and G0 /V is compact by Lemma 5.12. The quotient of a locally compact group is locally compact, and the extension of a locally compact group by a locally compact group is locally compact. (iv) ⇒ (iii): Since comp(G) = comp(H ) this is clear. (iii) ⇒ (ii): Trivial. (ii) ⇒ (iv): The locally compact abelian group G/V has a compact identity component (G/V )0 = comp(G/V )0 and thus has a compact open subgroup C. Let K be the full inverse image of C in G. Then K is an almost connected open subgroup of G to which the Vector Group Splitting Theorem applies. Thus K is the direct product of V and the unique maximal compact subgroup comp(K) of K. Then (G/ comp(K))0 = K/ comp(K) ∼ = V , this is an open divisible subgroup of G/ comp(K). But then G/ comp(K) is the direct sum of (K/ comp(K))0 and a discrete group H / comp(K) with a closed subgroup H of G containing comp(K) as an open subgroup. In particular, comp(K) ⊆ comp(H ) and comp(H ) is open in H . From (G/ comp(K)) ∼ = (K/ comp(K)) × (H / comp(K)) and K∼ = V × comp(K) we derive G∼ = V × H. As we have comp(G) = comp(H ), the implication (ii) ⇒ (iv) is proved.

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Compactly Generated Abelian Pro-Lie Groups The Vector Group Splitting Theorem tells us that each abelian pro-Lie group G is built up in a lucid fashion from a weakly complete vector group and a more special abelian pro-Lie group H which is turn is an extension of the characteristic closed subgroup comp(G) = comp(H ) by a prodiscrete compact-free factor group H / comp(H ) ∼ = G/(G0 + comp(G)). The examples 5.1 (ii)(b) and (ii)(e) (and the groups easily manufactured from these by passing to products, subgroups and quotients) indicate that we are not to expect very explicit information on comp(G) without further hypotheses, and a similar statement holds for prodiscrete compact-free groups (see 5.1 (ii)(c), (ii)(d) and 5.2). A topological space is called a Polish space if it is completely metrizable and second countable. It is said to be σ -compact, if it is a countable union of compact subspaces. It is said to be separable if it has a dense countable subset.  Countable products of Polish spaces are Polish. For instance, any product n∈N Ln of a countable sequence of second countable Lie groups is a Polish pro-Lie group; this applies in particular to RN or ZN . A mixture of topological and algebraic properties of topological groups is exemplified by the concepts introduced in 5.6, to which we return presently. Remark 5.22. (i) Every almost connected locally compact group is compactly generated. (ii) Every compactly generated topological group is σ -compact. (iii) A topological group whose underlying space is a Baire space and which is σ -compact is a locally compact topological group. (iv) A σ -compact Polish group is locally compact. (v) A compactly generated Baire group is locally compact. Proof. (i) Let K be a compact neighborhood of the identity. Then K is an open subgroup which has finite index in G. Let F be any finite set which meets each coset modulo K. Then K ∪ F is a compact generating set of G. def

(ii) If K is a compact generating set of G, C = KK −1 is a compact generating then ∞ −1 set satisfying C = C; then G = C = n=1 C n . (iii) A Baire space cannot be the union of a countable set of nowhere dense closed subsets. A topological group containing a compact set with nonempty interior is locally compact. (iv) By the Baire Category Theorem (see [26, Chapter 9, §5, no 3, Théorème 1]), every Polish space is a Baire space. (v) is clear from the preceding. The following remarks are straightforward from the definitions, from Proposition 5.8, and the Vector Group Splitting Theorem 5.20. Remark 5.23. Let G ∼ = V × H be an abelian pro-Lie group with a vector group complement V and H as in Theorem 5.20. Then the following statements are equivalent:

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(i) G is Polish iff both V and H are Polish. (ii) G is σ -compact iff V, comp(G), and H / comp(H ) are σ -compact. (iii) G is compactly generated iff V and H / comp(H ) are compactly generated and comp(G) is compact. (iv) G is separable iff V and H are separable. These simple remarks lend some urgency to a more detailed understanding of the situation of weakly complete topological vector spaces; we shall turn to this topic in the next section. Remark 5.24. For a discrete abelian group, the following statements are equivalent: (i) (ii) (iii) (iv)

G is finitely generated free. G is isomorphic to a closed additive subgroup of Rn for some natural number n. G is isomorphic to a closed additive subgroup of RJ for some set J . G is isomorphic to a closed additive subgroup of a weakly complete topological vector space.

Proof. For the equivalence of (i) and (ii) see for instance [102, Theorem A1.12 (i)]. Trivially (ii) ⇒ (iii) ⇒ (iv). Assume (iv), that is, that G is a closed discrete subgroup of a weakly complete vector group W . Since G is discrete, there is an identity neighborhood U1 of W such that W ∩ U1 = {0}. Let U be an open identity neighborhood of W such that U + U + U + U ⊆ U1 . Since lim N (W ) = 0 there is a V ∈ N (W ) such that V ⊆ U and thus U + V ⊆ U + U . By replacing U by U + V where necessary we assume that U + V = U and U + U ⊆ U1 . If u ∈ (G − U ) ∩ U then u = g − u for some 0 = g ∈ G and u ∈ U ; thus g = u + u ∈ G ∩ U + U ⊆ G ∩ U1 = {0}. Thus V is the complement of (G \ {0}) − U in G + V . So G ∼ = (G + V )/V is a discrete hence closed subgroup of the finite-dimensional vector space W/V . This result provides an alternative proof of the fact that the prodiscrete free abelian group F of infinite rank in 5.2 cannot be discrete. Proposition 5.25. Let G be an abelian compact-free pro-Lie group. (i) If N (G) has a basis M of subgroups such that G/M is compactly generated, then G is isomorphic to a closed subgroup of a product RI × ZJ and hence also of a weakly complete topological vector space RK . (ii) If G is compactly generated, then (i) applies. (iii) The group G is compactly generated and Polish iff it is isomorphic to Rm × Zn iff it is locally compact. Proof. (i) We assume that for N ∈ M we have G/N = VN ⊕ FN ⊕ tor G/N where VN is a finite-dimensional vector group, FN is finitely generated free and tor G/N is the finite torsion  group of G/N. It follows that G∼may  be identified with a closed subgroup of P = N∈M G/N = V ×F ×C where V = N ∈M VN is a weakly complete vector   ∼ ∼ group, F = N ∈M FN and C = N ∈M tor G/N . Then comp(P ) = {0} × {0} × C

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and G ∩ comp(P ) = {0} since G is compact-free. The projection P → V × F is a proper, hence closed morphism, with kernel comp(P ), mapping G onto a closed subgroup of F which is isomorphic to G/(G ∩ comp(P )) ∼ = G. Since V is a product of copies of R and F is a product of copies of Z, assertion (i) is proved. (ii) If G is compactly generated and N ∈ N (G) then N is a closed subgroup such that G/N is compactly generated, hence is of the form specified in the proof of (i). (iii) If G is Polish and compactly generated, then it is locally compact by 5.22 (i),(iv) and thus, being compact-free, is isomorphic to Rm × Zn . Corollary 5.26. Any compactly generated, compact-free prodiscrete group is isomorphic to a closed subgroup of a group ZJ . If it is not of finite rank, then it is not isomorphic to a subgroup of ZN . This situation is illustrated by the example in Proposition 5.2. In the proof of Proposition 5.8 it was only needed that the Lie group quotients of the abelian Lie group G in question were compactly generated. This together with Proposition 5.25 yields at once: Corollary 5.27. If G is an abelian pro-Lie group whose Lie group quotients are compactly generated, then comp(G) is compact and G/ comp(G) is embeddable in a weakly complete vector group. This applies, in particular, to all compactly generated abelian pro-Lie groups.

Weakly Complete Topological Vector Spaces Revisited We begin with an observation showing that the idea of topologically compactly generated pro-Lie groups may not be very restrictive. Remark 5.28. A weakly complete vector group is topologically compactly generated. A group of the form ZJ for any set J is topologically compactly generated. Proof. See Remark A2.15 of Appendix 2. Lemma 5.29. For a weakly complete topological vector space W , the following statements are equivalent: (A) (B) (C) (D)

W W W W

is σ -compact. is locally compact. is finite-dimensional. is compactly generated.

Proof. See Proposition A2.17 of Appendix 2. ∼ RJ by CorolLet W be a weakly complete topological vector space. Then W = lary A2.9 of Appendix 2. The cardinal card J is called the topological dimension of W . (See [103].)

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Lemma 5.30. For a weakly complete topological vector space W , the following statements are equivalent: (i) W ∼ = RJ with card J ≤ ℵ0 . (ii) W is locally compact or is isomorphic to RN . (iii) W is finite-dimensional or is isomorphic to RN . (iv) W is second countable. (v) W is first countable. (vi) W is Polish. Proof. See Appendix 2, Proposition A2.18. Lemma 5.31. For a weakly complete topological vector space W , the following statements are equivalent: (a) W is separable. (b) W contains a dense vector subspace of countable linear dimension over R. (c) W is isomorphic as a topological vector space to RJ with card J ≤ 2ℵ0 . They are implied by the equivalent statements of Lemma 5.30. Proof. See Proposition A2.19 of Appendix 2. For the following theorem we recall the definition of the characteristic closed subgroup G1 = G0 + comp(G) ∼ = V × comp(G) of an abelian pro-Lie group in Vector Group Splitting Theorem 5.20 (v) for a vector group complement V (see Definition 5.13 (ii)). Theorem 5.32 (The Compact Generation Theorem for Abelian Pro-Lie Groups). (i) For a compactly generated abelian pro-Lie group G the characteristic closed subgroup comp(G) is compact and the characteristic closed subgroup G1 is locally compact. (ii) In particular, every vector group complement V is isomorphic to a euclidean group Rm for some m ∈ N0 = {0, 1, 2, . . . }. (iii) The factor group G/G1 is a compactly generated prodiscrete group without compact subgroups. If G/G1 is Polish, then G is locally compact and G∼ = Rm × comp(G) × Zn . (iv) Assume that G is a pro-Lie group containing a finitely generated abelian dense subgroup. Then comp(G) is compact and G ∼ = Rm comp(G) × Zn , m, n ∈ N. In particular, G is locally compact. (v) A finitely generated abelian pro-Lie group is discrete. Proof. By Theorem 5.20, G ∼ = V × H such that H0 is compact. The factors V and H are compactly generated as homomorphic images of G. By Lemma 5.29, V ∼ = Rn for some nonnegative integer n. By Proposition 5.8, comp(H ) is compact and by 5.20, comp(G) = comp(H ) and H0 ⊆ comp(H ). Thus G1 ∼ = V × comp(G) is locally compact. Also, H / comp(H ) is totally disconnected, and by Theorem 4.28 (iii) this

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quotient is a pro-Lie group and hence is prodiscrete by Proposition 4.23. If the factor group G/G1 is Polish, then it is finitely generated free by 5.25 (iii), and the remainder of (iii) follows. For a proof of (iv) notice first that G is obviously abelian, containing a dense finitely generated abelian subgroup. All quotient groups of G satisfy the hypothesis of (iv), notably each vector group complement of G has a dense finitely generated subgroup and therefore has a dense finite-dimensional vector group; finite-dimensional vector groups are locally compact and therefore complete. Thus each vector group complement is isomorphic to Rm for some m ∈ N. def Also G/G1 contains a dense finitely generated subgroup. Set F = G/G1 ; then comp(F ) = {0}, and by 5.10 (ii), F may be considered as a closed subgroup of a weakly complete vector group V . Then R ⊗ F is algebraically isomorphic with the R-linear span E of F . So dimR E is finite. Every finite-dimensional topological vector space over R is locally compact and is isomorphic to Rp . Any closed free subgroup of Rp , however, is discrete (see for instance [102], Appendix 1, the Closed Subgroups and Quotients Theorem A1.12). This shows that G/G1 is discrete and thus is finitely generated free. Since free discrete factor groups are always direct summands, it now follows from the Vector Group Splitting Theorem 5.20 that G ∼ = Rm × comp(G) × Zn . So comp(G) contains a finitely generated dense subgroup g1  ⊕ · · · ⊕ gk . (For the structure of finitely generated abelian groups see for instance [102], Appendix 1, the Fundamental Theorem of Finitely Generated Abelian Groups A1.11.) By Weil’s Lemma for Pro-Lie Groups 5.3, for each j = 1, . . . , k, either gj ∈ comp(G) or def gj  ∼ = Z. Hence gj  is compact for all j and then K = g1  + · · · + gk  is compact, hence contained in comp(G). On the other hand, K being closed and containing g1 , . . . , gk , contains comp(G). Thus K = comp(G) and this shows that comp(G) is compact. This completes the proof of (iv). Conclusion (v) follows now from (iv): If the group G is finitely generated, then it is countable. A countable locally compact group is discrete by the Baire Category Theorem (or by the existence of Haar measure). Hence a finitely generated locally compact group is discrete. The observation 5.32 (iv) is (in an obvious sense) a generalization of Weil’s Lemma for Pro-Lie Groups 5.3. It is not known whether a compactly generated abelian prodiscrete compact-free group is finitely generated free. By Proposition 5.2, the free abelian group of countably many generators has a nondiscrete nonmetric pro-Lie group topology and fails to be compactly generated.

The Duality Theory of Abelian Pro-Lie Groups The structure theory results we discussed permit us to derive results on the duality of abelian pro-Lie groups. Recall that this class contains the class of all locally compact abelian groups properly.

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 = Hom(G, T) denote its dual with For any topological abelian group G we let G the compact open topology. (See e.g. [102, Chapter 7].) There is a natural morphism   given by ηG (g)(χ ) = χ(g) which may or may not of abelian groups ηG : G → G be continuous; information regarding this issue is to be found for instance in [102], notably in Theorem 7.7. We shall call a topological abelian group semireflexive if   is bijective and reflexive if ηG is an isomorphism of topological groups; ηG : G → G in the latter case G is also said to have duality (see [102, Definition 7.8]). There is an example of a nondiscrete but prodiscrete abelian torsion group due to Banaszczyk [5, p. 159, Example 17.11], which is semireflexive but not reflexive (see also Chapter 14, Example Ac.2. Therefore we know that the category of abelian pro-Lie groups is not self-dual under Pontryagin duality. Recall the following observations. Proposition 5.33. The dual of a weakly complete vector group V is naturally isomorphic to the additive group of its vector space dual V  endowed with the finest locally convex vector space topology. The dual of the additive group of a real vector space E with the finest locally convex vector space topology is naturally isomorphic to the additive group of its vector space dual E  with the weak ∗-topology, that is the topology of pointwise convergence. Both a weakly complete vector group and the additive group of a real vector space given its finest locally convex topology are reflexive. Proof. See Appendix 2, Theorem A2.8.  is naturally Proposition 5.34. Let A and B be topological abelian groups. Then A×B   isomorphic to A × B, and if A and B are reflexive, then A × B is reflexive. Proof. Exercise . Exercise E5.4. Prove Proposition 5.34. [Hint. The proof is straightforward.] Proposition 5.35. Let G be an abelian pro-Lie group and let V be a vector group complement. Then G is reflexive, respectively, semireflexive iff G/V is reflexive, re is isomorphic to a product E × A spectively, semireflexive. The character group G where E is the additive group of a real vector space with its finest locally convex topology and A is the character group of an abelian pro-Lie group whose identity component is compact. Proof. By the Vector Group Splitting Theorem 5.20, G is isomorphic to V × H for a weakly complete vector group V and a pro-Lie group H with compact identity component H0 . The vector group V is reflexive by Proposition 5.33 and its dual is the additive group of a vector space with its finest locally convex vector space topology. The assertion now follows from Proposition 5.34.

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This result reduces the issue of duality of abelian pro-Lie groups to the class of abelian pro-Lie groups whose identity component is compact. Keep in mind for the following corollary that every connected topological group is almost connected. Thus it applies to all connected abelian pro-Lie groups. Theorem 5.36. Every almost connected abelian pro-Lie group is reflexive, and its character group is a direct sum of the additive topological group of a real vector space endowed with the finest locally convex topology and a discrete abelian group. Pontryagin duality establishes a contravariant functorial bijection between the categories of almost connected abelian pro-Lie groups and the full subcategory of the category of topological abelian groups containing all direct sums of vector groups with the finest locally convex topology and discrete abelian groups. Proof. By the Vector Group Splitting Theorem 5.20 according to which an abelian pro-Lie group is almost connected if and only if it is a direct product of a weakly complete vector group V and a compact group C. By Proposition 5.33, V is reflexive and its Pontryagin dual is a vector group obtained as the additive topological group of a real vector space which is given its finest locally convex topology. By the Pontryagin duality between the categories of compact abelian groups and discrete abelian groups (see for instance [102], notably Chapters 7 and 8), C is reflexive and its dual is a discrete abelian group. Then the preceding Proposition 5.35 proves the first assertion. The remaining duality statement is then automatic. Exercise E5.5. Prove the following result. Proposition. If G is an abelian pro-Lie group then the characters of G separate the   ηG (g)(χ ) = χ(g) is injective. points, and the canonical morphism ηG : G → G, [Hint. If g = 0, find N ∈ N (G) not containing g. Let q : G → G/N be the quotient morphism. On the abelian Lie group G/N find a character χ not annihilating q(g) = 0. Then χ  q is a character of G not annihilating g. The kernel of ηG consists of all g which are annihilated by all characters and this is singleton.] The following lemmas deal with not necessarily abelian pro-Lie groups. Lemma 5.37. Assume that G is a pro-Lie group. Then G0 is the intersection of open subgroups. Proof. By Proposition 3.31, G/G0 is protodiscrete. Hence the filter basis N (G/G0 ) consists of open normal subgroups and converges to the identity. If g ∈ G \ G0 , then gG0 = G0 and so there is a normal subgroup N of G such that N/G0 ∈ N (G/G0 ) not containing gG0 . But then N is an open normal subgroup not containing g. A pair (G, H ) consisting of a topological group G and a subgroup is said to have enough compact sets if for each compact subset K of G/H there is a compact subset C of G such that (CH )/H ⊇ K. (See [102, Definition 7.12 (ii)].)

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Lemma 5.38. Let G be a topological group and N a closed normal subgroup. (i) Assume that the quotient morphism G → G/N has local cross sections. Then for every N ∈ N (G), the pair (G, N ) has enough compact sets. (ii) If N ∈ N (G), then G → G/N has local cross sections provided G is a pro-Lie group or G is protodiscrete. Proof. (i) Assume that the quotient morphism qN : G → G/N has local cross sections. Then there is an open identity neighborhood U satisfying U N = U and a continuous function σ : U/N → G such that (qN |U )  σ : U/N → U/N is the identity. Let V be a compact identity neighborhood in the Lie group G/N which is contained in U/N and let V ◦ denote its interior.  Now let K be a compact subset of G/N . Then K ⊆ k∈K kV ◦ . Thus by the compactness of K we find elements k1 , . . . , kn ∈ K such that K ⊆ k1 V ∪ · · · ∪ kn V . −1 We pick elements gj ∈ qN (kj ), j = 1, . . . , n and set C = g1 σ (V ) ∪ · · · ∪ gn σ (V ). Then C is compact as a finite union of compact sets, and qN (C) =

n  j =1

qN (gj )qN σ (V ) =

n 

kj V ⊇ K.

j =1

(ii) If N ∈ N (G) and G is a pro-Lie group then qN has local cross sections by Theorem 4.22 (iv). If G is protodiscrete, and N ∈ N (G), then G/N is discrete; so local (and even global cross) sections exist trivially. Lemma 5.39. Let G be an abelian proto-Lie group and let N ∈ N (G). Assume that the pair (G, N) has enough compact sets. Then:  → N ⊥ , λG,N (χ )(g) = χ(g+N ), is an isomorphism (i) The morphism λG,N : G/N of topological groups. (ii) N ⊥ is a compactly generated locally compact abelian group. (iii) The group N ⊥ is compact if and only if N is open. Proof. (i) If the pair (G, N ) has enough compact subsets, the claim that λG,N is an isomorphism of topological groups follows at once from [102, Lemma 7.17 (i)]. (ii) Since G/N is an abelian Lie group, it is isomorphic to Rp × Tq × D for a  is discrete group D (see for instance [102, Corollary 7.58 (iii)]). Thus N ⊥ ∼ = G/N p q   isomorphic to R × Z × D, where D is a compact abelian group and every compact  for a discrete abelian group. Now Rp × D  is an almost abelian group occurs as D connected locally compact abelian group, and every almost connected locally compact abelian group is of this form by the Vector Group Splitting Lemma. (iii) follows from the fact that G/N is discrete iff N is open and the facts that the character group of a discrete group is compact and vice versa.  its charProposition 5.40. Assume that G is an abelian proto-Lie group and A = G acter group. Then:

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(i) A is the directed union of the family of locally compact subgroups N ⊥ , N ∈ N (G), each of which is algebraically generated by a compact subset. (ii) If G is protodiscrete then A = comp(A). Proof. (i) Let U be a zero-neighborhood of T in which every subgroup is singleton. If χ : G → T is a character of an abelian pro-Lie group G, then all subgroups contained in χ −1 (U ) are annihilated. Since lim N (G) = 0, there is an N that  ∈ N (G) such ⊥ = χ (N ) ⊆ U and thus χ (N) = {0}, that is, χ ∈ N ⊥ . Thus G N ∈N (G) N . Since N (G) = {N : N ∈ N (G)} is a filter basis, {N ⊥ : N ∈ N (G)} is an upwards directed family of closed subgroups of A. By Lemma 5.39, every N ⊥ is a locally compact and compactly generated subgroup. (ii)A pro-Lie group G is totally disconnected iff it is prodiscrete by Proposition 4.23. If this condition is satisfied, then N (G) is a basis for the filter of identity neighborhoods. Thus by 5.39 (iii), every N ⊥ is compact. Hence by (i), we get (ii). def

The character group A = (ZN ) of a power of copies of Z (see Example 5.1 (ii)(b)) is a sum of copies of the circle group T with the compact open topology; it is an arcwise connected topological abelian group. From Theorem 5.20 we know connected abelian pro-Lie groups are isomorphic to V × C for a weakly complete vector group V and a compact connected group C, and thus we can say without even detailing the topology on A that this group is not a pro-Lie group. It does, however, satisfy A = comp(A).  its dual. Then Proposition 5.41. Let G be an abelian pro-Lie group and A = G G⊥ = comp(A). 0 0 is Proof. (i) By 3.31, G/G0 is protodiscrete, and thus by Proposition 5.40 (ii), G/G ⊥  the union of its compact subgroups. Now λ : G/G0 → G0 , λ((χ )(g)) = χ(g + G0 ) is a bijective morphism of topological abelian groups. Since a bijective morphism of topological groups maps each compact subgroup isomorphically onto its image, G⊥ 0 is ⊆ comp(A). a union of compact subgroups. Thus G⊥ 0 ⊥ ⊥  (ii) κ : A/G⊥ 0 → G0 , κ(χ +G0 ) = χ (h) maps A/G0 continuously and bijectively   onto A|G0 ⊆ G0 . But by Theorem 5.36 we know that G0 ∼ = E ×D where E is a vector space with its finest locally convex vector space topology, and D, as the character group of a compact connected abelian group is a discrete and torsion-free abelian group (see 0 ) is singleton. Therefore A|G0 and thus A/G⊥ [102, Corollary 8.5]). Hence comp(G 0 do not contain any nondegenerate compact subgroups. It follows that comp(A) ⊆ G⊥ 0. . Therefore comp(A) = G⊥ 0

The Toral Homomorphic Images of an Abelian Pro-Lie Group From the duality of compact abelian groups we cite some results which we shall immediately apply to abelian pro-Lie groups. Let C denote a compact abelian group.

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 contains a free subgroup of maximal rank such Then the (discrete) character group C  that C/F is a torsion group. The subgroup F is not unique. Indeed, one way of get → Q ⊗ C,  f (x) = 1 ⊗ x, ting F is as follows. We consider the morphism f : C where the tensor product is taken over the ring of integers. Then ker f is exactly  of C,  and the tensor product Q ⊗ C  is a rational vector the torsion subgroup tor C   contains a space which is spanned over Q by f (C). Thus by Zorn’s Lemma, f (C)   Q-basis B, whose Z-span in Q ⊗ C is a free subgroup B whose rank is dimQ (Q ⊗ C), def  Let A = f −1 (B) be the full inverse image of this the torsion-free rank of C. ∼ free abelian group. Then A/ tor C = B is a free abelian group. Since free groups  ⊕ F . Then are projective, there is a free subgroup F of A such that A = tor C f |F : F → B is an isomorphism. The factor group A/F is torsion as is the group ∼   is torsion.   which is a torsion group. Thus C/F C/A ⊆ (Q ⊗ C)/B = f (C)/B def  The character group T = F is a torus, that is, a product of circles, and the dual of the   gives a surjective morphism f: C  → T by duality, inclusion morphism j : F → C  ∼ and C = C. Hence we have a surjective morphism of compact groups q : C → T such  R) → Hom(T, R) is an isomorphism, since C → Q⊗R that Hom( q , T) : Hom(C,  and T → Q ⊗ T induce isomorphisms  R) → Hom(C,  R), Hom(Q ⊗ C, Hom(Q ⊗ T , R) → Hom(T , R). Since L(C) = Hom(R, C), respectively, L(T ) = Hom(R, T ), is isomorphic to  R), respectively, Hom(T , R), we conclude that L(q) : L(C) → L(T ) is an Hom(C, def

isomorphism. This is also seen from the fact that D = ker q is the character group of  , a torsion group and is, therefore, a totally disconnected compact abelian group. C/F Thus the exact sequence q 0 → D → C −−→ T → 0 gives rise to an exact sequence L(q)

0 → L(D) = {0} → L(C) −−−→ L(T ) → 0. For further details see [102, Chapter 8, Definition 8.12–8.20]. In particular, it is shown in [102, Corollary 8.18], that for a connected compact abelian group G, the subgroups D may be chosen arbitrarily small. These considerations motivate Definition 5.42. Let G be an abelian topological group. A closed subgroup D of G will be called a cotorus subgroup if G/D is a torus and the quotient morphism q : G → G/D induces an isomorphism L(q) : L(G) → L(C/D) of Lie algebras. Any group isomorphic to G/D for a cotorus subgroup D of G will be called a toral homomorphic image of G We noted in the discussion preceding this definitions that, in a compact abelian group, cotorus subgroups always exist, and that they are zero-dimensional. We will now show that an abelian pro-Lie group G always has cotorus subgroups, and indeed arbitrarily small ones if G is connected.

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243

Proposition 5.43. An abelian pro-Lie group G always has at least one cotorus subgroup D, and D is prodiscrete. If G is connected, then D may be chosen arbitrarily small. Proof. By Theorem 5.20, G has a unique largest compact connected subgroup comp G0 = (comp G)0 . Let D1 be a cotorus subgroup of comp G0 . Set G1 = G/D1 . Then (G1 )0 ∼ = R I × comp(G1 )0 ∼ = RI × TJ for suitable sets I and J , I ∪J ∼ and L(G) = R . Now we apply Theorem 5.19 and deduce that G1 is the direct product of (G1 )0 and G1 /(G1 )0 ∼ = G/G0 . Thus we obtain a quotient morphism q : G → RI × TJ × G/G0 . Now we set D = q −1 (ZI × {0} × G/G0 ). Then G ∼ R I × TJ × G/G0 ∼ I = T × TJ . = I D Z × {0} × G/G0 Moreover D/D1 ∼ = ZI × (G/G0 ). That is, we have an exact sequence 0 → D1 → G → ZI × (G/G0 ) → 0, yielding an exact sequence 0 → L(D1 ) → L(G) → L(Z)I × L(G/G0 ) → 0 by Theorem 4.20 and the limit hence product preservation of L. Now L(D1 ) = {0} and L(G/G0 ) = {0} by Proposition 3.30. It follows that L(D) = {0} and that therefore, in view of 4.20, L(G) → L(G/D) is an isomorphism. So D is a cotorus subgroup of G. If G is connected, we may take D = D1 and since D1 may be taken arbitrarily small by [102, Corollary 8.18], we are finished. Corollary 5.44. If G is an abelian pro-Lie group then there is a quotient morphism q : G → TK for some set K such that L(q) is an isomorphism and there is a commutative diagram L(q) L(G) −−−→ R⏐K ⏐ ⏐exp K expG ⏐   T G −−−→ TK . q

Proof. This is just a reformulation of Proposition 5.43. Later, in Chapter 6 and Chapter 8 we shall see that the additive group of L(G)  the simply connected universal group of G. In [102, Chapter 9] it is is written as G, discussed that the fundamental group π1 (C) of a compact group is naturally isomorphic  Z). Thus there are many reflexive topological abelian groups G (definition to Hom(C,  R) agrees with in this chapter in the discussion preceding 5.33) for which Hom(G,  = Hom(C,  R) ∼ C = Rdim C where dim C is the cardinal representing the torsion for which  Among these there are those like G = C  or (Q ⊗ C) free rank of C. (dim C) ∼  π1 (G) = {0} while the torus homomorphic image T = F = (Z ) ∼ = Tdim C has

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5 Abelian Pro-Lie Groups

the largest possible fundamental group π1 (T ) = Hom(T, Z) ∼ = Hom(Z(dim C) , Z) = dim C dim C ∼ Hom(Z, Z) . =Z To some extent this can be generalized to an arbitrary (not necessarily commutative) pro-Lie group G by considering cotorus subgroups D of the center Z(G). Proposition 5.45. Let G be a pro-Lie group and D a cotorus subgroup of its center. def Then there is a quotient map q : G → H = G/D such that L(q) : L(G) → L(H ) is an isomorphism and Z(H )0 is a torus. Moreover, if G is connected, Z(H ) itself is connected and is a torus, and D may be chosen arbitrarily small. Proof. By Theorem 4.20 we have an exact sequence L(q)

0 → L(D) → L(G) −−−→ L(H ) → 0. Since D is prodiscrete by Proposition 5.43, we conclude L(D) = {0} (see 4.23). Since bijective morphisms between weakly complete topological vector spaces are isomorphisms, we infer that L(q) is an isomorphism. Now let G be connected; consider the center Z(H ) of H and its full inverse image def

A = q −1 (Z(H )). Since Z(H ) is the center of H , we have comm(G × A) ⊆ D. Now comm(G × {x}) is connected for each x and contains 1. But D is totally disconnected, and thus comm(G × A) = {1}, that is, A ⊆ Z(G). Therefore Z(H ) = Z(G)/D = Z(H )0 since D is a cotorus subgroup of Z(G). Then, by Proposition 5.43, D may be chosen arbitrarily small. The main purpose of the cotorus subgroups, however, is that they permit us to prove for pro-Lie groups the existence of good substitutes for universal covering morphisms of Lie groups. The Resolution Theorem of Abelian Pro-Lie Groups Theorem 5.46. Let G be an abelian pro-Lie group and write G = V × H with a vector group complement V according to the Vector Group Splitting Theorem for Abelian ProLie Groups 5.20. Let D be a cotorus subgroup of H . Then D is a prodiscrete group and the morphism δ : L(G) × D → G,

δ(X, x) = (expG X)x,

is a quotient morphism inducing an isomorphism of Lie algebras and having the prodiscrete kernel {(X, exp −x) : exp X ∈ D} ∼ = exp−1 D ∼ = exp−1 (D ∩ comp(G0 )). Proof. Since L(G) = L(V ) × L(H ) and expV : L(V ) → V is an isomorphism, the assertion on G it true if it is true on H . Thus without loss of generality we now assume that V = {0}. Then G0 = (comp G)0 is compact and G = G0 D with a compact def

intersection  = D ∩ G0 , so that the morphism ϕ : G0 × D → G,

ϕ(g0 , d) = g0 d

The Toral Homomorphic Images of an Abelian Pro-Lie Group

245

is a quotient morphism with kernel {(−d, d) : d ∈ } ∼ = . The compact subgroup  is a cotorus subgroup of G0 . So the morphism μ : L(G) ×  → G0 , μ(X, x) = (exp X)x, is a quotient morphism by the Resolution Theorem for Compact Groups ([102, Theorem 8.20]). So we conclude that ν : L(G) ×  × D → G,

ν(X, d1 , d) = (expG X)d1 d

which essentially is the composition ϕ  ν of two quotient morphisms is itself a quotient morphism. Since  is compact, the map κ :  × D → D,

κ(d1 , d) = d1 d

is a quotient morphism. So idL(G) ×κ : L(G) ×  × D → L(G) × D is a quotient map. Now ν = δ  (idL(G) ×κ) and under these circumstances, δ is a quotient morphism. So by Theorem 4.20, L(δ) is a quotient morphism. The assertion about ker δ is straightforward; so ker δ is totally disconnected, and thus L(ker δ) = {0} by 4.23. Hence L(δ) is an isomorphism. A quotient morphism with a prodiscrete kernel is the next best thing we can have to a covering morphism. We have two quotient morphisms with a totally disconnected kernel at work, permitting us to formulate the following “Sandwich Theorem”. The Sandwich Theorem for Abelian Pro-Lie Groups Corollary 5.47. Let G and D be as in Theorem 5.46 and write  = D ∩ comp(G)0 . Then there are quotient maps with totally disconnected kernels δ

ρ

L(G) × D −−→G−−→

G0 D ×  

such that G0 / ∼ = RI × TJ for suitable sets I and J where L(G) ∼ = RI ∪J and that the composition D G0 × ρ  δ : L(G) × D →   is the obvious map, namely, the product of the exponential function L(G) → G0 / and the quotient morphism D → D/. Proof. The information on δ was established in Theorem 5.46, and the information on ρ in Proposition 5.43 and its proof.

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Postscript By contrast with the category of locally compact abelian groups, the larger category AbproLieGr of abelian pro-Lie groups is closed in the category of all abelian topological groups and continuous group morphisms under the formation of arbitrary limits and the passing to closed subgroups. Each locally compact abelian group is a proLie group and thus arbitrary products of locally compact abelian groups and arbitrary closed subgroups of such products are abelian pro-Lie groups. Therefore, the category AbproLieGr of abelian pro-Lie groups is rather large. In the first section we recorded some individual examples so that the reader can form a first impression of the type of topological abelian groups we face. This addresses in particular those readers whose intuition is trained by studying locally compact abelian groups which are the daily bread of classical harmonic analysis and whose structure was exposed in [102], Chapters 7 and 8. The most remarkable of the prodiscrete examples presented here is the example of the free abelian group on countably infinitely many generators which supports a nonmetric totally disconnected pro-Lie group structure with bizarre properties (Proposition 5.1). One of the most prominent examples of connected abelian pro-Lie groups is the additive topological group of the dual vector space of an arbitrary real vector space, where the dual E  = HomR (E, R) of a real vector space is given the topology of pointwise convergence; this topology is also called the weak ∗-topology. It makes the dual into a complete topological vector space which is called a weakly complete topological vector space. Therefore its additive topological group is called a weakly complete vector group, and whenever it occurs as a subgroup of a topological group, it is called a weakly complete vector subgroup. Weakly complete vector groups have a perfect duality theory as is discussed in Appendix 2. The character group and the vector spaces dual of a topological vector space are naturally isomorphic. The compact open topology on the dual of a weakly complete vector space is the finest locally convex topology on the vector space dual. Since the character group of a weakly complete vector group is a vector space and thus is a direct sum of copies of R, every weakly complete vector group is isomorphic to a vector group of the form RJ for some set J . Thus a weakly complete vector group is locally compact if and only if it is finite-dimensional. The transfinite topological dimension of RJ as discussed in [103] is card J . Therefore the topological dimension of a weakly complete vector group agrees with the linear dimension of its dual. Pontryagin duality establishes a dual equivalence between the category of compact abelian groups and the category of (discrete) abelian groups. Therefore an abelian pro-Lie group is a direct product of a weakly complete vector group and a compact abelian group if and only if its dual is a direct sum of a real vector space given its finest locally convex topology, and some discrete abelian group. The main result of this chapter is that every abelian pro-Lie group G is isomorphic to the direct product W ×H of a weakly complete vector subgroup W of G and an abelian pro-Lie group H whose identity component H0 is compact and is the intersection of

Postscript

247

open subgroups. In particular, a connected abelian pro-Lie group is the direct product of a vector group and a compact group. Thus for such abelian pro-Lie groups the structure theory is completely reduced to the theory of weakly complete topological vector spaces on the one hand and to the familiar compact group situation on the other hand. From this result we derive that a necessary and sufficient condition for a connected abelian pro-Lie group to be locally compact is that it is algebraically generated by a compact set. In a somewhat loose way one could describe the situation by saying that abelian pro-Lie groups are very well understood if they are connected (or even not far from being connected) while the area of totally disconnected, that is, prodiscrete abelian groups, is a wide open field. We have observed that almost connected abelian pro-Lie groups are reflexive and have a lucid duality theory which is completely known due to the known vector space duality of weakly complete topological vector spaces and the duality of compact abelian groups. The Lie theory of abelian pro-Lie groups is likewise lucid: Since L(G) = L(G0 ) for any topological abelian group, it deals with the connected part of G only, and if G0 is an abelian pro-Lie group then the Vector Group Splitting Theorem says that G0 is algebraically and topologically the direct sum G0 = V ⊕ comp(G) of a weakly complete vector subgroup complement V and the unique largest compact subgroup comp(G); hence L(G) = L(V ) ⊕ L(comp(G), where expV : L(V ) → V is an isomorphism of a weakly complete vector group and where the exponential function expcomp(G) : L(comp(G)) → comp(G) is very well understood from [102], Chapters 7 and 8. From the theory of the exponential function of compact groups we know that, except for the case of a torus group G, the exponential function expG is rarely if ever surjective. In [102, Chapter 8, Theorem 8.20] it was shown in the so called “Resolution Theorem for Compact Abelian Groups” that this defect could be compensated for by finding a suitable totally disconnected closed subgroup D and a quotient morphism δ : L(G) × D → G such that L(δ) is an isomorphism of Lie algebras. It is noteworthy, that the Resolution Theorem remains intact for arbitrary abelian pro-Lie groups. In the end we have a sandwich theorem of the following nature: There is a totally disconnected subgroup D intersecting G0 in a compact totally disconnected subgroup , and there are quotient morphisms G D ×   inducing isomorphisms on the Lie algebra level such that the composition is the product of the exponential function δ

ρ

L(G) × D −−→ G −−→

expG

−−−→ L(G) ⏐ ⏐ ∼ = RI ∪J

G0  ⏐ ⏐∼ =

−−−→ RI × TJ quot

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5 Abelian Pro-Lie Groups

and the quotient map D → D/. These results highlight once more the role of weakly complete topological vector spaces and compact groups play in the structure theory of abelian pro-Lie groups. We shall see more resolution theorems in the nonabelian environment in Chapter 12.

Chapter 6

Lie’s Third Fundamental Theorem

The concept of a prosimply connected pro-Lie group is introduced. It coincides with the purely topological concept of a simply connected topological group when the proLie group is in fact a Lie group. It is proved that for every pro-Lie algebra there is a prosimply connected pro-Lie group whose Lie algebra it is, and this prosimply connected pro-Lie group is unique up to isomorphism. In fact, all of this is done functorially and the universal properties arising in this way are studied. Prerequisites. We need the general theory of pro-Lie groups such as presented in Chapters 3 and 4. We also require from the reader a certain familiarity with basic category theory; the concept of an adjoint functor which was used extensively in Chapter 2 will be used essentially here as well.

Lie’s Third Fundamental Theorem for Pro-Lie Groups Consider a pro-Lie group G and the filter basis N (G) of closed normal subgroups N such that G/N is a Lie group. Then G ∈ N (G) and the singleton G/G is a simply connected Lie group. Let us define def

N S(G) = {N ∈ N (G) : G/N is simply connected}. Recall from Chapter 1, Definition 1.12ff. that a subset C of a directed set is cofinal if for each d ∈ D there is a c ∈ C such that d ≤ c. Definition 6.1. A pro-Lie group G is called prosimply connected if N S(G) is cofinal in N (G). We notice that for a prosimply connected pro-Lie group G, the set N S(G) is a filter basis and that G = limN ∈N S(G) G/N by the Cofinality Lemma 1.13 (or 1.21). From Corollary 4.22 (iii) it follows that a prosimply connected pro-Lie group is connected. A pro-Lie group G is a (finite-dimensional) Lie group if and only if {1} ∈ N (G). Accordingly, a Lie group is prosimply connected if and only if it is simply connected. For the simply connected Lie group R, the filter basis N (R) consists of R itself and all cyclic subgroups including {0}. For {0}, R = N ∈ N (R) the quotient R/N is a circle group and is therefore not simply connected. Therefore it would not have been a good idea, in Definition 6.1 to postulate that G/N is simply connected for all N ∈ N (G). Simple connectivity may be defined in several reasonable but nonequivalent ways, including firstly the way it was done in [102, Definition A2.6]. According to this

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6 Lie’s Third Fundamental Theorem

definition, a space X is called simply connected if it has the following universal property: For any covering map p : E → B between topological spaces, any point e0 ∈ E and any continuous function f : X → B with p(e0 ) = f (x0 ) for some x0 ∈ X there is a continuous map f: X → E such that p  f = f and f(x0 ) = e0 . Let us say that a space is loopwise simply connected if it is connected and π1 (X) = 0. (See e.g. [102, Definition 8.60.]) Examples 6.2. (i) Any product of simply connected finite-dimensional Lie groups is simply connected in any sense and is also a prosimply connected pro-Lie group. (ii) Let G be a compact connected abelian group; then G is a pro-Lie group (see [102, Corollary 2.43]). Every Lie group quotient G/N , N ∈ N (G) is a torus. Thus G is never prosimply connected.  Z). Thus if G is the 1-dimensional By [102, Theorem 8.62], π1 (G) ∼ = Hom(G, ∼ , then G solenoid Q = Q and π1 (G) = {0}, that is, G is loopwise simply connected. However, no compact connected abelian group is simply connected (see [102, Theorem 9.29]). While the definition of prosimple connectedness of a pro-Lie group G does not agree with any of the traditional concepts of simple connectivity, we shall show in Corollary 8.15 in a later chapter that it does agree with the concept of simple connectivity expressed in terms of the lifting property described in the paragraph preceding 6.2 above (see e.g. [102, Definition A2.6]), and the one expressed by the vanishing of the fundamental group π1 (G). Proposition 6.3. Let G be a pro-Lie group. (a) Assume N ∈ N (G). Consider the following two statements: (i) N ∈ N S(G). (ii) N is connected. Then (i) ⇒ (ii), and if G is prosimply connected, both statements are equivalent. (b) Assume that N (G) contains a cofinal subset C of connected subgroups N of G. Then G is locally connected. (c) Any prosimply connected pro-Lie group is locally connected. Proof. We begin by proving (a). (i) ⇒ (ii): Let N ∈ N S(G); then G/N is a simply connected Lie group. Since the identity component N0 of N is characteristic and closed, it is normal in G, and G/N0 is a pro-Lie group by the Quotient Theorem for Pro-Lie Groups Revisited 4.28 (i). Now N/N0 is a pro-Lie group as a closed subgroup of G/N0 by the Closed Subgroup Theorem 3.35. Since N/N0 is a totally disconnected normal subgroup of a connected group, it is central by Lemma 12.55. Thus N/N0 is a totally disconnected abelian proLie group and so is prodiscrete by Proposition 4.23. Hence there is an open subgroup M of N such that M/N0 , being central in G/N0 , is normal in G/N0 , whence M is normal in G. The quotient morphism q : G/M → G/N has the discrete kernel N/M and therefore is a covering morphism. Since G/N is simply connected, q is an

Lie’s Third Fundamental Theorem for Pro-Lie Groups

251

isomorphism, that is M = N. Therefore, the prodiscrete group N/N0 has no open normal subgroups and thus is singleton. Hence N = N0 which is what we had to show. (ii) ⇒ (i): Assume that G is prosimply connected and that N is connected. Since N S(G) is cofinal in N (G), there is an M such that M ⊆ N. Thus there is a quotient morphism q : G/M → G/N. Now assume that ϕ : (G/N ) → G/N is the universal covering morphism of connected Lie groups. Since G/M is simply connected, by the lifting property, there is a morphism f : G/M → (G/N ) such that ϕ  f = q. Let qM : G → G/M denote the quotient morphism. Since ϕ is a covering morphism and f is the lifting of a quotient morphism, f and thus f  qM are quotient morphism. Also (G/N), being the universal covering of the Lie group G/N , is a Lie group. Thus def N1 = ker(f  qN ) ∈ N (G) and N1 ⊆ N. Also, N/N1 ∼ = ker ϕ, and the kernel ker ϕ of the universal covering of G/N is discrete. Hence N1 is open in the connected group N, whence N1 = N. Thus ϕ is an isomorphism and so G/N is a simply connected Lie group, that is, N ∈ N S(G). (b) Assume the existence of C as stated in (b). We have to show that G has arbitrarily small connected identity neighborhoods. Indeed G has arbitrarily small identity neighborhoods V containing an N ∈ C such that V N = V and that V /N is an open cell neighborhood of the identity in the Lie group G/N . Let W be any nonempty open closed subset of V . Since N is connected, for all w ∈ W we get wN ⊆ W . Hence W N = W . Then W/N is nonempty open closed in the open cell V /N and thus W = V . So V is connected, and (b) is proved. (c) is now an immediate consequence of (a) and (b). If G = SO(3)N , then the members of N (G) are precisely the cofinite partial products of G, and all of these are connected. But N S(G) = {G}. One notices directly, that G is locally connected. At any rate, for any prosimply connected pro-Lie group G, by Proposition 6.3, all members of N S(G) are connected. The occurrence of prosimply connected pro-Lie groups is not a rare event. Indeed, Theorem 6.4 below asserts the existence of a plethora of prosimply connected pro-Lie groups. Subsequent results will place the construction into a more functorial frame. But first we recall a basic result of Lie group theory utilizing the universal property of simple connectivity: Lemma L. Let S be a simply connected Lie group and L a Lie group, and assume that there is a morphism α : L(S) → L(L). Then there is a unique morphism of Lie groups a : S → L such that L(a) = α. The following diagram commutes: α

L(S) −−→ L(L) ⏐ ⏐ ⏐exp expS ⏐   L S −−→ L.

(SC)

a

Proof (Indication). The proof is standard Lie theory: Let B be a convex zero neighborhood of L(S) such that expS induces a homeomorphism expS |B : B → V onto an identity neighborhood of S. Then expL  α  (expS |B)−1 : B → L is a local morphism

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6 Lie’s Third Fundamental Theorem

which by the simple connectivity of S extends uniquely to a morphism a : S → L such that the diagram (SC) commutes (see e.g. [102, Corollary A2.26]; note that U is assumed to be connected there). This completes the indication of proof of the lemma. Theorem 6.4 (Lie’s Third Fundamental Theorem for Pro-Lie Algebras). Let g be a pro-Lie algebra. Then there is a prosimply connected pro-Lie group (g) for which there is an isomorphism ηg : g → L((g)). Proof. For each i ∈  (g) there is (up to isomorphism) a unique simply connected Lie group Gi such that there is an isomorphism ϕi : g/i → L(Gi ). Since all Gi are simply connected, for i ⊇ j in  (g) by Lemma L, the quotient map quotij : g/j → g/i induces a unique surjective morphism fij : Gj → Gi such that the following diagram commutes: quotij

←−−−−− g/j g/i ⏐ ⏐ ⏐ ⏐ϕ ϕi   j L(G −−−− L(G ⏐ i ) ←− ⏐ j) L(fij ) ⏐expG expG ⏐  i j Gi ←−−−−− Gj . fij

Now {fij : Gj → Gi | (i, j) ∈  (g) ×  (g), j ⊇ i} def

is a projective system of finite-dimensional Lie groups. Let (g) = limi∈ (g) Gi be its limit and let fi : (g) → Gi be the limit maps. We claim that (g) is a prodef

Lie group. Indeed the filter basis N S(G) = {ker fi | i ∈  (g)} converges to 1 by Theorem 1.27 (ii), and (g) is complete as the limit of complete groups, whence γGN S(G) : (g) → GN S(G) is an isomorphism by 1.30 and 1.33, inverting the natural isomorphism η : GN S(G) → (g) of 1.27 (ii). For each i ∈  (g), the following diagram commutes: η / (g) GN S(G) MMM MMM MMMfi νker fi fi MMM MMM M &   / Gj , (g)/ ker fi  fi

where η = γG−1 . The filter basis N S(G) is cofinal in N ((g)). The fi are quotient N S(G) morphisms by 1.27 (iii). Therefore each fi : (g)/ ker fi → Gi is an isomorphism of finite-dimensional Lie groups. Thus all quotients (g)/ ker fi are simply connected. Now GN is a pro-Lie group by 1.29 and 1.40. Thus (g) is a pro-Lie group as claimed. By Definition 6.1 and what we showed above, it is simply connected. Since L is a continuous functor by 2.25 (ii) we have an isomorphism α : L((g)) → limi∈ (g) L(Gi ) and thus we obtain an isomorphism ηg as a composition

Lie’s Third Fundamental Theorem for Pro-Lie Groups γg

253

α −1

limi∈ (g) ϕi

g −−→ limi∈ (g) g/i −−−−−−→ limi∈ (g) L(Gi ) −−→ L(limi∈ (g) Gi ) = L((g)). This completes the proof. Exercise E6.1. Prove the following assertion: Let g be an abelian pro-Lie algebra, that is, a weakly complete topological vector space; then the additive group (g) of g is a simply connected pro-Lie group such that L((g)) = Hom(R, (g)) ∼ = g. This theorem has a functorial enhancement of considerable elegance. We have to refer to the concept of adjoint functors. (See e.g. [102, Appendix 2, Definition A3.29].) Lie’s Third Fundamental Theorem, Functorial Version Theorem 6.5. The construction  of Theorem 6.4 assigning to a pro-Lie algebra a prosimply connected pro-Lie group extends to a functor  : proLieAlg → proLieGr which is left adjoint to the Lie algebra functor L : proLieGr → proLieAlg. Proof. By TheoremA3.28 of [102] we have to verify the universal property best pursued by the following diagram. proLieAlg ηg

proLieGr

g ⏐ ⏐ ∀f 

−−−−→

L((g)) ⏐ ⏐  L(f )

L(H )

−−−−→

L(H )

idL(H )

(g) ⏐ ⏐  ∃!f

()

H

For each morphism of topological Lie algebras f : g → L(H ) for a pro-Lie group H we have to find a unique morphism of topological groups f  : (g) → H such that L(f  )  ηg = f . Thus assume we are given f : g → L(H ). For each N ∈ N (H ) let qN : H → H /N be the quotient map. Then incl

L(qN )

0 → L(N ) −−−−→ L(H ) −−−−→ L(H /N ) is exact since L preserves kernels (see e.g. 3.22). Since dim L(H /N ) < ∞ we have def

L(N) ∈  (L)(H ). Then i(N ) = f −1 (L(N )) ∈  (g) and f induces a morphism of topological Lie algebras fN : g/i(N ) → L(G/N ). By definition, (g) = limi∈ (g) Gi for a projective system of simply connected Lie groups Gi such that there is an isomorphism ϕi : g/i → L(Gi ). Since Gi(N ) is simply connected there is a unique morphism of topological groups gN : Gi(N ) → H /N such that the following diagram is commutative: −1 ϕi(N) fN L(G⏐i(N ) ) −−−−→ g/i(N ) −−−−→ L(H⏐/N ) ⏐ ⏐exp expG  H /N i(N)  Gi(N ) −−−−→ Gi(N ) −−−−→ H /N; idGi(N)

gN

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6 Lie’s Third Fundamental Theorem

−1 consequently, L(gN ) = ϕi(N )  fN . Let

pi : αi : βi :

(g) −→ Gi , limj∈ (g) g/j −→ g/i, limj∈ (g) L(Gj ) −→ L(Gi )

be the limit morphisms. Then we have, in particular, a commutative diagram in which lim is short for limi∈ (g) : λg

limi ϕi

L(qN )

id

g −−−−→ lim −−−−→ lim L(G g ⏐ ⏐i ⏐ i ) ←−−− ⏐ ⏐β αi(N) ⏐ =   i(N) id g g − − − − → − − − − → L(G i(N ⏐ ⏐) ⏐i(N ) ) ←−−− ϕi(N) quot ⏐ ⏐ ⏐L(g ) f fN   i(N) id L(H ) −−−−→ L(H /N ) −−−−→ L(H /N ) ←−−− α

L((g)) ⏐ ⏐L(p (N ))  i L(G⏐i(N ) ) ⏐L(g )  i(N) L(H /N ).

def

For each N ∈ N (H ) we set fN = gN pi(N ) : (g) → H /N. The map α −1 (limi ϕi )λg from the top left corner of the diagram to the top right corner is ηg by the definition of ηg in the proof of 6.4. Thus the outside of the diagram yields L(fN )  ηg = L(qn )  f.

(1)

If N1 ⊇ N2 in N (H ) we straightforwardly see that fN 2 = qN1 N2 fN 1 with the obvious quotient morphism qN1 N2 : H /N2 → H /N1 . Then the universal property of the limit H = limN∈N (H ) H /N attaches to the cone {fN : L((g)) → L(H /N ) | N ∈ N (H )} a unique morphism f  : (g) → H such that qN  f  = fN for all N ∈ N (H ) and thus L(qN )  L(f  ) = L(fN ). (2) So from (1) and (2) we have L(qN )L(f  )ηg = L(qN )f for all N ∈ N (H ). Since L is limit preserving, the L(qN ) form a limit cone and thus separate the points. Therefore L(f  )  ηg = f. The question of the uniqueness of f  remains. Assume also that f  : (g) → H satisfies L(f  )  ηg = f = L(f  )  ηg . Since ηg is an epic (being even an isomorphism) this means L(f  ) = L(f  ). Now consider the two superimposed commutative diagrams L(f  )

L((g)) ⏐ ⏐ exp(g) ⏐  (g)

−−−−→ −−−−→  L(f ) f

−−−−→ −−−− →  f

L(H ⏐ ) ⏐ ⏐expH  H

and conclude f   exp(g) = f   exp(g) . Since the exponential function of a pro-Lie group is generating, we conclude f  = f  , and this completes the proof.

Lie’s Third Fundamental Theorem for Pro-Lie Groups

255

From purely category theoretical considerations, we obtain the following version of the universal properties of the adjoint functors L and  (see e.g. [102, Appendix 3, Proposition A3.36]). There is a natural morphism πG : (L(G)) → G of pro-Lie groups with the following universal property: Given a pro-Lie group G and any morphism f : (h) → G for some pro-Lie algebra h, there is a unique morphism f  : h → L(G) of pro-Lie algebras such that f = πG  (f  ). proLieAlg

proLieGr πG

L(G)  ⏐ ∃!f  ⏐

(L(G))  ⏐  ⏐(f )

−−−−→

G  ⏐∀f ⏐

h

(h)

−−−−→

(h).

id(h)

(⊥)

However, with the information from Chapter 4 that L preserves quotient morphisms it serves a useful purpose if we elaborate on this situation in the concrete case of pro-Lie groups. The Theorem on the Reflection into the Category of Prosimply Connected Pro-Lie Groups Theorem 6.6. (i) Let G be a pro-Lie group. Then there is a prosimply connected  and a pro-open morphism pro-Lie group G  −−−→ G πG : G which induces an isomorphism of pro-Lie algebras ∼ =

 −−−→ L(G). L(πG ) : L(G)  extends to a functor proLieGr → proLieGr onto the (ii) The assignment G  → G full subcategory proSimpConLieGr of prosimply connected pro-Lie groups, which is right adjoint to the inclusion functor proSimpConLieGr → proLieGr. In particular,  → G is a natural transformation with a prodiscrete kernel P (G) def = ker πG . πG : G (iii) The following statements (a) and (b) are equivalent:  → G is an isomorphism. (a) πG : G (b) G is prosimply connected. (iv) The image of πG is dense in G0 . (v) Let f : G → H be a morphism of connected pro-Lie groups such that L(f ) is an isomorphism and H is prosimply connected. Then f is an isomorphism.

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6 Lie’s Third Fundamental Theorem

(vi) By a slight abuse of notation denote the corestriction of the functor  to the category of prosimply connected pro-Lie groups by  : proLieAlg → proSimpConLieGr and the restriction of the functor L to the category of prosimply connected pro-Lie groups by L : proSimpConLieGr → proLieAlg. Then the two functors  : proLieAlg → proSimpConLieGr,

L : proSimpConLieGr → proLieAlg

implement an equivalence of categories. (vii) If f : G → H is a bijective morphism of connected pro-Lie groups, then  → H  is L(f ) : L(G) → L(H ) is an isomorphism of pro-Lie algebras and f: G an isomorphism of prosimply connected pro-Lie algebras. In particular, a bijective morphism of prosimply connected pro-Lie groups is an isomorphism.  = (g) be the pro-Lie Proof. (i) Set g = L(G); then g is a pro-Lie algebra. We let G  group constructed in Theorem 6.1; then there is an isomorphism ηg : L(G) → L(G).  We construct πG : G → G. Let M ∈ N (G) and note that the quotient morphism qM : G → G/M induces an exact sequence L(qM )

incl

0 → L(M) −−−−→ L(G) = g −−−−→ L(G/M) → 0 ∼ limi∈ (g) g/i and L(qM ) induces is exact by Corollary 4.16. Hence L(M) ∈  (g) = †  = limi∈ (g) Gi for a family of an isomorphism qM : g/L(M) → L(G/M). Recall G simply connected finite-dimensional Lie groups Gi with an isomorphism ϕi : g/i → L(Gi ). In particular, there is an isomorphism ϕL(M) : g/L(M) → L(GL(M) ). Then † −1 qM  ϕL(M) : L(GL(M) ) → L(G/M) is an isomorphism. Hence the two Lie groups GL(M) and G/M are locally isomorphic (see e.g. [102, Theorem 5.42]; this theorem does not depend on the linearity of the Lie groups involved); thus there is an open connected identity neighborhood U of GL(M) , an open identity neighborhood V of G/M, and a homeomorphism μ0 : U → V satisfying μ0 (uu ) = μ0 (u)μ0 (u ) for u, u , uu ∈ U . Since GL(M) is a finite-dimensional simply connected Lie group, there is a unique morphism μM : GL(M) → G/M extending μ0 and implementing a local isomorphism (see e.g. [102, Corollary A2.26], as in the proof of Lemma L used in the proof of Theorem 6.4); the following diagram is commutative: L(G ⏐ i) ⏐ expG L(M)  GL(M)

L(μM )

−−−−→ L(G/M) ⏐ ⏐exp  G/M −−−−→ G/M, μM

 → Gi be a limit morphism. Then for each and L(μM ) =  Let pi : G  → G/N , αM = μM  pL(M) . The cone M ∈ N (G) we have a morphism αM : G property qM1 M2  αM2 = αM1 for M1 ⊇ M2 in N (G) is straightforwardly verified. Since G is a pro-Lie group, λG : G → limN ∈N (G) G/N is an isomorphism, and by  → G such that the universal property of the limit, there is a unique morphism πG : G qM  πG = αM , i.e., that the following diagram commutes for all M ∈ N (G): † qM

−1 ϕL(M) .

Lie’s Third Fundamental Theorem for Pro-Lie Groups

 G ⏐ ⏐ pM  GL(M)

πG

−−−−→ −−−−→ μM

G ⏐ ⏐q M G/M.

257

(3)

Then πG is pro-open and for each M there is a commutative diagram L(πG )

 −−−−→ L(⏐G) ⏐ L(pM ) L(GL(M) ) −−−−→ L(μM )

L(G) ⏐ ⏐L(q )  M L(G/M).

(4)

Recall that L(μM ) is an isomorphism of Lie algebras. The functor L preserves limits. Thus L(γG )

g = L(G) −−−−→

lim

M∈N (G)

† −1 limM∈N (G) (qM )

L(G/M) −−−−−−−−−−→

lim

M∈N (G)

L(G)/L(M)

is an isomorphism. This implies that the filter basis {L(M) ∈  (g) | M ∈ N (G)} converges to 0 in g. From Proposition 1.40 it follows that it is cofinal in  (g) and that,  may be identified with limM∈N (G) GL(M) by the Cofinality Lemma 1.21, therefore G in such a way that the pL(M) are the limit maps. As a consequence, in diagram (4), the vertical maps are the respective limit maps, and since the L(μM ) are isomorphisms it follows that L(πG ) is an isomorphism. This concludes the construction of πG and finishes the proof of (i).  = (L(G)) with the back adjunction πG (see e.g. [102, We have reconstructed G Definition A3.37]) in (⊥).  extends to a right adjoint functor to the inclusion (ii) In order to prove that G  → G  functor and that πG : G → G is a natural transformation, it again suffices to verify the universal property in a fashion analogous to that in the proof of 6.5: proSimpConLieGr

 G  ⏐  f ⏐ H

proLieGr πG  −−− G −→  ⏐  ⏐f

H

−−−−→ idH

G  ⏐f ⏐ H

We have to show that for each morphism of pro-Lie groups f : H → G from a simply  of topological connected pro-Lie group H there is a unique morphism f  : H → G groups such that π  f  = f . So let f : H → G be given and let M ∈ N (G). Since L(μM ) is an isomorphism, μM : GL(M) → G/M implements a covering morphism of the identity component (G/M)0 . Since H is simply connected, by Lemma L, qM f

258

6 Lie’s Third Fundamental Theorem

has a unique lifting νM : H → GL(M) such that H ⏐ ⏐ f G

νM

−−−−→ GL(M) ⏐ ⏐μ  M −−−−→ G/M

(5)

qM

is commutative for all M ∈ N (G). We recall that {L(M) | M ∈ N (G)} is cofinal in  (g) and that therefore  = lim GL(M) G M∈N (G)

up to a natural identification, where the limit maps are denoted pM . If M1 ⊆ M2 then straightforwardly pM1 M2 νM2 = νM1 . Thus by the universal property of the limit, there  such that pM f  = νM for all M ∈ N (G). Then is a unique morphism f  : H → G    qM πG f = μM pM f = μM fM = qM f for all M ∈ N (G). Since the qM separate points, πG f  = f . Thus we have the existence of f  ; it remains to show that it is unique. If also πG f  = f = πG f  , then κ : H → ker πG , κ(h) = f  (h)(f  (h)−1 , is a continuous function from a connected space to ker πG . But L(ker πG ) = ker L(πG ) = {0} since L(πG ) is an isomorphism. Hence ker πG is totally disconnected, equivalently, prodiscrete by 3.30 (iv). Thus κ is constant and this proves f  = f  and completes the proof of (ii).  is prosimply connected. Let us prove (b) ⇒ (a): (iii) Trivially, (a) ⇒ (b) since G  such that We apply (i) with H = G and f = idG . This gives us a ν : G → G   being the πG ν idG . Thus G is a retract of G in proLieGr, the semidirect factor of G totally disconnected group ker πG . As a topological direct factor of a connected space, it must be singleton. Thus πG is a monic retraction and therefore is an isomorphism. (See [102, Remark A3.13 (ii)].)  → L(G) is an isomorphism. Then (iv) From (i) we know that L(πG ) : L(G) 0 ) by 4.22 (ii). Since G 0 = G  as G  is prosimply connected, the assertion G0 = πG (G is proved. The following commutative diagram may be helpful to visualize the situation: L(πG )

 −−−−→ L(G) L(⏐G) ⏐ ⏐ ⏐exp expG˜   G  G. G −−−−→ πG

(v) Let g = L(G); since H is prosimply connected we may write H = (h) and identify h and L((h)) via the isomorphism ηh in () in the proof of 6.5. Then by the universal property expressed in (), there is a unique morphism f ∗ : H = (h) → G such that L(f ∗ ) = L(f )−1 . Now L(f ∗  f ) = L(f ∗ )  L(f ) = idL(G) = L(idG ), and by the uniqueness in the universal property, we have f ∗  f = idG . Similarly, f  f ∗ = idH . Hence f ∗ = f −1 and f is an isomorphism. (vi) Let g be a pro-Lie algebra. Then ηG : g → L((g)) is an isomorphism for each pro-Lie algebra g by 6.4, and πG : (L(G)) → G is an isomorphism for each

Lie’s Third Fundamental Theorem for Pro-Lie Groups

259

prosimply connected pro-Lie group G by (iii) above. These statements together prove Claim (vi). (vii) By Theorem 2.25, the functor L preserves kernels; hence the injectivity of f implies the injectivity of L(f ). By the connectivity of G and by Theorem 4.22 (ii)(b), the morphism L(f ) is a surjective continuous and open morphism of pro-Lie algebras and is therefore an isomorphism. As does any functor, the functor  : proLieAlg → →H  proSimpConLieGr preserves isomorphisms. Since = L we obtain that f: G is an isomorphism. The last assertion then follows from this and (iii) above. We shall call the kernel P (G) of G the Poincaré group of G. In the case that G is a Lie group, this notation agrees with the classical one and thus correctly extends this  → G as the universal morphism of G; in the concept. We also shall refer to πG : G  will be called case of Lie groups this is the universal covering morphism. The group G the universal group. (Cf. Theorem 8.21 below.)  is isomorphic to the additive group of L(G) If G is a compact abelian group, then G and πG is equivalent to expG : L(G) → G which is a morphism of topological groups  and G will be quite different in almost all aspects; in this case. Therefore, in general, G this is a fact one knows from Chapter 8 in [102], on compact abelian groups. We note that 6.6 (iv) says that the categories proLieAlg of pro-Lie algebras and continuous Lie algebra morphisms and the full subcategory of the category of topological groups and continuous group homomorphisms consisting of prosimply connected proLie groups are faithful images of each other; whatever is true in one of them, grosso modo, holds in the other as well, and the two functors  and L achieve the translation. Since  : proLieAlg → proLieGr is a left adjoint, it preserves colimits, it preserves cokernels. Before we exploit this useful fact, let us quickly review the concept of a cokernel. In a pointed category (that is, a category with a terminal and initial object and thus with zero morphisms; see e.g. [102, Appendix 3]) the kernel of a morphism f : X → Y is morphism e : K → X which is the equalizer of f and the zero morphism X → Y (in the category of groups, for instance, the constant morphism). As an equalizer, a kernel K is a limit. Dually, the cokernel of a morphism f : X → Y is a morphism c : Y → C which is the coequalizer of f and the zero morphism. In the category of Hausdorff topological groups, the cokernel c is the quotient morphism c : Y → Y /N where N is the smallest closed normal subgroup containing f (X). In fact, whenever ϕ : Y → Z is a morphism of topological groups such that ϕ f : X → Z is the constant morphism, then f (X) ⊆ ker ϕ and thus N ⊆ ker ϕ. So ϕ factors uniquely through the quotient morphism c : Y → Y /N. The Strict Exactness Theorem for  Theorem 6.7. Assume that

e

f

n −−−→ g −−−→ h is a strict exact sequence of morphisms of pro-Lie algebras; that is, assume that im e = ker f . Then (f ) (e) (n) −−−−→ (g) −−−−→ (h)

260

6 Lie’s Third Fundamental Theorem

is a strict exact sequence of morphisms of pro-Lie groups, that is, the image of (e) : (n) → (g) is the kernel of (f ) : (g) → (h) and this kernel is prosimply connected. Proof. The morphism f factors through g/ ker f and ker f is also the kernel of the quotient morphism g → g/ ker f . We may therefore replace f by the quotient morphism g → g/ ker f and will therefore assume, without losing generality, that f is itself a quotient morphism. It is then the cokernel of e and since  preserves cokernels, (q) is the cokernel of (e) in proLieGr. We let K = ker (f ). Then the quotient morphism p : (g) → (g)/K is the cokernel of (e) in TopGr. Let Q ⊇ (g)/K be the completion and j : (g)/K → Q the inclusion. Then the coextension j  p : (g) → Q is the cokernel of (e) in proLieGr. By the uniqueness of cokernels, there is an isomorphism ι : (h) → Q such that ι  (f ) = j  p: (f ) (g) −−−−→ ((h) ⏐ ⏐ ⏐ ⏐ι p (∗)  (g)/K −−−−→ Q. j

By the Strict Exactness Theorem for L 4.20, the map L(j ) : L((g)/K) → L(Q) is an isomorphism. Trivially, L(ι) : L((g)) → L(Q) is an isomorphism. Since L preserves kernels, we have L(K) = ker L((f )) ⊆ L((g)). In the commutative diagram e

f

L((e))

L((f ))

n −−−−→ g −−−−→ h ⏐ ⏐ ⏐ ⏐ ⏐ ηh ηg ⏐ ηn    L((n)) −−−−→ L((g)) −−−−→ L((h)) the vertical arrows are isomorphisms by 6.5 and thus, if we identify pro-Lie algebras def

via the natural automorphism η, we see that k = L(K) equals e(n). This means that the corestriction of (e) to K is the unique morphism e : (n) → K such that L(e ) : n = L((n)) → k = e(n) is the corestriction of e to k. The situation is illustrated by the following commutative diagram L(e )

−−−−→ ⏐k n ⏐ exp(n) ⏐ expK ⏐   (n) −−−− → K  e

−−−−→

L(i)

f

i

(f )

g −−−−→ ⏐ ⏐exp  (g) −−−−→ (g) −−−−→

h ⏐ ⏐exp  (h) (h),

where i : K → (g) denotes the inclusion morphism. Thus if we identify L((g)/K) with g/k (via 4.20), and η with the identity then L(p) identifies with the quotient map g → g/k and we obtain from (∗) upon applying the functor L the commuting diagram f

g −−−−→ ⏐ ⏐ quot g/k −−−−→ L(j )

h ⏐ ⏐L(ι)  L(Q).

Lie’s Third Fundamental Theorem for Pro-Lie Groups

261

We claim that K is connected, that is K0 = K. Let j ∗ : (g)/K0 → Q∗ denote completion. Then L(j ∗ ) : L((g)/K0 ) = g/k → L(Q∗ ) is an isomorphism and the surjective morphism s : (g)/K0 → (g)/K, s(xK0 ) = xK induces a unique extension s ∗ : Q∗ → Q which in turn induces an isomorphism L(s) : L(Q∗ ) → L(Q). But Q∗ is a connected pro-Lie group since (g)/N0 is connected, and Q is prosimply connected since Q ∼ = (h). As a consequence s ∗ is an isomorphism of topological groups by 6.6 (v). Since the morphism s is a restriction of s ∗ , it is injective. As it is also surjective it is bijective. But this means K = K0 as asserted. Now we apply the Closed Subgroup Theorem for Projective Limits 1.34 for (g) in place of G, for N S((g)) in place of N and K in place of H . Then N ∼ = limM∈N S((g)) K/(K ∩ M), where K/(K ∩ M) ∼ = KM/M. Now for each M ∈ N S((g)), the quotient group K/(K ∩ M) is a quotient of a connected pro-Lie group, and is therefore a connected proto-Lie group by 4.1; on the other hand, KM/M is a subgroup of the Lie group (g)/M and therefore has an identity neighborhood, in which the singleton subgroup is the only subgroup. Thus KM/M ∼ = K/(K ∩ M) is a normal Lie subgroup of the simply connected Lie group (g)/M and is therefore simply connected (see e.g. [17, Chap. 3, §6, no 6, Proposition 14] or [85, p. 224, III.3.17]). So K is a projective limit of simply connected Lie groups and is therefore prosimply connected by 6.1. Hence πK : (k) → K is an isomorphism of topological groups by 6.6 (iii). If k : k → g is the inclusion, then e = k  L(e ) and we note a commutative diagram (L(e ))

(i)

e

incl

−−−−→ (k) −−−−→ (n) ⏐ ⏐ ⏐ ⏐ π K0  id (n) −−−− → K0 −−−−→ 

(g) ⏐ ⏐ id (g).

This means that im (e) = K. Thus im (e) = ker (f ) and this is exactly the claim of the theorem. Theorem on the Quotient Preservation in Lie’s Third Theorem Corollary 6.8. (i) Assume that f : g → h is a surjective morphism of pro-Lie algebras. Then the morphism of topological groups (f ) : (g) → (h) is a quotient morphism. In particular, it is surjective. Moreover, ker (f ) is connected. (ii) Assume that G is a pro-Lie group and N a closed normal subgroup. Let F : G → def : G →H  H = G/N be the quotient morphism and j : N → G the inclusion. Then F is a quotient morphism of prosimply connected pro-Lie groups whose kernel is the  in G,  that is, there is a strict exact sequence image of N j˜

 F

−−−−→G −−−−→H  → 1. 1→N If H is a pro-Lie group, then there is a commutative diagram whose rows are strict exact sequences

262

6 Lie’s Third Fundamental Theorem ˜



j

F

j F   −−− →1 1→N −→ H ⏐ −−−−→ G ⏐ ⏐ ⏐ ⏐ ⏐ pN  pG  pH  1 → N −−−−→ G −−−−→ H → 1.

(iii) Let G a prosimply connected pro-Lie group and N a closed normal subgroup. Then the following two statements are equivalent: (a) G/N is prosimply connected. (b) N is prosimply connected. (c) N is connected. (iv) Assume that F : G → H is a surjective morphism of pro-Lie groups and that : G →H  is a quotient morphism. G is connected. Then F Proof. (i) The surjectivity of f implies the bijectivity of the induced map f  : g/n → h. In the category of weakly complete topological vector spaces, however, a bijective morphism, as the adjoint of a bijective morphism of vector spaces, is an isomorphism. (Compare this with [102, Theorem 7.30 (iv)], cited below as Theorem 7.7 (iv).) Hence f : g → h is a quotient morphism and if n denotes the kernel of f , then f is in fact the cokernel of the inclusion morphism e : n → g. Applying  to the exact sequence f

e

{0} → n −−−→ g −−−→ h → {0}, by the Strict Exactness Theorem for , 6.7, we get an exact sequence of morphisms of pro-Lie groups (f )

(e)

const

{1} → (n) −−−−→ (g) −−−−→ (h) −−−−→ {1}. In particular im (e) = ker (f ), which is a closed connected normal subgroup N of def

G = (g), and im (g) = ker const = (h). Thus (f ) is a surjective morphism of connected pro-Lie groups with connected kernel. It is also the cokernel of (e) in proLieGr since  preserves cokernels. In TopGr, the quotient morphism q : G → G/N is the cokernel of (e). By Theorem 4.1 the factor group G/N is a proto-Lie group and its completion Q ⊇ G/N is a pro-Lie group. Let j : G/N → Q be the inclusion. Then j  q : G → Q is the cokernel of (e) in proLieGr. By the uniqueness of cokernels, there is an isomorphism ι : (h) → Q such that G ⏐ ⏐ q G/N

(f )

−−−−→ (h) ⏐ ⏐ι  −−−−→ Q j

commutes. Since (f ) is surjective and j is injective, it follows that j is bijective. That is, G/N = Q is complete and therefore is a pro-Lie group. Then (f ) and q are equivalent, and thus (f ) is in fact a quotient morphism.

Lie’s Third Fundamental Theorem for Pro-Lie Groups

263

(ii) By the Strict Exactness Theorem 4.20 for L, we have a strict exact sequence L(F )

incl

0 → n −−−−→ g −−−−→ h → 0 with a quotient morphism L(F ) and with n = ker L(F ) ∼ = L(N ) by the kernel preservation of L. Now by the Strict Exactness Theorem 6.7 for  we obtain a strict exact sequence (incl)

L(F )

1 → (n) −−−−→ (g) −−−−→ (h) → 1, where L(F ) is a quotient morphism by (i) above. Now we recall from Theorem 6.6 and its proof that  · =   L and that πG :  g → G is that natural morphism inducing an isomorphism on the Lie algebra level. Thus (ii) follows. (iii) We let F : G → H be the quotient morphisms as in (ii). The group G is prosimply connected iff pG is an isomorphism by Theorem 6.6 (iii). If, in the commuting diagram in (ii), two of the three vertical morphisms pN , pG , and pH are isomorphisms, so is the third one. Hence by 6.6 (iii) again, (a) and (b) are equivalent.  with G by the isomorphism Trivially (b) implies (c). Finally assume (c). We identify G  pG and consider both N and N as subgroups of G via the inclusions j˜ and j . Then pN  ⊆ N . (Note that this is consistent with F = pH  F  is an inclusion map, that is N     and N = ker F ⊆ ker(pH  F ) = ker F = N.) Since L(pN ) : L(N ) → L(N ) is an  = N0 . Our assumption, however, isomorphism, from Corollary 4.22 we conclude N is N0 = N, and so pN is the identity map, whence (b) holds. (iv) By the connectivity of G and by Theorem 4.22 (ii)(b), the morphism L(F ) is a  = (L(F )) is a quotient quotient morphism of pro-Lie algebras; then by (i) above, F morphism of pro-Lie groups. Theorem on the Preservation of Embeddings in Lie’s Third Theorem Corollary 6.9. (i) If j is any closed ideal of a pro-Lie algebra g, then the inclusion i : j → g induces an embedding (i) : (j) → (g). That is, (j) may be considered as a closed normal subgroup of (g). (ii) If N is the kernel of a morphism F : G → H of pro-Lie groups and j : N → G →G  is an embedding so that N  may be identified the inclusion morphism, then j˜ : N  with the kernel of F . i

q

Proof. (i) The exact sequence 0 → j −−→ g −−→ g/j → 0 by the Exactness Theorem 6.7 gives an exact sequence i

(q)

{1} → (j) −−−→ (g) −−−→ (g/j) → {1}. If K = ker (q), then K is a closed normal subgroup and thus a pro-Lie group by the Closed Subgroup Theorem 3.35. Since L preserves kernels, L(K) = j. By the Exactness Theorem 6.7, the corestriction ε : (j) → K of (j ) is a bijective morphism inducing an isomorphism L(ε) : ((j) → L(K) = j) which becomes the identity when L((j)) is identified with j via ηj . It follows from the uniqueness in

264

6 Lie’s Third Fundamental Theorem

the universal property of  (see 6.6) that ε is none other than the canonical morphism  → K. In 6.8, however, we saw that K was prosimply connected. πK : (L(K)) = K But then by 6.6 (iii), ε = πK is an isomorphism. But this means that (i) is an embedding algebraically and topologically, as asserted. (ii) Since the functor L preserves kernels, the Lie algebra n = L(N ) is (up to natural isomorphism) the kernel of L(F ) : L(G) → L(H ). Now we apply (i) above to n in place of j and recall  · =   L again to obtain the assertion.

Semidirect Products We discussed in Chapter 1 the concept of a semidirect product of topological groups in Exercise E1.5. The concept of a semidirect product in the category of topological groups is very fruitful, both traditionally in classical Lie group theory and in the theory of pro-Lie groups – as we still have to see. Let us review the concept with particular emphasis on the functorial aspects of it. Remember that a self map f of a set is called idempotent if f 2 = f . Also recall that for topological groups N and H and a morphism α : H → Aut(N ) such that (h, n)  → α(h)(n) : H × N → N is continuous, the product N × H becomes a topological group N α H with the multiplication (n, h)(n , h ) = (nα(h)(n ), hh ), called the semidirect product of N by H (with respect to α). If n and h are topological Lie algebras and δ : h → Der(n) is a morphism of Lie algebras such that (Y, X)  → δ(Y )(X) : h × n → n is continuous, then n × h becomes a Lie algebra n ⊕δ h with respect to componentwise scalar multiplication and addition and the bracket multiplication [(X, Y ), (X , Y  )] = (δ(Y )(X  ) − δ(Y  )(X), [Y, Y  ]), called the semidirect sum of n with h (with respect to δ). Exercise E6.2. Verify that the bracket multiplication of the semidirect sum is antisymmetric and satisfies the Jacobi identity (see Definition A1.1 (ii), p. 624). Proposition 6.10. (a) Let G be a topological group and N a closed normal subgroup. Then the following conditions are equivalent: (i) There is an idempotent endomorphism ϕ of G such that ker ϕ = N . (ii) There are morphisms q : G → K and σ : K → G such that q  σ = idK and ker q = N. (iii) There is a closed subgroup H such that the function (n, h)  → nh : N × H → G is an isomorphism of the semidirect product N I H of N by H with respect to the morphism I defined by inner automorphisms I (h)(n) = hnh−1 onto G. If these conditions are satisfied, H = ϕ(G) = σ (K).

Semidirect Products

265

(b) Let g be a topological Lie algebra and n a closed ideal. Then the following conditions are equivalent: (i) There is an idempotent endomorphism  of g such that ker  = n. (ii) There are morphisms p : g → k and ρ : k → g such that p  ρ = idk and ker p = n. (iii) There is a closed subalgebra h such that the function (X, Y )  → X+Y : n×h → g is an isomorphism of the semidirect sum n ⊕ad h of n by h with respect to the morphism ad defined by the adjoint representation ad(Y )(X) = [Y, X] onto g. If these conditions are satisfied, h = (g) = ρ(k). Proof. We prove part (a): (i) ⇒ (ii): We let K = ϕ(G) and let q be the corestriction of ϕ to its image. Define σ : K → G be the inclusion map. Then ker q = ker ϕ = N, and if k ∈ K, then k = ϕ(g) for some g ∈ G and (q σ )(k) = q(ϕ(g)) = ϕ 2 (g) = ϕ(g) = k. (ii) ⇒ (iii): Define H = σ (K). Let μ : N I H → G be defined by μ(n, h) = nh. Then μ is easily seen to be a morphism of topological groups. The function g  → (gσ (q(g)), q(g)) is readily verified to be the inverse map μ−1 of μ. (iii) ⇒ (i): Assume that μ : N I H → G is an isomorphism of topological groups, μ(n, h) = nh. Let i : H → G be the inclusion morphism and pr H : N I H → H the projection onto H . Define ϕ : G → G by ϕ = i  pr 2 μ−1 . Assume g = nh. Then μ−1 (g) = (n, h), and μ−1 (h) = (1, h), whence ϕ 2 (g) = i pr 2 μ−1 i pr 2 μ−1 (g) = h = i pr 2 μ−1 (g) = ϕ(g). Part (b) is proved in a completely analogous fashion and is left as an exercise. Exercise E6.3. Spell out the details of the proof of Part (b) of Proposition 6.10. Theorem 6.11 (Preservation of Semidirect Products). Let g be the semidirect sum of def the closed ideal n by the closed subalgebra h. Then G = (g) is the semidirect product of a closed normal subgroup N naturally isomorphic to (n) and a subgroup H isomorphic to (h). Proof. By Proposition 6.10 (b), the hypothesis is equivalent to the existence of an idempotent endomorphism  of g with kernel n and image h. We apply the functor  def

and obtain and idempotent endomorphism ϕ = () : G → G. Let i : n → g denote the inclusion morphism. The exact sequence i



0 → n −−→ g −−→ g gives rise to an exact sequence (i)

(ϕ)

1 → (n)−−−−→(g)−−−−→(g) by the Exactness Theorem for  6.7, and (j ) is an embedding by Corollary 6.6. Set N = (j)((n)). Then (n) ∼ = N = ker () = ker ϕ. It now follows from

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6 Lie’s Third Fundamental Theorem def

Proposition 6.10 that G is the semidirect product of N by H = ϕ(G). It remains to identify im ϕ. def

Now let p : g → h = (g) be the corestriction of  to its image. Then p is a quotient morphism inducing an isomorphism g/n → h. Then the exact sequence p

g −−→ h → 0 induces an exact sequence (p)

G = (g) −−−−→ (h) → 0 def

by the Exactness Theorem for  6.7, and q = (p) : G → (h) is a quotient morphism by Corollary 6.8. If we let ρ : h → g denote the inclusion, we have pρ = idh , whence def

def

q  σ = id(h) for σ = (ρ). Then H = σ ((h)) is isomorphic to (h). Also  = ρ  p and thus ϕ = () = (ρ)  (p) = σ  q whence im ϕ = im σ = H .

Postscript From classical Lie group theory we know that if G is a finite-dimensional Lie group, then there exists a unique (up to isomorphism) simply connected finite-dimensional Lie  such that their Lie algebras L(G) and L(G)  are isomorphic; further, there is group G,  onto G implementing a local isomorphism. an open continuous homomorphism of G When one moves out of the world of finite-dimensional Lie groups or rather out of the world of manifolds, the first task is to be clear about what “simply connected” should mean. For a connected topological space X with a base point x0 several definitions of simple connectivity are in use. For instance, • X is called simply connected if it has the following universal property: For any covering map p : E → B between topological spaces, any point e0 ∈ E and any continuous function f : X → B with p(e0 ) = f (x0 ) for some x0 ∈ X there is a continuous map f: X → E such that p  f = f and f(x0 ) = e0 . f

X ⏐ −−−−→ ⏐ idX 

E ⏐ ⏐p 

X −−−−→

B.

f

Alternatively, • X is called simply connected if π1 (X, x0 ) = {0}, that is if every loop attached to x0 is contractible.

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267

No nondegenerate connected compact abelian group is simply connected with respect to the first definition. Any connected compact abelian group G such that  Z) = {0} (such as e.g. G = Q ) is simply connected with respect to the second Hom(G, definition. For arcwise connected, locally arcwise connected, and locally arcwise simply connected spaces, in particular for manifolds, the two definitions agree (see e.g. [102, Proposition A2.10]). In [102], we opted for the first definition because of its universal properties which are practically built into the definition. Taking these complications into consideration, we ask what should be the right concept of simple connectivity for pro-Lie groups? We say that • a pro-Lie group G is prosimply connected if N (G) contains a cofinal subset N S(G) such that G/N is a simply connected Lie group for each N ∈ N S(G). This definition takes the characteristic property G ∼ = limN ∈N (G) G/N into account. But is it useful for the structure theory of pro-Lie groups? With the aid of the structure theory of pro-Lie groups that we will develop in later chapters, we will show that a pro-Lie group G is prosimply connected if and only it is simply connected, and that it satisfies π1 (G) = {0} whenever it is prosimply connected. That is a reassuring fact. But recall that no compact connected abelian group is prosimply connected. That may make us a bit nervous. However, within the full category of pro-Lie groups in the category of topological groups, the full subcategory of prosimply connected pro-Lie groups has precisely the right universal properties. Firstly, for every profinite Lie algebra g there is a unique prosimply connected pro-Lie group G (up to isomorphism) such that L(G) = g. (Lie’s Third Fundamental Theorem). Thus there is essentially a “bijection” between the possible Lie algebras of pro-Lie groups and the prosimply connected specimens in proLieGr. If g is abelian, then G is the additive group of g; this is reasonable and has no analog in such categories as the category of locally compact groups. Secondly, for each connected pro-Lie group G there is a (functorially attached)  and a pro-open natural morphism πG : G →G prosimply connected pro-Lie group G  inducing an isomorphism L(πG ) : L(G) → L(G) of Lie algebras, and every morphism  of G into a prosimply connected pro-Lie group uniquely lifts to a morphism from G, factoring through πG . All of this very satisfactorily generalizes familiar facts from classical Lie group theory. Yet none of these facts is exactly trivial. For a compact  is naturally isomorphic to the additive group of L(G) and πG : G →G abelian group, G identifies with expG : L(G) → G. With our theorems, among other things we are saying: the category proLieAlg of profinite Lie algebras has “a precise copy”, namely, proSimpConLieGr, the full subcategory of the category proLieGr of pro-Lie groups containing precisely the prosimply

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connected pro-Lie groups. Moreover, the category of connected Lie groups allows a left-reflection into this category, that is a left adjoint functor which behaves like a retraction onto proSimpConLieGr. We have seen that Lie’s Third Fundamental Theorem in reality complements the Lie algebra functor to an adjoint situation. This is of course true in the finite-dimensional situation, but as a rule, less emphasis is placed on this fact in elementary Lie theory. Among other reasons, the most compelling one is that in the finite-dimensional case we are not dealing with complete categories. Adjoint situations are most appropriate where limits are present, because right adjoint functors (such as L in our case) preserve all limits. The functor  : proLieAlg → proLieGr that is left adjoint to the Lie algebra functor L : proLieGr → proLieAlg has excellent exactness properties, as we have seen in Theorem 6.7 and its two corollaries. This is a parallel to the Exactness Theorem for L 4.20 in Chapter 4. The theorems on the preservation of quotients 6.8 and of embeddings 6.9 are particularly relevant since quotients are somewhat problematic as we have seen in Chapter 4, and since we cannot have an Open Mapping Theorem for pro-Lie groups in general. The general covering theory for topological groups introduced by Berestovskii and Plaut fits very well with our theory for pro-Lie groups (see [6], [7], [8]). For a  connected pro-Lie group G, the Berestovskii–Plaut-cover agrees with G.

Chapter 7

Profinite-Dimensional Modules and Lie Algebras

In the previous chapters we recognized the significance of the category of topological groups with profinite-dimensional Lie algebras. This category can be understood only if one first comprehends the nature of pro-Lie algebras, that is, profinite-dimensional Lie algebras themselves. In this chapter we develop the structure theory of pro-Lie algebras by utilizing the duality between the category of weakly complete topological vector spaces and the category of vector spaces we review completely in Appendix 2. The use of this duality is made possible through representation theory or, equivalently, module theory of a fixed Lie algebra L. In fact we shall consider profinite-dimensional modules which emulate the definition of pro-Lie algebras in such a fashion that the adjoint module of a pro-Lie algebra is exactly a profinite-dimensional module. While pro-Lie algebras themselves do not dualize in an obvious way, profinite-dimensional modules do, and we shall make ample use of this fact. Prerequisites. In this chapter we shall make use of the duality theory of weakly complete topological vector spaces as it is presented in Appendix 2. We shall collect the essential features in Theorem 7.7. The representation theory of Lie algebras used here is simple. But we do resort to the basic structure theory of finite-dimensional Lie algebras presented in source books that are widely used such as [16]. In this chapter we do not use any information from Chapters 3, 4 and 5.

Modules over a Lie Algebra We begin with some elementary facts on the linear algebra of modules over a Lie algebra. Definition 7.1. (i) Let L be a Lie algebra and E a vector space. Then E is an L-module if there is a bilinear map (x, v)  → x · v : L × E → E

satisfying

[x, y] · v = x · (y · v) − y · (x · v)

for all x, y ∈ L and v ∈ E. A function f : E1 → E2 between L-modules is said to be a morphism of L-modules if it is linear and satisfies (∀x ∈ L, v ∈ E1 )

f (x · v) = x · f (v).

(1)

(ii) If L is a Lie algebra and V is a topological vector space, then V is said to be a continuous L-module if the underlying vector space is an L-module in the sense of (i) above and the functions v  → x · v : V → V are continuous for all x ∈ L.

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(iii) If L is a topological Lie algebra, then a topological vector space V is said to be a topological L-module if it is a continuous L-module and the maps x  → x · v : L → V are continuous for all v ∈ V ; in other words, if (x, v)  → x · v : L × V → V is continuous in each variable separately. Let us consider some prominent examples. Example 7.2. For any vector space E, let gl(E) denote the Lie algebra of all endomorphisms of E with the Lie bracket [ϕ, ψ] = ϕψ − ψϕ. If L is any Lie algebra, let |L| denote the underlying vector space. If we define ad x : L → L by (ad x)(y) = [x, y], then the Jacobi identity shows that ad : L → gl(|L|) is a morphism of Lie algebras, called the adjoint representation. If for x ∈ L and v ∈ |L| we set x · v = (ad x)v, then |L| becomes an L-module called the adjoint module. If L is a topological Lie algebra then the underlying topological vector space |L| becomes a topological L-module, again called the adjoint module. In particular, if g is a pro-Lie algebra and |g| is the underlying weakly complete topological vector space, then |g| is a topological g-module via the adjoint operation. If E is an L-module, for x ∈ L we define xE : E → E by xE (v) = x · v. Then x  → xE : L → gl(E) is a morphism of Lie algebras. Conversely, if π : L → gl(E) is a representation, i.e. a morphism of Lie algebras, then E becomes an L-module via x · v = π(x)(v). Thus the set of representations of L on V is in bijective correspondence with the L-module structures on E. If V is a topological vector space and an L-module, then each xV is an endomorphism of the topological vector space V , and if gl(V ) denotes the Lie algebra of all endomorphisms of the topological vector space V with the Lie bracket [ϕ, ψ] = ϕψ − ψϕ, then x  → xV : L → gl(V ) is a representation. Conversely, every representation of L into gl(V ) yields on the topological vector space V the structure of an L-module. If L is a topological Lie algebra and V a topological L-module, then x  → xV : L → gl(V ) is a morphism of topological Lie algebras if gl(V ) is given the topology of pointwise convergence, sometimes called the strong operator topology on V . We shall now make use of the duality of topological vector spaces. If V is a topological vector space, its topological dual is the vector space V  = Hom(V , R) of all continuous linear functionals. In accordance with this definition, V  is a vector space without additional structure, but it may be equipped with many relevant vector space topologies; we shall consider two of them here. The coarser of the two is the weak *-topology, also called the topology of pointwise convergence, that is the topology induced from the product RV . If V  is equipped with this topology, we denote it by V  . The second topology we consider is the topology of uniform convergence on compact subsets of V also called the compact open topology. (See [102, Proposition 7.1ff].) If the topological dual V  is endowed with the compact open topology, we denote  have the same underlying vector . The topological vector spaces V  and V it by V . For a space V  , but the topology of V  may be properly coarser than that of V  is defined to be TopGr(G, T) = Hom(G, T) topological group G the character group G with the compact open topology; of course, this works well only when G is abelian. If V

Modules over a Lie Algebra

271

is a topological vector space, then the natural morphism Hom(V , R) → Hom(V , R/Z) is an isomorphism of topological vector spaces when both hom-groups are given the compact open topology (see [102, Proposition 7.5 (iii)]). This justifies our notation of  for V  with the compact open topology. V Now let L be a Lie-algebra and V a topological vector space. If ω ∈ V  , then ω  xV : V → R is a continuous linear functional. Thus if we define x · ω by x · ω = −ω  xV ∈ V  we have (∀x ∈ L, v ∈ V )

(x · ω)(v) = −ω(x · v).

(2)

Lemma 7.3. (i) Let L be a Lie algebra and V a topological vector space and L→V , module. With respect to the bilinear map (x, ω) → x · ω = −ω  xV : L × V  the topological vector space V is an L-module. (ii) If L is a topological Lie algebra and V a topological L-module, then V  is a topological L-module. Proof. (i) First we compute ([x, y] · ω)(v) = −ω([x, y] · v) = −ω(x · (y · v)) + ω(y · (x · v)) = −y · ω(x · v) + x · ω(y · v) = x · (y · ω)(v) − y · (x · ω)(v). Thus V  is an L-module. Now we observe the required continuity property, namely, that xV  , defined by xV  (ω) = x · ω = −ω  xV , is continuous for all x ∈ L. But that is immediate from the continuity of all xV and ω. (ii) Let ω ∈ V  and v ∈ V . All functions x  → x · v : L → V are continuous. Hence all functions x  → (x · ω)(v) = −ω(x · v) : L → R are continuous and that means that the linear function x  → x · ω : L → V  is continuous with respect to the topology of pointwise convergence on V  . Lemma 7.4. (i) Let L be a Lie algebra and V a topological vector space and L-mod→V , the ule. With respect to the bilinear map (x, ω)  → x · ω = −ω  xV : L × V  is an L-module. topological vector space V (ii) If L is a topological Lie algebra and V a topological L-module such that  is a topological L-module. (x, v)  → x · v : L × V → V is continuous, then V Proof. (i) After 7.1 (ii) we have to verify only that for all x ∈ L the linear maps →V  are continuous. For a proof we assume that a compact subset C of V and xV : V a zero neighborhood U of R are given and thus determine a basic zero neighborhood W (C, U ) = {ω ∈ V  | ω(C) ⊆ U }. Since xV is continuous, −x · C = −xV (C) is compact, and every functional ω satisfying ω(−x · C) ⊆ U also satisfies ω(−x · C) = (x · ω)(C) ⊆ U . Thus xV maps W (−x · C, U ) into W (C, U ) →V , (ii) For a proof of the continuity of the functions x → x · ω = −ω  xV : V we again assume that a compact subset C of V and an open zero neighborhood U of R are given. We must find a zero neighborhood W of L such that x ∈ W implies (x · ω)(C) ⊆ U i.e., ω(−x · C) ⊆ U . Now ω−1 (U ) is an open identity neighborhood

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of V . If the function (x, v) → x · v : L × V → V is assumed to be continuous, then an open symmetric zero neighborhood W of L exists such that W · C ⊆ ω−1 (U ), and this proves the assertion. Definition 7.5. 7.5. The L-module V  is called the dual module of V . Example 7.6. Let L be a topological Lie algebra and let |L| denote the underlying topological vector space. Define L to be the topological dual |L| of |L| in the weak *-topology. Then the dual module of the adjoint module |L| on L is a topological  x)(ω) = L-module which is called the coadjoint module, written Lcoad . If we set (ad   x · ω = ω  ad x, then ad : L → gl(L) is a continuous morphism of Lie algebras if one endows gl(|L|) with the topology of pointwise convergence.  denotes the topological dual of |L| with the compact open topology, then L  is If L a topological L-module with respect to the coadjoint action.

Duality of Modules We consider two categories of topological vector spaces which are dual to each other under the passage to the topological dual with the compact open topology. (See [102, Proposition 7.1ff].) Namely, we consider firstly the category Vect of vector spaces and linear maps between them, and secondly, the category WCVect of weakly complete topological vector spaces and continuous linear maps between them. Any real vector space can be considered as a topological vector space with respect to the finest locally convex vector space topology (see Appendix 2, Proposition A2.3 and the passage that precedes it), and any linear map between vector spaces is automatically continuous with respect to this topology on the domain and the range space. An abelian topological group A is called reflexive if the natural evaluation morphism  η: A →  A into the double dual is an isomorphism of topological groups. (See [102, Definition 7.8ff].) We denote by RefTopGr the full subcategory of TopGr whose objects are all reflexive abelian topological groups. Note that a morphism f : V → W of topological groups between the additive groups of topological vector spaces is always linear. To see this note that if r = m n with m, n ∈ N, then n · f (r · v) = f (m · v) = m · f (v), whence f (r · v) = r · f (v), and this relation extends by continuity to all r ∈ R. the set of all ω ∈ V  If S is a subset of a topological vector space V , then S ⊥ denotes   ⊥ such that ω(S) = {0}. If S is a subset of V , then S = ω∈S ker ω. This will not cause confusion if we always keep track of the vector space in which S is located. The vector subspace S ⊥ is called the annihilator of S. For easy reference, we review the duality of vector spaces and weakly complete topological vector spaces.

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Theorem 7.7 (Duality of Real Vector Spaces). Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then   is an isomorphism of topological vector spaces. (i) E is reflexive; i.e. ηE : E → E  and E  is a weakly complete Thus E belongs to RefTopGr. Moreover, E  = E, topological vector space.   is an isomorphism of topological vector spaces, (ii) V is reflexive; i.e. ηV : V → V  has the finest locally convex topology, and thus V belongs to RefTopGr. The dual V   ∼   and V = V = V . (iii) Vect and WCVect are subcategories of RefTopGr, and the contravariant functor  · : RefTopGr → RefTopGr exchanges the categories Vect and WCVect. (iv) Every closed vector subspace V1 of V is algebraically and topologically a direct summand; that is there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces. Every surjective morphism of weakly complete topological vector spaces f : V → W splits; that is, there is a morphism σ : W → V such that f  σ = idW . (v) For every closed vector subspace H of E, the relation H ⊥⊥ = H ∼ = (E  /H ⊥ )  ⊥ ⊥  holds and E /H is isomorphic to H . The map F  → F is an antiisomorphism of the complete lattice of vector subspaces of E onto the lattice of closed vector subspaces of E  . Proof. See Appendix 2, notably Theorems A2.8, A2.11, and A2.12. A subset M of an L-module V is called a submodule if L · M ⊆ M. If M is a closed submodule then V /M is an L-module with respect to the bilinear map (x, v + M)  → x ∗ (v + M) = x · v + M. Definition 7.8. Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. (i) V is called a profinite-dimensional L-module if it is complete as a topological vector space and the filter basis M of closed submodules M ⊆ V such that dim V /M < ∞ converges to 0. (ii) E is called a locally finite-dimensional L-module if for each finite subset S of E there is a finite-dimensional submodule F of E containing S. We specifically draw the reader’s attention to the facts that according to our definitions, a profinite-dimensional L-module is a topological vector space with a module structure, while a locally finite-dimensional L-module is merely a vector space with an L-module structure. It is possible at all times to consider it as a topological vector space with a module structure by equipping it with the finest locally convex topology, but doing that is not likely to give us additional insights. Thus a locally finite-dimensional module is a purely algebraic object even if L is a topological Lie algebra. Let g be a finitely generated topological Lie algebra on an infinite-dimensional vector space. Then the underlying vector space |g| with the adjoint module structure cannot be locally finite-dimensional, since a finite set of generators of g is not contained

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in a finite-dimensional subalgebra, let alone an ideal, and thus it is not contained in a finite-dimensional submodule. A simple example is as follows: Let s = so(3) and ej ∈ s, j ∈ Z(3) = Z/3Z, a basis of s such ' that [e & j , ej +1 ] = ej +2 for j = 0, 1, 2 mod 3. Then [s, e0 ] contains e1 and e2 and s, [s, e0 ] = s. Now set g = sN and x = (xn )n∈N with xn = e0 for all n. In ' &N the adjoint module we have g·x = [s, e0 ]N and g·(g·x) = s, [s, e0 ] = sN = g. Thus the adjoint module is singly generated algebraically and yields a profinite-dimensional g-module whose underlying vector space does not yield a locally finite-dimensional g-module, since {x} is not contained in a finite-dimensional submodule. Proposition 7.9. For an L-module on a topological vector space V , the following statements are equivalent: (i) V is a profinite-dimensional L-module.  = limM∈M V /M (see 1.29) is an isomorphism. (ii) The morphism γV : V → V Every profinite-dimensional L-module is weakly complete. Proof. The assertion follows from Theorems 1.29 and 1.30. Proposition 7.10. For an L-module E the following conditions are equivalent: (i) (ii) (iii) (iv)

E is locally finite-dimensional. Each element of E is contained in a finite-dimensional submodule. E is the union of its finite-dimensional submodules. E is a sum of a family of finite-dimensional submodules.

Proof. Exercise E7.1. Exercise E7.1. Prove Proposition 7.10. [Hint. (i) ⇒ (ii) ⇒ (iii) ⇒ (i) and (iii) ⇒ (iv) are trivial. If (iv) holds, then E = 5 . Let F denote the family j ∈J Vj for a family of finite-dimensional submodules Vj of all finite sums of the submodules Vj . Then (iv) says E = F and this implies (iii).] The Duality Theorem for Profinite-Dimensional and Locally Finite-Dimensional L-Modules Theorem 7.11. Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. Then the following conclusions hold.  = E  is weakly complete. (ia) The underlying vector space of the dual module E   of E  is E (up to natural isomorphism). If E is a locally finiteThe dual module E  is a profinite-dimensional L-module. dimensional L-module, then E (ib) If L is a topological Lie algebra and E is a topological L-module then the dual  = E  is a topological L-module. module E   has the finest locally convex topology. Its dual (V ) = V  is (ii) The dual module V  V (up to natural isomorphism). If V is a profinite-dimensional L-module, then V is a locally finite-dimensional L-module.

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(iii) The functor  : Vect → WCVect maps the category of locally finite-dimensional L-modules to the category of profinite-dimensional topological L-modules. The functor  : WCVect → Vect maps the category of profinite-dimensional topological L-modules to the category of locally finite-dimensional L-modules. (iv) Every closed L-submodule V1 of V is a direct topological vector space summand; that is, there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces. Every surjective morphism of profinite-dimensional topological L-modules f : V → W splits in terms of topological vector spaces; i.e. there is a morphism of topological vector spaces σ : W → V such that f  σ = idW . (v) A vector subspace H of E is an L-submodule iff its annihilator H ⊥ in E  is an L-submodule of E  . A closed vector subspace K of V is an L-submodule of V iff its annihilator in V  is an L-submodule of V  . For every L-submodule H of E, the . The map relation H ⊥⊥ = H ∼ = (E  /H ⊥ ) holds and E  /H ⊥ is isomorphic to H F  → F ⊥ is an antiisomorphism of the complete lattice of L-submodules of E onto the lattice of closed L-submodules of E  , and F ⊥⊥ = F for each submodule F of E and each closed submodule F of V .  and E  is a weakly complete topological vector Proof. (i) By 7.7 (i) and (iii), E  = E  space. By 7.3, E is an L-module, and if L is a topological Lie algebra and E is a  ∼ topological L-module, then E  is a topological L-module as well. The relation E =E follows from 7.7 (i). Now assume that E is a locally finite-dimensional L-module.  is complete and that the filter basis of closed vector Since we know that E  = E  subspaces N with dim E/N < ∞ converges to 0, in order to verify condition 7.8 (i)  we have to show that every closed vector subspace N of E  such that E/N  is for V = E  finite-dimensional contains an L-submodule M such that E/M is finite-dimensional. By the Annihilator Mechanism 7.7 (v), the annihilator N ⊥ of N in E is finite-dimensional. Since E is a locally finite-dimensional L-module, there is a finite-dimensional L-module F of E containing N ⊥ . Then we let M be the annihilator F ⊥ of F in E  .  we have   ⊥∼ Since annihilators of submodules are submodules and E/M = E/F =F  dim E/M < ∞. Thus the proof of (i) is complete.  has the finest locally convex topology and by (i) above and 7.7 (i) (ii) By 7.7 (iii), V  ) = V ∼ and (ii), we have (V = V . Now we assume that V is a profinite-dimensional L-module. We claim that V  is a locally finite-dimensional L-module. So let S be a finite subset of V  . Let S ⊥ be the annihilator of this set in V . Then by 7.7 (v), the vector space V /S ⊥ ∼ = (span S) is finite-dimensional. Since V is a profinite-dimensional L-module, there is a submodule M such that M ⊆ S ⊥ and that dim V /M < ∞. def ∼ Then F = M ⊥ , the annihilator of M in V  is a submodule and since F = V /M we ⊥ ⊥⊥ know that it is finite-dimensional. Also S ⊆ M implies S ⊆ S ⊆ M⊥ = F . Statement (iii) is now a consequence of (i) and (ii), and conditions (iv) follows directly from 7.7 (iv). (v) Let H be a vector subspace of E. Then x · H ⊆ H for x ∈ L iff ω(H ) = {0} implies x · ω(H ) = −ω(x · H ) = {0} for all ω ∈ E  since H = H ⊥⊥ by 7.7 (v). Thus x · H ⊆ H iff x · H ⊥ ⊆ H ⊥ . For a closed vector subspace K of V the same argument

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applies since for a closed vector subspace K of V one has V = V ⊥⊥ by 7.7 (v). The remainder of (v) is an immediate consequence of 7.7 (v). Corollary 7.12. Let L be a topological Lie algebra and V a weakly complete topological L-module. Then the following statements are equivalent: (i) V is a profinite-dimensional L-module. (ii) V  is a locally finite-dimensional L-module. Proof. Since E = V  with the finest locally convex topology and V are duals of each other by 7.7, the corollary is an immediate consequence of Theorem 7.11. The theory of profinite-dimensional modules is primarily developed to deal with profinite-dimensional Lie algebras. This is why the following example is crucial. Exercise E7.2. Prove the following proposition: Let g be a weakly complete topological Lie algebra. Then the following statements are equivalent. (i) g is a pro-Lie algebra. (ii) The coadjoint module gcoad of g is locally finite-dimensional. [Hint. With g = L and V = |g| (the underlying topological vector space), this is but a special case of Corollary 7.12.] As we progress further into duality theory we have reached the point to recall that categorical concepts such as limits and their various manifestations such as products, pullbacks, equalizers projective limits come in pairs, as each concept has its dual concept. Technically, if D : J → C is a diagram its colimit is the limit of the diagram D : J → C op in the opposite category. However, in order to have things self-contained in this book as much as possible, we explicitly formulate the definitions of a colimit and colimit cone. Definition 7.13. Let J and C be categories. A diagram D : J → C is said to have a colimit colim D ∈ ob C if there is a cone λ : D → const(colim D), called the colimit cone1 such that for each cone α : D → const A there is a unique morphism α  : colim D → A such that α = const(α  )  λ. Exercise E7.3. Formulate the definitions of coproduct, coequalizer, pushout, and injective limit. [Hint. Dualize the concepts introduced in 1.4.] of vector Lemma 7.14. (i) Let E be a vector space and {Fj | j ∈ J } a directed family  subspaces of E; i.e., J is a directed set and j ≤ k implies Jj ⊆ Jk . Then E = j ∈J Fj and E = colimj ∈J Fj are equivalent statements. 1 The

“cones” in this definition are “upside down” and should be called “cocones”. However, there are limits to the grammatical and etymological pliability of language. For instance, it is not true that a coconut is the dual of a nut. In fact, it is the bidual of a nut as the conut is its dual. For further references on philological analysis see the footnote in [102] regarding the terminology of Definitions 9.30.

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(ii) Let V be a profinite-dimensional L-module and M any filter basis of submodules M such that dim V /M < ∞ for all M ∈ M and that M converges to 0. Then the dual module V  is the union of the directed set of finite-dimensional submodules M ⊥ , M ∈ M. Proof. Exercise E7.4. Exercise E7.4. Verify Lemma 7.14. [Hint. (i) E = colimj ∈J Fj means that for every family of linear maps fj : Fj → F into a vector space F such that j ≤ k implies fk |Fj = fj there is a unique linear map f : E → F such that f |Fj = fj . Prove that this implies and is implied by E = colimj ∈J Fj . (ii) follows from (i) and the Duality Theorem 7.11.]

Semisimple and Reductive Modules Definition 7.15. (i) An L-module E is said to be simple if {0} and E = {0} are its only submodules. (ii) An L-module E is called semisimple if every submodule is a direct module summand. (iii) Let V be a profinite-dimensional topological vector space and an L-module. Then the module is said to be reductive if its dual module is semisimple. (iv) A pro-Lie algebra g is called reductive if its adjoint module is reductive. Note that a simple abelian module is isomorphic to R with the zero module operation x · r = 0 for all x ∈ L and r ∈ R. Recall that the direct sum  j ∈J Ej of a family Ej , j ∈ J of vector spaces is the set of all families (xj )j ∈J ∈ j ∈J Ej of elements xj ∈ Ej vanishing for all but a finite number of exceptions.5If each Ej is a vector subspace of a vector space E, then we shall say that the sum j ∈J Ej ⊆ E is direct if the morphism 6 8 6 xj : Ej → Ej (xj )j ∈J  → j ∈J

j ∈J

j ∈J

is an isomorphism of vector spaces. The Structure Theorem of Semisimple Locally Finite-Dimensional Modules Theorem 7.16. Let L be a Lie algebra and E a locally finite-dimensional L-module. Then the following statements are equivalent: (i) (i ) (ii) (iii)

E is a semisimple L-module. Every submodule is a direct module summand. Every finite-dimensional submodule of E is semisimple. E is the union of finite-dimensional semisimple submodules.

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(iv) E is (isomorphic to) a direct sum simple submodules Ej .

j ∈J

Ej of a family of finite-dimensional

Proof. (i) ⇔ (i ): Definition 7.15(ii). (i) ⇒ (ii): Let F be a finite-dimensional submodule of E and F1 a submodule of F . We must show that F1 has a complementary module summand in F . Since E is semisimple, there is a submodule E2 of E such that E = F1 ⊕ E2 . Set F2 = F ∩ E2 . Then F = F1 ⊕ F2 is readily checked (“Law of Modularity”). (ii) ⇒ (iii): Since E is a locally finite-dimensional module, every element is contained in a finite-dimensional submodule F . Hence this implication is trivial. (iii) ⇒ (iv): We consider sets S of finite-dimensional simple 5 submodules of E such that the morphism ε : F → E sending (x ) to F F ∈S F ∈S F ∈S xF ∈ E is injective. The set F of all sets S is partially ordered under inclusion. We claim that F is inductive. def  S and assert that For a proof let S be a totally ordered subset of F . We set5S = S ∈ F . Indeed we claim that the morphism (xF )F ∈S  → 5 x F ∈S F : F ∈S F → E is injective. Indeed let (xF )F ∈S be in its kernel. Then F ∈S xF = 0. The set T = {F ∈ S : xF = 0} is finite, hence is contained in some Sx ∈ S since S is the union of all S in S. But by definition of F the function 6 8 yF : F →E (yF )F ∈Sx  → F ∈Sx

F ∈Sx

is injective. Now (xF )F ∈ Sx is in the kernel of this map, and this implies xF = 0 for all F ∈ Sx . But xF = 0 for F ∈ S \ T and thus (xF )F ∈S is zero. This proves the claim and shows the assertion that S ∈ F 5. Hence F is indeed inductive. Now let S be a maximal element in F and set E0 = F ∈S F . If E0 = E we are finished. Suppose that E0 = E. Then by (ii) there is a finite-dimensional semisimple submodule W of E which is not contained in E0 . The submodule E0 ∩ W has a direct complement E1 in W , and thus E0 + E1 is a direct sum. Since E is locally finite-dimensional, so is E1 and thus there is a finite-dimensional nonzero submodule S of minimal dimension. Clearly S is a simple module. Since S ⊆ E1 , the sum E0 + S is direct. Hence the family S ∪ {S} is a member of F and is properly larger than S. This is a contradiction to the choice of S. Hence E0 = {0} and the proof is5 complete. (iv) ⇒ (i): We assume that E is a direct sum j ∈J Ej of submodules. Let F be a submodule of E. We must find a module complement of F5in E. For a fixed j ∈ J we either have Ej ⊆ F or Ej ∩ F = {0}. Let F1 = j ∈J, Ej ⊆F Ej and 5 E . Then E = F ⊕ F , and by the modular law we have F2 = 1 2 j ∈J, Ej ∩F ={0} j F = F1 ⊕ (F ∩ F2 ). Thus if we find a module complement of F ∩ F2 in F2 we are done. Thus in order to simplify notation, we might just as well 5 assume that F ∩Ej = {0} for all j ∈ J . The set  of all subsets I ⊆ J such that F ∩ 5j ∈I Ej = {0} is seen to be inductive again due to the fact that for every element x = j ∈J xj only finitely 5 many xj are nonzero. By Zorn’s Lemma let us pick a maximal subset I and set F = j ∈I Ej . Then clearly F ∩ F =5 {0}. We claim that E = F + F . Suppose that this is not the case. Then there is an x = j ∈J xj ∈ E \ (F + F ). In particular, there must be an i ∈ J \ I

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such that xi ∈ / F + F . We may say x = xi ∈ Ei . It follows that (F + F5) ∩ Ei = {0} which in turn entails F ∩ (F + Ei ) = {0}5because otherwise 0 = y = j ∈I ∪{i} yj ∈ F ∩ (F + Ej ) would imply yi = y − j ∈I yj ∈ Ei ∩ (F + F ) = {0} and thus 0 = y ∈ F ∩ F which is not possible. But now I ∪ {i} ∈  contradicting the assumed maximality of I . This contradiction proves the claim and finishes the proof. Exercise E7.5. Prove the following result. Proposition. In the class of L-modules, the following statements hold: (i) A submodule of a semisimple module is semisimple. (ii) A quotient of a semisimple module is semisimple. (iii) A direct sum of locally finite-dimensional semisimple modules is semisimple. [Hint. (i) Let F be a submodule of a semisimple module E and assume that F1 is a submodule of F . Since E is semisimple, there is a submodule E2 of E such that def

E = F1 ⊕ E2 . Define F2 = F ∩ E2 . Show that F = F1 ⊕ F2 . (ii) Let f : E → F be a surjective module morphism. Since E is semisimple, we find a submodule F1 of E such that E = ker f ⊕ F1 . Show that f |F1 : F1 → F is an isomorphism of modules and invoke (i) above to show that F1 is semisimple. (iii) By Theorem 7.16 every summand is a direct sum of simple submodules and thus the direct sum is a direct sum of simple submodules. Hence by Theorem 7.16 again the direct sum is semisimple.] Corollary 7.17. Assume that E is a locally finite-dimensional L-module. Let Ess be the union of all finite-dimensional semisimple submodules. Then: (i) Ess is the unique largest semisimple submodule. (ii) Ess is a direct sum of finite-dimensional simple modules. Proof. (i) By Theorem 7.16, the sum of two finite-dimensional semisimple submodules is a finite-dimensional semisimple submodule. Hence the set of all finite-dimensional semisimple submodules is directed. Therefore its union is a submodule. By Theorem 7.16 again it is semisimple, and it is obviously the largest semisimple submodule. (ii) This is an immediate consequence of (i) and Theorem 7.16. Theorem 7.18. (a) Let V be a profinite-dimensional L-module for a Lie algebra L. Then the following statements are equivalent: (i) V is reductive. (i ) Every closed submodule is a direct module summand algebraically and topologically. (ii) Every finite-dimensional quotient module of V is reductive. (iii) V is the projective limit of finite-dimensional reductive module quotients. (iv) V is isomorphic to a product of finite-dimensional simple modules. (b) Every profinite-dimensional L-module has a unique smallest submodule V ss such that V /V ss is reductive.

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Proof. Via duality 7.11, (a) follows from 7.16 and (b) from 7.17. In any L-module E we have y · (x · v) = yx · v − xy · v + x · (y · v) − x · v + x · v = [x, y] · v + x · (y · v − v) + x · v; hence the linear span of all x · v as (x, v) ranges through L × E is a submodule Eeff . Definition 7.19. For an L-module E define E0 = {v ∈ E | (∀x ∈ L) x · v = 0}; this set is clearly a submodule, called the maximal zero submodule. The submodule Eeff is called the effective submodule. In a locally finite-dimensional L-module E the submodule Ess is called the semisimple radical. Clearly, in E0 , every vector subspace is a submodule. If E is a locally finite-dimensional L-module, then E0 is just a vector space and thus is isomorphic to a direct sum R(I ) for some set I ; if V is a profinite-dimensional L-module, then V0 is just a weakly complete topological vector space and thus is isomorphic to RI for some set I (Corollary A2.9). In a locally finite-dimensional L-module, E0 ⊆ Ess . Proposition 7.20. (i) Assume that E is a simple L-module. Then E agrees with E0 if and only if it is zero or one-dimensional; if this is not the case, then E agrees with Eeff . In the latter case, if 0 = v ∈ E, then vL = E. (ii) 5If E is a semisimple locally finite-dimensional L-module, then E is the direct sum j ∈J Ej of a family of simple submodules Ej of E, and 6 E0 = Ej , (3) j ∈J dim Ej =1

Eeff =

6

Ej .

(4)

j ∈J dim Ej >1

In particular, E = E0 ⊕ Eeff .

(5)

Proof. (i) Let E be a simple L-module. Then E0 is either {0} or E. In the latter case, every vector subspace being a submodule, its dimension is either 0 or 1. In the former case, E = Eeff . Let 0 = v ∈ E. Then L · v is a submodule. Since E0 = {0} it is nonzero. Hence L · v = E 5 (ii)5 From 7.16 we deduce the direct 5 sum representation E = j ∈J Ej . Define F0 = j ∈J, dim Ej =1 Ej , and F1 = j ∈J, dim Ej >1 Ej . Let 0 = vj ∈ Ej . Then " {0} if dim Ej = 1, L · xj = Ej if dim Ej > 1 by 5 (i) above and thus F0 ⊆ E0 , F1 ⊆ Eeff . Now let v ∈ E0 . Then 0 = x · v = j ∈J x · vj for all x ∈ L, and thus x · vj = 0 for all x ∈ L and j ∈ J . Hence vj ∈ E0 for all j and thus v ∈ F0 . Therefore F0 = E0 .

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But Eeff = (F0 ∩ Eeff ) ⊕ F1 . Now let 5 F0 ⊕ F1 = E and F1 ⊆ Eeff imply5 v = j ∈J,dim Ej =1 vj ∈ F0 ∩ Eeff . Then v = nm=1 xm · wm with xm ∈ L, wm ∈ E. 5 Now wm = j ∈J wmj with wmj ∈ Ej , and v=

6

xm · wmj =

m=1,...,n j ∈J

6

xm m=1,...,n j ∈J dim Ej >1

· wmj ∈ F1 .

But then v ∈ F0 ∩ F1 = {0}, and therefore v = 0. It follows that F0 ∩ Eeff = {0}, and thus that F1 = Eeff . Proposition 7.21. Let V be a profinite-dimensional L-module and V  its dual module.  and (V )⊥ = V  . Then (V0 )⊥ = Veff eff 0 Proof. Consider an ω ∈ V  ; then ω ∈ (Veff )⊥ iff ω(x · v) = 0, for all x ∈ L and v ∈ V iff x · ω = 0 for all x ∈ L iff ω ∈ V0 . Thus V0 = (Veff )⊥ . Similarly  )⊥ by duality. Applying ⊥, in view of the Duality Theorem 7.11 (v) we get V0 = (Veff ⊥  )⊥⊥ = V  = V  since every vector subspace of V  is closed. V0 = (Veff eff eff The Structure Theorem of Reductive Profinite-Dimensional L-Modules Theorem 7.22. Let V be a reductive profinite-dimensional L-module. Then  V is isomorphic to a product V0 × Veff , and V0 ∼ = RI for some set I and Veff ∼ = j ∈J Vj where each Vj is a simple submodule of V such that 1 < dim Vj < ∞. Proof. Since the module V is reductive and profinite-dimensional, its dual V  is semisimple and locally finite-dimensional. Thus by Proposition 7.20 we have V  =  . Then Theorem 7.11 (v) and Proposition 7.21 imply V = (V  )⊥ ⊕ (V  )⊥ = V0 ⊕ Veff eff 0 ⊥ = V  /V  ∼ V  , and by V0 ⊕ (V0 )⊥⊥ = V0 ⊕ Veff . The dual of Veff is V  /Veff 0 = eff 7.20 (ii), this module is a direct sum of simple modules with finite dimension greater 1. Thus Veff by duality is a product of simple modules of finite dimension greater than 1.

Reductive Pro-Lie Algebras Let us apply some of these results to pro-Lie algebras. def

Lemma 7.23. Let g be a pro-Lie algebra and i a closed ideal such that h = g/i is simple. Then dim h < ∞ and the annihilator i⊥ in the coadjoint module gcoad of g is a finite-dimensional simple submodule and is isomorphic to the coadjoint module of h.

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Proof. We consider the adjoint module of g and its dual, the coadjoint module. The vector space  h is isomorphic to i⊥ (see [102, Theorem 7.30 (v)]) and thus i⊥ is isomorphic to the coadjoint module hcoad of h which is a simple Lie algebra and therefore has a simple adjoint module, whence the coadjoint module i⊥ is simple. Proposition 7.24. Let g be a pro-Lie algebra and  (g) the filter basis of closed ideals i such that dim g/i  < ∞. Further let gcoad = Hom(|g|, R) denote the coadjoint module. Then gcoad = i∈ (g) i⊥ , and i⊥ is a finite-dimensional module which is the coadjoint module of the finite-dimensional Lie algebra g/i. Proof. From the Duality Theorem 7.11 we know that (i⊥ ) ∼ = g/i and (g/i) = i⊥ ⊥ for i ∈  =  (g). Since  is a filter basis, {i | i ∈  } is a directed family of finitedimensional vector subspaces of  E. As g = limi∈ g/i we have E = colimi∈ i⊥ , and by 7.15 this means exactly E = i∈ i⊥ . Definition 7.25. (a) Let g be a topological Lie algebra. Then its center z(g) is the set {x ∈ g | (∀y ∈ g) [x, y] = 0}. The span of all elements [x, y], x, y ∈ L is a subalgebra of L, called the commutator subalgebra [g, g]. The notation g = [g, g] is also used. The closure [g, g] is called the closed commutator algebra. If [g, g] = g, then g is called perfect. (b) A pro-Lie algebra g is called reductive if its adjoint module gad is a reductive g-module. It is called semisimple if it is reductive and its center z(g) is zero. In terms of the adjoint module gad of g, the center is the maximal zero submodule (gad )0 , the commutator subalgebra [g, g] is the effective submodule (gad )eff . In particular, the center and the commutator algebra are ideals. We shall see presently that the commutator subalgebra of a reductive pro-Lie algebra is closed. The following lemma uses a piece of information on finite-dimensional simple Lie algebras which we provide in Appendix 3. Indeed by Corollary A3.3 we know that Lemma BR. In any finite-dimensional semisimple real Lie algebra, each element is the sum of at most two brackets. This allows us to prove the following lemma. Lemma 7.26. Let {s j | j ∈ J } be any family of finite-dimensional real semisimple Lie algebras. Then s = j ∈J sj is its own algebraic commutator algebra s = [s, s], that is, the linear span of all elements [X, Y ] for X, Y ∈ s. In particular, every semisimple pro-Lie algebra is perfect. Proof. For subsets A and B of a Lie algebra g we define brack (A, B) = n

n 6

 [Xj , Yj ] | Xj ∈ A, Yj ∈ B, j = 1, . . . , n .

j =1

By Lemma BR above we have sj = brack 2 (sj , sj ). Since  brack n (s, s) = brack n (sj , sj ) for n ∈ N j ∈J

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we note in particular that brack 2 (s, s) =

 j ∈J

brack 2 (sj , sj ) =



sj = s.

j ∈J

Clearly, brack 2 (g, g) ⊆ [s, s]. Thus s = [s, s] follows, and this says that s is perfect. The Structure Theorem of Reductive Pro-Lie Algebras Theorem 7.27. (a) For a pro-Lie algebra g the following conditions are equivalent. (i) g is reductive. (i ) Every closed ideal of g is an ideal direct summand algebraically and topologically. (ii) g is the product of a family of finite-dimensional simple or one-dimensional ideals of g. (b) Let g be a reductive pro-Lie algebra. Then the commutator algebra [g, g] is closed and is a product of finite simple real Lie algebras. Further g ∼ = z(g) ⊕ [g, g] algebraically and topologically, and z(g) ∼ = RI for some set I . (c) Every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. Proof. (a) The Lie algebra g is reductive iff the adjoint module gad is reductive, and by Theorem 7.18, this is the case iff gad is a product of finite-dimensional simple gmodules. Now a vector subspace of g is a submodule of the adjoint module gad iff it is an ideal of g. This remark completes the proof of the equivalence of (i), (i ) and (ii).  By (a) above g ∼ = RI × j ∈J sj with a family of simple finite-dimensional real Lie algebras sj . By the preceding Lemma 7.26, the second factor s satisfies [s, s] = s, and the first factor is abelian. The assertion follows. (c) This is a consequence of Theorem 7.18 (b) applied to the adjoint module of g. We shall call ncored (g) the coreductive radical. We shall say more about ncored (g) later in Theorem 7.66 and Theorem 7.67. From Definition 7.25 (b) and Theorem 7.27 (b) we get at once the following conclusions. Corollary 7.28. Any semisimple pro-Lie algebra is perfect. Structure of Semisimple Pro-Lie Algebras Corollary 7.29. For a pro-Lie algebra, the following statements are equivalent. (I) g is semisimple. (II) g is the product of a family of finite-dimensional simple ideals of g.

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Lemma 7.30. Assume that E is a locally finite-dimensional L-module and S is a semisimple submodule such that S0 = {0} and E/S is a zero module. Let K denote the set {x ∈ L : (∀v ∈ E) x · v = 0}. Then S = Eeff and if L/K is semisimple, then E is semisimple. In particular, (6) E = S ⊕ E0 . Proof. By Proposition 7.20, S = Seff , and therefore Seff ⊆ Eeff . But Eeff /S ⊆ (E/S)eff = {0} implies Eeff ⊆ S, and so S = Eeff . It remains to show that E is semisimple. Let F be a finite-dimensional submodule. By Theorem 7.16 it suffices to show that F is semisimple. Now S ∩ F is contained in a def

finite sum D = Ej1 ⊕ · · · ⊕ Ejn of simple submodules of dimension > 1; by replacing F by F +D we may assume that F ∩S = D. Then F /(F ∩S) ∼ = (F +S)/S ⊆ (E/S)0 . We are reduced to the finite-dimensional case and may assume that dim E < ∞. Let F1 be a submodule of E containing S such that dim F1 /S = 1; if we can show that F1 is semisimple, then F1 = S ⊕ R · x1 with an element x1 such that L · x1 = {0}. Since this works for arbitrary one-dimensional submodules of E/S we will obtain E = S ⊕ R · x1 ⊕ · · · ⊕ R · xk , k = dim E/S, which will finish the proof. Hence we may as well assume that dim E/S = 1. We may assume that K = {0} because E is an L/K-module in the obvious way, and submodules and quotient modules are the same. But then, as we assume that L is semisimple, the assertion follows from [16, §6, no 3, Définition 3]. Proposition 7.31 (Levi–Mal’cev Theorem for Reductive Pro-Lie Algebras). Let g be a pro-Lie algebra such that g/z(g) is semisimple. Then g is reductive and g is an ideal direct sum, algebraically and topologically, g = z(g) ⊕ [g, g].

(7)

Proof. We apply Lemma 7.30 with L = g, E = gcoad , S = z(g)⊥ , K = z(g). Then, by duality, we have the module isomorphisms S ∼ = (g/z(g)) and E/S ∼ = z(g) . By hypothesis, g/z(g) is semisimple and thus, by Definitions 7.15 and 7.27 (i), S is a semisimple module. Also, since the g-submodule z(g) of the adjoint module g is the zero module, its dual E/S is a zero module. The module L/K = g/z(g) is semisimple. Thus Lemma 7.30 applies and shows that E is semisimple and this means by Definition 7.15 (iv), g is reductive. So 7.28 implies g = z(g) ⊕ [g, g]. Now [g, g] ∼ = g/z(g) is a semisimple pro-Lie algebra, and then g is a reductive one. Thus (7) follows from Theorem 7.27.

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Transfinitely Solvable Lie Algebras The familiar concept of solvability of a finite-dimensional Lie algebra requires some thought in the absence of dimensional restrictions. We are dealing with topological Lie algebras, and thus there are added complications due to the possible definition of “topological solvability” which again is not apparent in the situation of finite-dimensional Lie algebras. Similar comments apply to nilpotency which we shall treat in a way that is parallel to that of solvability. Definition 7.32. Let g be a Lie algebra. Set g(0) = g and define sequences of ideals g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g(α) is defined for α < β.  (i) If β is a limit ordinal, set g(β) = α 0)(∀t ∈ K)

(|t| > r) ⇒ (v ∈ t · U ).

A balanced set is absorbing if every vector is contained in a multiple of the set. Recall that a topological vector space over K (or over any topological field K, for that matter) is an abelian topological group E with a continuous scalar multiplication (t, v)  → t · v : K × E → E. Exercise EA2.1. Show that the filter of zero neighborhoods U in a topological vector space satisfies the following:  (0) U = {0}. (i) (∀U ∈ U)(∃V ∈ U) V − V ⊆ U . (ii) (∀U ∈ U)(∃V ∈ U) D · V ⊆ U . (iii) Every U ∈ U is absorbing. Conversely show that, if a filter U satisfies (i), (ii), (iii), then the set O of all subsets U of E such that for v ∈ U there is a W ∈ U with v + W ⊆ U is a vector space topology whose filter of identity neighborhoods in U. If it also satisfies (0), then O is a Hausdorff topology. (We shall always assume that.) A subset P of a topological group is called precompact, if for every open subset U there is a finite subset F such that P ⊆ F U . We call an abelian topological group G locally precompact if there is a precompact identity neighborhood. For topological vector spaces over the reals (or indeed locally compact fields), the finite-dimensional ones form a topologically distinguished class. Proposition A2.2. (i) On a one-dimensional K-vector space E, outside the indiscrete topology {Ø, E} there is only one vector space topology. For each 0 = v ∈ E the map r  → r · v : K → E is an isomorphism of topological vector spaces. (ii) A locally compact subgroup H of a Hausdorff topological group G is a closed subset. (iii) A finite-dimensional K-vector space admits one and only one vector space topology. If E is a K-vector space with dim E = n and E → Kn is an isomorphism then it is an isomorphism of topological vector spaces where Kn has the product topology. (iv) A locally precompact topological vector space over K is finite-dimensional. Proof. (i) Observe that {0} is a vector subspace. Since dim E = 1 there are only two possibilities: {0} = E, yielding the indiscrete topology, or {0} = {0} which we assume. Then the topology on E is Hausdorff. Let 0 = e ∈ K, set f : K → E, f (t) = t · e. Let def

V = f −1 (U(E)) be the inverse image of the filter of zero neighborhoods of E. Since f is linear, V has a basis of balanced and absorbing sets. In K a set V is balanced iff it is of the form V = {r ∈ K : |r| < ε} or of the form V = {r ∈ K : |r| ≤ ε}. The

Character Groups of Topological Vector Spaces

631

only one among these which is not absorbing is {0}. Note that V intersects in {0} and conclude that V is the neighborhood filter of 0 in K. (ii) It is no loss of generality to assume G = H ; show H = G. Let K be a compact identity neighborhood of H and let U be an open identity neighborhood of G with U ∩ H ⊆ K. Consider g ∈ G; we must show g ∈ H . Since H is dense in G there is an h ∈ H ∩ U −1 g, say h = u−1 g. Since U is open, U ∩ H is dense in U . Thus U ∩ K is dense in U , i.e. U ⊆ K = K (as K is compact in a Hausdorff space). Thus u ∈ U ⊆ K ⊆ H , whence g = uh ∈ H H ⊆ H . (iii) Let e1 , . . . , en be a basis of E and set f : Kn → E, f (x1 , . . . , xn ) = x1 · e1 + · · · + xn · en . Prove by induction on n that f is an isomorphism of topological vector spaces. In (i) one dealt with n = 1. Consider vector subspaces def

def

N = K · e1 ⊕ · · · ⊕ K · en and H = K · en+1 . Show that (n, h)  → n + h : N × H → E is an isomorphism of topological vector spaces. This map is a continuous algebraic isomorphism. We must show that its inverse is continuous. By the induction hypothesis N ∼ = Kn , H ∼ = K, and so both vector spaces are locally compact hence closed by (ii). Then E/N is Hausdorff, and hence isomorphic to K by (i). Thus h  → h + N : H → E/N is an isomorphism of topological vector spaces by (i). Hence the projection of pH : E → H , pH (x1 · e1 + · · · + xn+1 · en+1 ) = xn+1 · en+1 , factoring through the quotient E → E/N and the isomorphism E/N → H is continuous. Hence the projection pN = id −pH onto N is continuous. So (n, h)  → n + h has a continuous inverse. (iv) Let U be a balanced zero neighborhood of E such that U +U is precompact. The sets U +u, u ∈ U form an open cover of U +U by translates of an identity neighborhood. By precompactness there is a finite subset F ⊆ U such that U + U ⊆ U + F . Let E1 = spanK F . Then U + U ⊆ U + E1 . Since the vector space E1 is finite-dimensional and therefore locally compact by (iii), it is closed by (ii). Hence E/E1 is Hausdorff. Set V = (U + E1 )/E1 , then V is a balanced 0-neighborhood of E/E1 satisfying V + V ⊆ V = −V . Then V is a vector space. However, V is precompact since U is precompact. Claim: V is a singleton. Suppose it is not a singleton; then by (i) the K-vector space contains a vector subspace isomorphic to K. Hence K would have to be precompact. But then the subspace N ⊆ K would have to be precompact, but it is not because it cannot be covered by a finite number of translates of a disc of radius 21 . Thus V is singleton and E = E1 . For a vector space E over K we shall denote the set of all finite-dimensional vector subspaces by Fin(E). For a topological vector space E over K we denote the set of cofinite-dimensional closed vector subspaces (i.e. closed vector subspaces M with dim E/M < ∞) by Cofin(E). A topological vector space E over K is called locally convex if every zero neighborhood contains a convex one. Now let E be any real vector space and let B(E) denote the set of all balanced, absorbing and convex subsets of E. Let us observe that there are plenty of those, in fact enough to allow only 0 in their intersection. Let F be a basis

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of E and ρ : F → ]0, ∞[ any function. Then the set 6  U (F ; ρ) = re · e : |re | < ρ(e) e∈F

is balanced, absorbing and convex. We call it a box neighborhood with respect to F . The box neighborhoods with respect to a single basis already intersect in 0. Thus the filter of all supersets of sets from B(E) satisfies (0), (i), (ii), and (iii) of E7.12. If we set O(E) = {W ⊆ E | (∀w ∈ W )(∃U ∈ B(E)) w + U ⊆ W }, then O(E) is a locally convex vector space topology. From its definition it is immediate that it contains every other locally convex vector space topology. It is clearly an algebraic invariant in so far as it depends only on the vector space structure of E. A convex subset U of E belongs to O(E) if and only if for every u ∈ U and every x ∈ E the set {r ∈ R | u + r · x ∈ U } is an open interval of R containing 0. It follows that a convex subset U of E belongs to O(E) if and only if for each finite-dimensional vector subspace F and each v ∈ E the intersection F ∩ (U − v) is open in F (in the unique vector space topology of F ). Let us record some of the basic properties of O(E). We shall see that O(E) is, in essence, a purely algebraic invariant of the real vector space E. Proposition A2.3. Let E be an arbitrary vector space over K. (i) If E1 and E2 are vector spaces, T : E1 → E2 is a linear map, and E2 is a locally convex topological vector space, then T is continuous for the topology O(E1 ). In particular, every algebraic linear form E → K is O(E)-continuous; i.e. the algebraic dual E ∗ = HomK (E, K) is the underlying vector space of the topological  (which is considered to carry the compact open topology). dual E  = Hom(E, K) ∼ =E (ii) Every vector subspace of E is O(E)-closed and is a direct summand algebraically and topologically. Moreover, the topology induced on each vector subspace is its finest locally convex topology. (iii) Let F be a linearly independent subset of E. Then there is a zero neighborhood U ∈ O(E) such that {v + U | v ∈ F } is a disjoint open cover of F . In particular, any linearly independent subset of (E, O(E)) is discrete. (iv) If C is an O(E)-precompact subset, then spanK (C) is finite-dimensional. Proof. (i) If U is any balanced and convex zero neighborhood of E2 then T −1 (U ) is balanced, absorbing and convex and thus belongs to B(E1 ). This shows the continuity of T with respect to O(E1 ). The remainder of (i) then follows at once. (ii) Let E1 be an arbitrary vector subspace of E and let E2 be a vector space complement; i.e. E = E1 ⊕E2 . The function x  → (pr 1 (x), pr 2 (x)) : E → E1 ×E2 is a vector space isomorphism and then is continuous by (i). The function α : E1 ×E2 → E, α(x, y) = x + y, is its inverse. Since it is the restriction of the continuous addition (x, y)  → x + y : E × E → E to the subspace E1 × E2 it is continuous. Hence α is an isomorphism of topological vector spaces. This proves assertion (ii).

Character Groups of Topological Vector Spaces

633

(iii) Let F be a linearly independent subset. Since by the Axiom of Choice, F can be supplemented to a basis {ej | j ∈ J } we may just as well assume that F is this basis. Let σ : F → ]0, ∞[ be the function with the constant value 21 . We claim that for different elements e ∈ F we have   (e + U (F ; σ )) ∩ f ∈F, f =e f + U (F ; σ ) = Ø. Indeed if ω : E → K is the linear functional defined by ω(f ) = 0 for e = f ∈ F and by ω(e) = 1, then    ' 1 ' ω  f ∈F, f =e f + U (F ; σ ) ⊆ 0, 2 and |ω(e + U (F ; σ )| ⊆

&1

2, 1

'

.

This proves (iii). (iv) Let K be a precompact subset of (E, O(E)). We want to show that dim spanK K < ∞. It is no loss of generality to assume that E = spanK K. Then we can select a basis B = {ej | j ∈ J } of elements ej ∈ K. Then B is precompact. By (iv) we find an open zero neighborhood U such that ej + U is a disjoint cover of B by translates of U . Since B is precompact this implies that J is finite. Hence dim E = card J < ∞. Our main interest will be with vector spaces dual to those we just discussed. Their topology was determined by the finite-dimensional vector subspaces. Dually we may consider vector space topologies which are determined by the cofinite-dimensional closed vector subspaces. Proposition A2.4. Let E be a topological vector space. Then for M, N ∈ Cofin(E) with N ⊆ M, there is a canonical quotient map qMN : E/N → E/M. Since Cofin(E) is a filter basis, there is an inverse system and, in the category of topological vector spaces, there is a projective limit ECofin(E) = limM∈Cofin(E) E/M, the vector subspace of all  E/M (vM + M)M∈Cofin(E) ∈ M∈Cofin(E)

such that N ⊆ M implies vN − vM ∈ M. The function γE : E → ECofin(E) , γE (v) = (v + M) M∈Cofin(E) is a morphism of topological vector spaces which is injective if and only if Cofin(E) = {0}. Proof. This is straightforward. The Definition of Weakly Complete Topological Vector Spaces Definition A2.5. The topology O(E) is called the finest locally convex vector space topology on E. A topological vector space E is called weakly complete if γE : E → ECofin(E) is an isomorphism of topological vector spaces. The topology O  (E) on E making γE : E → ECofin(E) a topological embedding is called the weak topology on E.

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The weak topology on a K-vector space is the smallest topology making all linear functionals f : E → K continuous. All finite-dimensional vector spaces are weakly complete. On a weakly complete topological vector space, the continuous functionals separate points. Our definition in terms of the projective limit establishes completeness in the weak topology as the name of a “weakly complete topological vector space” suggests. Recall from A2.3 (i) that the algebraic dual E ∗ of a real vector space E is at the same time the topological dual E  , consisting of all continuous linear functionals on E when E is endowed with the finest locally convex topology. On the basis of bare linear algebra one always has the weak ∗-topology on E ∗ , that is, the topology of pointwise convergence induced by the natural inclusion E ∗ → RE . The first item in the following lemma will show that this topology agrees with the topology of uniform convergence on compact sets which is the topology we consider in order to have the isomorphism  according to A2.1. E ∼ =E Lemma A2.6. Let E be a real vector space endowed with its finest locally convex vector space topology.  is the weak∗ -topology, i.e. the (i) Then the compact open topology on E  ∼ = E topology of pointwise convergence (ii) Every continuous linear functional  : E  → R is of the form ω  → ω(x) :  E → R for a unique x ∈ E. (iii) F  → F ⊥ : Fin(E) → Cofin(E  ) is an order reversing bijection. (iv) E  is weakly complete. Proof. (i) The compact open topology on E  is generated by the basic zero-neighborhoods VE (C, U ) for a compact subset of E and some zero neighborhood U of R and it is always equal to or finer than the weak∗ -topology. It is no loss of generality to assume that C is convex balanced, since C is contained in a finite-dimensional subspace by A2.3 (iv), where the closed convex circled hull C ∗ of a compact set C is compact and then VE (C ∗ , U ) ⊆ VE (C, U ). Assume C is convex balanced now. There is an ε > 0 such that ] − ε, ε[ ⊆ U , and then , 1 VE C, ] − 1, 1[ = VE (C, ] − ε, ε[) ⊆ VE (C, U ). ε Thus we may assume that the filter of zero-neighborhoods of E  for the compact open topology is generated by basic zero neighborhoods of the form VE (C, ] − 1, 1[) as C ranges through the compact convex and balanced subsets of E. Now span C is def

finite-dimensional; then there is a basis e$1 , . . . , en of span C such that C ⊆ K = # 5n r · e | |r | ≤ 1, j = 1, . . . , n , where K is the convex balanced hull of j j j j =1 {e1 , . . . , rn }. Then VE ({e1 , . . . , en }, ] − 1, 1[) = VE (K, ] − 1, 1[) ⊆ VE (C, ] − 1, 1[). Hence the topology of E  and the weak∗ -topology agree.

Character Groups of Topological Vector Spaces

635

(ii) Let  : E  → E be a continuous linear functional. By (i), its continuity implies the existence of a finite set F ⊆ E of vectors in E such that (VE (F, ] − 1, 1[) ⊆] − 1, 1[. Since F ⊥ ⊆ E  is contained in VE (F, ] − 1, 1[) we have (F ⊥ ) ⊆] − 1, 1[, and since (F ⊥ ) is a vector space, we conclude F ⊥ ⊆ ker(). Therefore we have a linear functional  : E  /F ⊥ → R such that  =   q for the quotient map q : E  → E  /F ⊥ . If x∈ E then x ⊥ = {ω ∈ E  : ω(x) = 0} is a closed hyperplane in E  , and thus F ⊥ = x∈F x ⊥ is a finite intersection of hyperplanes. Thus F ⊥ ∈ Cofin(E  ) and thus E  /F ⊥ is a finite-dimensional vector space. Its dual may be identified with span F in the sense that every linear functional of E  /F ⊥ is of the form ω + F ⊥  → ω(x) for an x ∈ span F . We apply this to  and find some vector x ∈ span F such that  (ω) = ω(x). Then (ω) = ω(x) for all ω ∈ E  , and this is what we had to show. (iii) If F ∈ Fin(E), then F ⊥ ∈ Cofin(E  ) as we saw in the proof of (i). Conversely let M ∈ Cofin(E  ). Then M is the intersection of finitely many closed hyperplanes. Each one of these is the kernel of a continuous functional ; we saw in (ii) that each one of these is of the form ω  → ω(x). Hence M is the annihilator M = F ⊥ of some F ∈ Fin(E). Hence F  → F ⊥ : Fin(E) → Cofin(E  ) is a containment reversing bijection. (iv) The function κE,F : E ∗ /F ⊥ → F ∗ given by κE,F (ω + F ⊥ ) = ω|F for ω ∈ E ∗ is a bijection by an elementary annihilator mechanism of linear algebra. (The map is clearly seen to have zero kernel and to be surjective, since by the Axiom of Choice E = F ⊕ F1 with a vector space complement so that every functional of F extends such has F1 in its kernel.) In order for (ωF + F ⊥ ))F ∈Fin(E) ∈  that the∗ extension ⊥ ∗ ⊥ F ∈Fin(E) E /F to be in lim F ∈Fin(E) E /F it is necessary and sufficient that F1 ⊆ F2 in Fin(E) implies ωF2 |F1 = ωF1 . Thus in view of (i), (ii) above there is an isomorphism ϕ:

lim

M∈Cofin(E ∗ )

E ∗ /M →

lim

F ∈Fin(E)

F ∗,

ϕ((ωF + F ⊥ )F ∈Fin(E) ) = (ωF |F )F ∈Fin(E) .

Now ψ : E ∗ → limF ∈Fin(E) F ∗ , ψ(ω) = (ω|F )F ∈Fin(E) is a morphism which is injective, because E = F ∈Fin(E) F , whence ω|F = 0 for all F implies ω = 0. Moreover, if (ωF )F ∈Fin(E) ∈ limF ∈Fin(E) F ∗ , then F1 ⊆ F2 in Fin(E) implies ωF2 |F1 = ωF1 . Thus we may define unambiguously ω ∈ E ∗ by setting ω(x) = ωR·x (x) and obtain ψ(ω) = (ωF )F ∈Fin(E) . Then ψ is surjective and so is an isomorphism of vector spaces. We observe ψ = ϕ ◦ γE and thus γE ∗ : E ∗ → limM∈Cofin(E ∗ ) E ∗ /M is an isomorphism of vector spaces. Let Fin1 (E) be the set of one-dimensional subspaces  , j (ω) = (ω|F )F ∈Fin (E) is an embedding of of E. Then j : E ∗ → F ∈Fin1 (E) F 1 ∗ topological vector spaces if E is given the weak ∗-topology. Consider the projection π:

 F ∈Fin(E)

→ F

 F ∈Fin1 (E)

, F

π((ωF )ωF ∈Fin(E) ) = (ωF )F ∈Fin1 (E) .

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Appendix 2 Weakly Complete Topological Vector Spaces

The diagram

 F limF ∈Fin(E) ⏐ ⏐ incl  ∗ F ∈Fin1 (E) F

ψ −1

−−−−→ −−−−→ π



E⏐∗ ⏐j  F ∈Fin1 (E) F

∗

is commutative. Since j is an embedding and j ◦ ψ −1 = π ◦ incl is continuous, it follows that ψ −1 is continuous and so that ψ is an isomorphism of topological vector spaces. Hence γE ∗ is an isomorphism of topological vector spaces. This shows that E ∗ is a weakly complete topological vector space in the weak ∗-topology, and thus E  is weakly complete in the compact open topology by (i). Let E be a locally convex topological vector space over K and E  its topological dual. If ηE : E → E  , ηE (x)(ω) = ω(x), denotes the evaluation morphism, then for  def each subset H ⊂ E we set H ◦ = {ω ∈ E  : |ω(H )| ⊆ [0, 1]} = h∈H ηE (h)−1 B1 with B1 = {r ∈ K : |r| ≤ 1} and call this set the polar of H in E  . Similarly for a def

subset  ⊆ E  we define the polar of  in E to be ◦ = {x ∈ E : |(x)| ⊆ [0, 1]} =  −1 ω∈ ω (B1 ). Again as in the case of annihilators of subsets of abelian topological groups one must specify where the polars are taken. Polars are always closed. For the following recall that E is called semireflexive if the morphism ηE : E → E  , ηE (x)(ω) = ω(x) is bijective. It is called reflexive if ηE is an isomorphism of topological groups. Lemma A2.7 (The Bipolar Lemma). Let E be a locally convex vector space and U be a convex balanced subset of E. Let  be a convex balanced subset of E  . Then (i) U  = U , and (ii) if E is semireflexive,  = . Proof. (i) The taking of polars is containment reversing. Hence U ⊆ U  and since polars are closed we have U ⊆ U  . In order to prove the converse containment let x ∈ U  . This means that |U ◦ (x)| ⊆ [0, 1]. We claim that this implies x ∈ U . Suppose it does not. Then by the Theorem of Hahn and Banach (see e.g. [15]), there is a real linear functional ρ : E → R with ρ(x) > 1 and ρ(U ) ⊆ [−1, 1]. If K = R, then this says that ρ ∈ U ◦ , contradicting |U ◦ (x)| ⊆ [0, 1]. If K = C, then ω(y) = ρ(x) − iρ(i · x) defines a complex linear functional such that |ω(x)| ≥ Re |(ω(x))| = ρ(x) > 1, while, on the other hand, for 0 = u ∈ U we define z with |z| = 1 by zω(u) = |ω(u)|. Then zu ∈ U and |ω(u)| = zω(u)ω(zu) = Re ω(zu) = ρ(zu) = |ρ(zu)| ≤ 1. Thus ω ∈ U ◦ . This proves the claim. (ii) If E is semireflexive, then E may be identified with the vector space of all continuous linear functionals of E  . Since E  is locally convex, the proof of part (i) applies here and proves the assertion. We shall stay in the category AbTopGr of all abelian topological groups, and we shall denote the full subcategory of all reflexive abelian topological groups by RefTopGr,

Character Groups of Topological Vector Spaces

637

 η (x)(χ ) = χ(x), is  that is all abelian topological groups G such that ηG : G → G, G an isomorphism of abelian topological groups. In the following theorem we shall see that for each real vector space E endowed with its finest locally convex topology, the additive group of E belongs to RefTopGr, and that for each weakly complete topological vector space V , the additive group of V belongs to RefTopGr. Duality of Real Vector Spaces Theorem A2.8. Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then:   is an isomorphism of topological vector (i) E is reflexive; that is, ηE : E → E spaces.   is an isomorphism of topological vector (ii) V is reflexive; that is, ηV : V → V spaces. (iii) The contravariant functor  · : RefTopGr → RefTopGr exchanges the full subcategory of real vector spaces (given the finest locally convex topology) and the full subcategory of weakly complete topological vector spaces.  are isomorphic by 7.5 (iii). Proof. (i) The topological dual E  and the character group E Since the linear functionals separate the points, ηE is injective. By A2.6 (ii), ηE is   may be identified. surjective. As vector spaces, therefore, E, E  , and E  ∼  is compact if and only if it is closed in E  and for each compact A subset K in E = E subset C of E and each ε > 0 there is an M ∈ B(E) such that |K(C ∩ M)| ⊆ [0, ε[ (see for instance [102, Proposition 7.6]). In view of A2.3 (iv) we may express this as follows: (∀F ∈ Fin(E), ε > 0)(∃M ∈ B(E))

K(F ∩ M) ⊆ [0, ε].

(∗)

We shall now show that B(E) has a basis of basic zero neighborhoods VE  (K, Bε ) = {x ∈ E : |K(x)| ⊆ [0, ε]} for the compact open topology on E when E is identified   For a proof of the claim let U ∈ B(E). Set K def = {ω ∈ E  : |ω(U )| ⊆ with E  ∼ = E. ◦ [0, 1]} = U . We claim that K is compact. Clearly, K is closed in E  since E  has the topology of pointwise convergence by (i) above. Let F ∈ Fin(E) and ε > 0. Then |K(F ∩ ε · U )| ⊆ ε · [0, 1] by the definition of K. Hence (∗) is satisfied and K is compact as asserted. Now we observe that VE  (K, B1 ) = U  . By the Bipolar Lemma A2.7 we have VE  (K, B1 ) = U . Since the filter basis B(E) has a basis of closed sets, we have shown that it has a basis of sets VE  (K, B1 ). Thus the compact open topology on E is finer than or equal to the given topology of E which is the finest locally convex topology. But since the sets VE  (K, B1 ) are convex, the two topologies agree and thus E is the dual of E  and E is reflexive. (ii) Set E = V  . Then each nonzero ω ∈ E has a closed hyperplane as kernel, and each closed hyperplane is the kernel of such an ω. It follows that M  → M ⊥ :

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Appendix 2 Weakly Complete Topological Vector Spaces

Cofin(V ) → Fin(E) is an order reversing bijection. Now let  ∈ E  and consider F ∈ Fin(E). Set M = F ⊥ . Now F may be canonically identified with the dual of V /M so that ω, v + M = ω(v) for ω ∈ F = M ⊥ , v ∈ V (see the Annihilator Mechanism [102, Lemma 7.17 (i)]). But F and V /M are finite-dimensional vector spaces which are reflexive. Hence there is a unique element M ∈ V /M such that ω, |F  = ω, M  for all ω ∈ F . Moreover, if F1 ⊆ F2 then M2 = F2⊥ ⊆ F1⊥ = M1 and since map qM1 M2 : V /M2 → V /M1 maps M2 to M1 . (|F2 )|F1 = |F1 the quotient  Hence (M )M∈Cofin(V ) ∈ M∈Cofin(V ) V /M is contained in limM∈Cofin(V ) V /M. By hypothesis, V is weakly complete; hence there is an element v ∈ V such that v + M = def

M . Let ω ∈ E. Then F = K · ω ∈ Fin(E). Thus, letting M = F ⊥ we get (ω) = ω, |F  = ω, M  = ω, v + M = ω(v). Hence  = ηV (v). This suffices to show that V is semireflexive. Now we investigate the compact open topology of E and show that it agrees with the finest locally convex topology. Let U be in the set B(E) of all balanced, absorbing, def

and convex subsets of E. Set U ◦ = {x ∈ V : |U (x)| ⊆ [0, 1]} and consider 0 = ω ∈ E. Set M = ω−1 (0). Since U is absorbing, Bε · ω ⊆ U for some def

ε > 0, Bε #= {r ∈ K : |r| ≤ ε}. $Thus U ◦ ⊆ (Bε · ω)◦ = {x ∈ V : |(Bε · ω)(x)| ⊆ [0, 1]} = x ∈ V : |ω(x)| ≤ 1ε = ω−1 B1/ε . Since V = limM∈Cofin(V ) V /M, the sets ω−1 (Br ), ω ∈ E and r > 0 are subbasic zero neighborhoods of V (meaning that the collection of all finite intersections of these form a basis of the filter of zero neighborhoods), and the topology of V is the smallest making all ω ∈ E continuous. E\{0} is an embedding and U ◦ is mapped onto a Thus x  → (ω(x)) ω∈E\{0} : V → K closed subset of ω∈E\{0} Br(ω) for a family of positive numbers r(ω). Hence U ◦ is compact in V . Now U  = {ω ∈ E : ω(U ◦ ) ⊆ B1 } is a zero neighborhood for the compact open topology. Since V is semireflexive, by the Bipolar Lemma A2.7 (ii), U  = U . Thus the closed sets which are members of B(E) are zero neighborhoods of the compact open topology. This says that the compact open topology of E and the finest locally convex topology of E agree. Now E  carries the topology of pointwise convergence by A2.6 (i); by the characterisation of the topology of V just derived we know that ηV : V → V  = E  is an algebraic and topological embedding. Since we have seen ηV to be bijective, it is an isomorphism of topological vector spaces. (iii) is a consequence of (i) and (ii). Since we know, using the Axiom of Choice, that a real vector space has a basis and thus is a direct sum of copies of R, duality instantly gives us the following corollary Weakly Complete Topological Vector Spaces Explicitly Corollary A2.9. For a topological vector space V the following conditions are equivalent: (i) V is a weakly complete topological vector space. (ii) There is a set J such that V is isomorphic as a topological vector space to RJ .

Duality: Structure Theory of Weakly Complete Topological Vector Spaces

639

Proof. The dual of a sum, that is a coproduct of vector spaces in the category of real vector spaces is a product in the category of topological vector spaces. The cardinal card J is called the topological dimension of W . (See [103].) Thus the topological dimension of a weakly complete topological vector space is the linear dimension of its dual. In this spirit, the weakly complete topological vector spaces are generalisations of the familiar euclidean vector spaces Rn , and we provide enough evidence in this book that they are the correct generalisation. As an exercise on the duality of a vector spaces we present a lemma which we use in Chapter 3 where we amply use limits of projective systems of Lie groups and Lie algebras. Corollary A2.10. Let g = limk∈J gk be a projective limit of a projective system {γj k : gk → gj | j ≤ k, (j, k) ∈ J × J } of finite-dimensional real vector spaces in the category of weakly complete topological vector spaces. Let γj : g → gj denote the limit maps. Then for each j ∈ J there is an index kj ≥ j such that γj kj (gk ) ⊆ γj (g). Proof. Since gj is finite-dimensional γj (g) is a closed vector subspace of gj . By the Duality Theorem A2.8 (iii), the corollary is equivalent to the following assertion: (∗) Let E = colimk∈J Ek be the direct limit of a direct system {ηj k : Ej → Ek | j ≤ k, (j, k) ∈ J × J } of real vector spaces. Fix an index j ∈ J . Then there is an index kj ≥ j such that ηj kj vanishes on ker ηj . Now E is the directed union of the images ηk (Ek ). If x ∈ Ej is such that ηj k (x) = 0 for all k, then ηj (x) = 0. Thus for each x ∈ ker ηj there is a kx ≥ j such that ηj kx (x) = 0. Since dim ker ηj ≤ dim Ej is finite, ker ηj is finitely generated. Statement (∗) follows.

Duality at Work for the Structure Theory of Weakly Complete Topological Vector Spaces Duality: Vector Subspaces Theorem A2.11. Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then (i) Every closed vector subspace V1 of V is algebraically and topologically a direct summand; that is there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces.

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Appendix 2 Weakly Complete Topological Vector Spaces

(ii) For every closed vector subspace H of E, the relation H ⊥⊥ = H ∼ = (E  /H ⊥ )  ⊥ ⊥  holds and E /H is isomorphic to H . The map F  → F is an antiisomorphism of the complete lattice of vector subspaces of E onto the lattice of closed vector subspaces of E  . E   = E/H  H   ∼ /H ⊥ = V  {0}

 = E/H



{0}    H⊥   V.

Proof. (i) Let E = V  . Then V  is a vector space with its finest locally convex topology by (iii). Let E1 = V1⊥ . Then there is a vector subspace E2 such that E = E1 ⊕ E2 algebraically and topologically by A2.3 (ii). Set V2 = E2⊥ and conclude V = V2 ⊕ V1 2 and V2 ∼ 1 . This proves (ii). with V1 ∼ =E =E (ii) We consider a closed vector subspace H of E and note that E is reflexive by (i) above and that the linear functionals of E/H , all of which are continuous by A2.3 (i), separate the points. Then the Annihilator Mechanism applies (see for instance [102, Lemma 7.17 (iii)]) and shows H = H ⊥⊥ . Since E is reflexive and the continuous functionals, and thus the characters of E/H , separate points, [102, 7.17 (v)] shows that (E  /H ⊥ ) and H are isomorphic topological vector spaces. By A2.3 (ii) we have E = H ⊕ K, and by A2.8 (ii) and its proof we obtain E  = K ⊥ ⊕ H ⊥ with a  of E  . Then closed and hence weakly complete topological vector subspace K ⊥ ∼ =H  ⊥ ⊥ E /H ∼ = K is a weakly complete topological vector space and thus is reflexive  (see for instance [102, Lemma 7.17 (vi)]). In by (ii). This implies that E  /H ⊥ ∼ =H particular, E  /H ⊥ is reflexive by (ii). The last assertion of (iii) is now a consequence of this. Duality: Morphisms Theorem A2.12. (a) Let f : V → W be a morphism of weakly complete topological vector spaces. Assume that f has a dense image. Then f : V → W splits; that is, there is a morphism σ : W → V such that f  σ = idW . (b) Let f : V → W be a morphism of weakly complete topological vector spaces. Then f (V ) is a closed vector subspace of W , and the natural bijection V / ker f → f (V ) is an isomorphism of topological vector spaces. V ⏐ ⏐ q V / ker f

f

−−−→

W  ⏐j ⏐ −−−→ f (V )  f

(c) (The Second Isomorphism Theorem) If V and W are closed vector subspaces of a weakly complete topological vector space U , then V + W is closed, and the function

Duality: Structure Theory of Weakly Complete Topological Vector Spaces

641

f : V /(V ∩ W ) → (V + W )/W , f (v + (V ∩ W )) = v + W is an isomorphism of topological vector spaces. Proof. (a) Let f : V → W be a morphism of weakly complete topological vector spaces with dense image. Then the morphism f is an epimorphism in the category of →V  is a monomortopological vector spaces since it has a zero cokernel. Then f: W phism of real vector spaces with their finest locally convex topology by A2.8 (iii). Thus fhas a zero kernel and so fis injective. Now the linear algebra of vector spaces  → W  such provides, with the use of the Axiom of Choice, a linear function τ : V that τ  f = idW . By A2.3 (i), the map τ is continuous and is thus a morphism of     . By topological vector spaces. From (iii) we know that f  τ is the identity map of W (ii) we get a morphism σ : W → V such that   σ = τ and that f  σ = idW . (b) In the category of weakly complete topological vector spaces, we have a canonical decomposition f V −−−→ W ⏐  ⏐ ⏐j q ⏐ V / ker f

−−−→ f (V )  f

f  (v

+ ker f ) = f (v). After replacing f by f  where q(v) = v + ker f , j (w) = w, we may assume without loss of generality that f is injective and has a dense image. Then f is both a monic and an epic in the category of weakly complete topological vector spaces. By (a) it is also a retraction, and a monic retraction is an isomorphism. In particular, it is surjective and thus (b) is proved. (c) We define a morphism of weakly complete topological vector spaces F : V /(V ∩ W ) → U/W

by F (v + (V ∩ W )) = v + W.

By Part (a), the vector subspace (V +W )/V = F (V ) is closed in U , and thus V +W , the full inverse image under the quotient morphism U → U/W is closed in U . Moreover, the corestriction of the injective morphism F to the image. Namely, f : V /(V ∩W ) → (V + W )/W , is an isomorphism of weakly complete topological vector spaces. Filter Bases of Closed Linear Subspaces Theorem A2.13. Let W be a weakly complete topological vector space and F a filter basis of closed linear subspaces. (a) Assume that F is a closed vector subspace of W . Then   F+ F = F + H. (∗) H ∈F

If, in addition, dim E/F < ∞, then   F+ F = (F + H ). H ∈F

(∗∗)

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Appendix 2 Weakly Complete Topological Vector Spaces

(b) The following conditions are equivalent: (i)  lim F = 0. (ii) F = {0}. (c) Assume that F ⊆ N (W ). Then the following conditions are equivalent: (i) lim F = 0. (ii) F is a basis of N (W ). Proof. (a) By duality of weakly complete topological vector spaces according to Theorems A2.8 (iii) and A2.11 (ii) by passing to annihilators, the assertion is equivalent to the following assertion in a vector space E with a vector subspace S and a directed (ascending) set D of vector subspaces   S ∩ U. S∩ D= U ∈D

This relation holds for elementary set-theoretic reasons. Now assume that dim W/F < ∞. All vector subspaces of a finite-dimensional (Hausdorff topological) vector space are closed. Hence every vector subspace of W containing F is the full inverse image under the quotient morphism W → W/F of a vector subspace of W/F and thus is closed. Thus (∗∗) is equivalent to (∗) in the present situation. (b) Let U denote the filter of zero neighborhoods of W . Then by definition, (i) is equivalent to (i) The filter F  generated by F contains U. This in turn is equivalent to (i) (∀U ∈ U)(∃V ∈ F ) V ⊆ U .   Now (i) ⇒ (ii) is clear since F ⊆ U = {0}. We prove (ii) ⇒ (i) . Let U be a zero neighborhood. Since W is weakly complete the filter basis V of cofinite-dimensional vector spaces V converges to 0. Hence we may assume that there is a cofinite-dimensional vector space V contained in U , and we may even assume that V is the only vector space containing V and being contained  in U . By (a) we have V  ∈F (V + V  ) = V + F = V + {0} = V . Now {(V + V  )/V : V  ∈ F } is a filter basis of vector subspaces of the finite-dimensional vector space W/V intersecting in {0}, and thus there is a V  ∈ F such that (V + V  )/V is zero, that is, V  ⊆ V ⊆ U . This proves (i) . (c) Trivially, (ii) implies (i) because lim N (W ) = 0. For a proof of (i) implies (ii), let V ∈ N (W ). Then W/V is finite-dimensional, and by the continuity of the quotient map W → W/V the filter basis {B + V /V : B ∈ B} of vector subspaces of W/V converges to zero. Since W/V has no small subgroups, there is a B ∈ B such that (B + V )/V = V /V , that is B + V = V or, equivalently, B ⊆ V .

Topological Properties of Weakly Complete Topological Vector Spaces

643

In lattice theoretic terminology conclusion (a) reads: The lattice of closed vector subspaces of V is meet continuous. For the following, recall that an affine subspace A of a vector space W is a subset of the form A = g + V for some vector subspace V . The affine subspace is linear iff g ∈ V. Filter Bases of Affine Subspaces Theorem A2.14. Let W be a weakly complete topological vector space and F a filter  basis of closed affine subspaces. Then F = Ø. Proof. (a) We write the affine members of F in the form gj + Vj with closed vector subspaces Vj and elements gj , j ∈ J . We claim that the set {Vj : j ∈ J } is a filter basis. Indeed let i, j ∈ J , then there is a k ∈ J such that gk + Vk ⊆ (gi + Vi ) ∩ (gj + Vj ), since F is a filter basis. Therefore gi + Vi = gk + Vi and gj + Vj = gk + Vj . Now gk +Vk ⊆ (gk +Vi )∩(gk +Vj ), and hence Vk ⊆ Vi ∩Vj . Let V = j ∈J Vj . Then W/V is a weakly complete topological vector space and F /V = {(gj +V )+Vj /V : j ∈ J } is a filter basis of closed affine subsets. It clearly suffices to show that F /V has a nonempty intersection. Thus we assume from here on that V = {0}, that is the filter def

basis V = {Vj : j ∈ J } has the intersection {0}. But then lim V = 0 in W by (b) above. This implies that F is a Cauchy filter: Let U be an identity neighborhood; then there is a j ∈ J such that Vj ⊆ U . Then (gj + Vj ) − (gj + Vj ) = Vj ⊆ U . Since W is a complete topological vector space, every Cauchy filter basis converges. Let g = lim F . Since all gj + Vj are closed, we have g ∈ gj + Vj for all j ∈ J and this completes the proof of the lemma. In terms of a terminology that has been used for situations resembling the one we have in the previous theorem, this result can be expressed in the following form: Weakly complete topological vector spaces are linearly compact.

Topological Properties of Weakly Complete Topological Vector Spaces A topological group G is topologically compactly generated if there is a compact subset C ⊆ G such that G = C. It is compactly generated if there is a compact subset C ⊆ G such that G = C. We begin by observing that the idea of a weakly complete topological vector space to be a topologically compactly generated pro-Lie group is not very restrictive. Remark A2.15. Any weakly complete topological vector group is topologically compactly generated.

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Appendix 2 Weakly Complete Topological Vector Spaces

Proof. For the purposes of the proof we may and will assume that W = RJ for some set J . For any subset I of J we identify RI naturally with a subgroup of RJ . The def  may and will be identified with R(J ) , the set of all f : J5→ R with finite dual E = W support, in such a fashion that f ∈ E and g ∈ W gives us f, g = j ∈J f (j )g(j ). Let K = {δj ∈ RJ : j ∈ J } ∪ {0}. Let V be a cofinite-dimensional vector subspace def  of W . Then V ⊥ is a finite-dimensional vector subspace of the dual E = W . Let Fin(J )  denote the set of finite subsets of J . Since E = I ∈Fin(J ) R(I ) and since V ⊥ is finite-dimensional, there is an I ∈ Fin(J ) such that V ⊥ ⊆ R(I ) and thus V ⊆ (R(I ) )⊥ = R J \I . Hence K \ V = {δi : i ∈ I } is finite. Therefore K is compact. On the other hand, W = R(J ) = [0, 1] · K and [0, 1] · K is a compact subset of R(J ) . Hence W is topologically compactly generated. Since δj ∈ ZJ ⊆ RJ , the assertion on ZJ follows analogously, as ZJ = Z(J ) = K. In Chapter 5 we specified the following concepts: A topological space is called a Polish space if it is completely metrizable and second countable. It is said to be σ -compact, if it is a countable union of compact subspaces. It is said to be separable if it has a dense countable subset. Countable products of Polish spaces are Polish. For instance, RN is Polish. Remark A2.16 (= 5.22). (i) Every almost connected locally compact group is compactly generated. (ii) Every compactly generated topological group is σ -compact. (iii) A topological group whose underlying space is a Baire space and which is σ -compact is a locally compact topological group. (iv) A σ -compact Polish group is locally compact. (v) A compactly generated Baire group is locally compact. Proof. See proof of 5.22. Proposition A2.17. For a weakly complete topological vector space W , the following statements are equivalent: (A) (B) (C) (D)

W W W W

is σ -compact. is locally compact. is finite-dimensional. is compactly generated.

Proof. The equivalence of (B) and (C) was shown in A2.2 (iii) and (iv). Locally compact connected groups are compactly generated by A2.16 = 5.22 (i) and so (B) implies (D); and (D) implies (A) by A2.16 = 5.22 (ii). In order to prove that (A) implies (C), let W be a weakly complete σ -compact topological vector space. Its dual is a vector space E and W is finite-dimensional iff E is finite-dimensional. Suppose that E is infinite-dimensional. Selecting from a basis an infinite countable subset we get a vector subspace F with a countable basis. Then

Topological Properties of Weakly Complete Topological Vector Spaces

645

W/F ⊥ is isomorphic to the dual of F ∼ = R(N) and therefore W/F ⊥ is a homomorphic image of W which is isomorphic to RN and therefore is a Polish topological vector space. Since it is also σ -compact as a homomorphic image of a σ -compact group, it is locally compact by A2.16 = 5.22. But then it is finite-dimensional, a contradiction. Proposition A2.18. For a weakly complete topological vector space W , the following statements are equivalent: (i) (ii) (iii) (iv) (v) (vi)

W W W W W W

∼ = RJ with card J ≤ ℵ0 . is locally compact or is isomorphic to RN . is finite-dimensional or is isomorphic to RN . is second countable. is first countable. is Polish.

Proof. By the remarks preceding the proposition, for each cardinal ℵ, there is, up to isomorphy of topological vector spaces and of topological groups one and only one weakly complete topological vector space of topological dimension ℵ, namely, Rℵ . Conditions (i), (ii), (iii) are ostensibly all equivalent to saying that ℵ is countable. The weight w(W ), that is the smallest cardinal representing the cardinality of a basis for the topology of W ∼ = Rℵ is ℵ0 if ℵ is countable, and is ℵ if ℵ is uncountable (see e.g. [102, Exercise EA4.3 following Proposition A2.8]), so (iv) is likewise equivalent to (ii), and implies (v). If the weakly complete topological vector space W is first countable, then the filter basis  (W ) of cofinite-dimensional closed vector subspaces has and thus W ∼ = limV ∈ (W ) W/V is a closed vector subspace of  a countable basis, N ∼ W/V R and thus (vi) implies (iv). If (iv) is satisfied then the complete = V ∈ (W ) topological vector space W is metrizable (see e.g. [102, Theorem A4.16]) and thus (vi) follows; trivially (vi) implies (iv). Proposition A2.19. For a weakly complete topological vector space W , the following statements are equivalent: (a) W is separable. (b) W contains a dense vector subspace of countable linear dimensions over R. (c) W is isomorphic as a topological vector space to RJ with card J ≤ 2ℵ0 . These conditions are implied by the equivalent statements of Proposition A2.17. Proof. A second countable space is always separable: It suffices to pick a point in every set in a countable basis for the topology: this yields a countable dense set. What remains therefore is to see the equivalence of (a), (b), and (c). We may safely assume that W is infinite-dimensional, since the finite-dimensional case is clear. (a) ⇒ (b): Let C be a countable dense subset of RJ . Then the real linear span of C is dense vector subspace of RJ whose linear dimension is countable. (b) ⇒ (c): Assume that ι : R(N) → RJ is a linear map between vector spaces such that im(ι) = RJ . We give R(N) the finest locally convex topology. The vector space

646

Appendix 2 Weakly Complete Topological Vector Spaces

dual of R(N) may be identified with RN , and that of RJ with R(J ) . The morphism ι is both an epic (and a monic) in the category of (Hausdorff) topological vector spaces. Its adjoint morphism ι : R(J ) → RN is a monic (and epic) and is therefore an injection (with dense image). Thus card(J ) ≤ dimR RN = 2ℵ0 . (c) ⇒ (a): Let W = RJ with card(J ) = 2ℵ0 . We shall show that W is separable; since RI with card(I ) ≤ card(J ) is a homomorphic image of RJ , this will entail the implication. The topological vector space dual of RJ may be identified with R(J ) and then there is a linear bijection β : R(J ) → RN . If we give R(J ) the finest locally convex topology and RN the product topology, then β is an epic (and a monic) in the category of topological vector spaces and thus its adjoint β  : R(N) → RJ has a dense image (and is injective). Even in the finest locally convex topology, Q(N) is dense in R(N) , and Q(N) is countable. Hence RJ is separable as asserted.

Summability in Weakly Complete Topological Vector Spaces Lemma A2.20. Let V be a weakly complete topological  vector space and F a filter basis consisting of closed vector subspaces satisfying F = {0}. Let J be an infinite index set and let {aj : j ∈ J } be a family of elements of V such that for5 each finite / E. Then j ∈J aj is subset E ⊆ J we find an F ∈ F such that aj ∈ F whenever j ∈ summable in V . Proof. Let Fin(J ) denote the directed set of finite subsets of J . Since V is a complete topological vector 5 space it suffices to show that the net (sE )E∈Fin(J ) of all finite partial sums sE = j ∈E aj is a Cauchy net. That is, we have to show that for each zero neighborhood U of V there is an EU ∈ Fin(J ) such that for all E, E  ⊇ EU we have sE  − sE ∈ U . Let a zero neighborhood U of V be given. By Theorem A2.13 (b) we have lim F = 0. Hence there is an F ∈ F such that F ⊆ U . By hypothesis on the family of the aj we find a finite set EU ∈ Fin(J ) such that j ∈ J \ EU implies aj ∈ F .    Now let E, E  ∈ Fin(J 5) and E, E ⊇5EU . Then (E \ E) ∩ EU = Ø = (E \ E ) ∩ EU and thus sE  − sE = j ∈E  \E aj + j ∈E\E  (−aj ) ∈ F ⊆ U and this is what we had to show.

Continuous Symplectic Maps on Weakly Complete Topological Vector Spaces Lemma A2.21. Let U , V and W be weakly complete topological vector spaces and b : U × V → W a continuous bilinear map. Assume dim W < ∞. Then there are cofinite-dimensional vector subspaces E of U and F of V such that b vanishes on E × V and U × F .

Continuous Symplectic Maps on Weakly Complete Topological Vector Spaces

647

Proof. Let W ∗ be a zero neighborhood of W not containing any vector subspaces other than {0}. We have b({0} × V ) = {0}. Hence for each y ∈ V there are open zero neighborhoods Uy and Vy in U and V , respectively, such that b(Uy ×(y+Vy )) ⊆ W ∗ for all y ∈ V . Now by the definition of the topology of weakly complete topological vector spaces there are cofinite-dimensional vector subspaces Ey and Fy contained in Uy and ∗ Vy , respectively, whence b(Ey × (y + Fy )) ⊆ W and therefore b(Ey × (y + Fy )) = b(Ey × {y}) + x∈Ey b({x} × Fy ) = {0} because U does not contain any nonsingleton linear subspaces. Since this set is stable under scalar multiplication we have in fact (∀y ∈ V ) b(Ey × (R · y + Fy )) = {0}.

(∗)

Vy . We find vectors y2 , . . . , yn in V such Let y5 1 = 0; then Fy1 is cofinite-dimensional in that nj=1 (R · yj + Fyj ) = V , and set E = nj=1 Eyj . As an intersection of finitely many cofinite-dimensional vector subspaces, E is cofinite-dimensional. Therefore (∗) implies b(E × V ) =

n 6

b(E × (R · yj + Fyj )) ⊆

j =1

n 6

b(Eyj × (R · yj + Fyj )) = {0}.

j =1

Exchanging the roles of U and V yields the second part of the assertion. If E and F are vector spaces, then a function s : E × E → F is called symplectic if it is bilinear and satisfies s(x, y) = −s(y, x) for all x, y ∈ E. Proposition A2.22. For each weakly complete ; topological vector space V there is a weakly complete topological vector space 2 V and a continuous symplectic map (v, w)  → v ∧ w : V × V →

2 >

V

such that for every continuous symplectic map b : V × V → W of weakly ; complete topological vector spaces there is a unique continuous linear map b : 2 V → W such that b(v, w) = b (v ∧ w). ; The linear span of {v ∧ w : v, w ∈ V } is dense in 2 V . Proof. For  a finite-dimensional V with ; dimension n, the construction is classical yielding an n2 -dimensional vector space 2 V with the required properties. For an arbitrary weakly complete topological vector space let  (V ) denote the filter basis of closed cofinite-dimensional vector subspaces; then V ∼ = V (V ) = limj∈ (V ) V /j. For j ⊇ k in ;  (V ), let pj,k : V /k → V /j be the quotient morphism. Since 2 • is a functor of the category of finite-dimensional vector spaces to itself, 2 >

pjk :

2 >

V /k →

2 >



V /j | k ⊆ j in  (V )

is a projective system. We define 2 >

V = lim

j∈ (V )

2 >

V /j.

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Appendix 2 Weakly Complete Topological Vector Spaces

Since the inclusion functor from topological vector spaces to topological spaces preserves limits, the limit property for limj∈ (V ) V /j in the category of topological spaces ; gives us a fill in map ∧ : V × V → 2 V filling in correctly in the infinite diagram ;2 ∧ −−−−→ V V ×V ⏐ ⏐ ;2 ⏐ ⏐ pj ×pj   pj ;2 V /j × V /j −−−−→ V /j. ∧V /j

We must verify the universal property: Let b : V ×V → W be a continuous symplectic map. Pick any j ∈  (W ). From Lemma A2.21 we deduce the existence of a k in  (V ) such that the symplectic map qj

b

V × V −−→ W −−→ Wj factors through V /k × V /k via V ×V ⏐ ⏐ pk ×pk 

b

−−−−→ fjk

;2 W ⏐ ⏐;2  qj ;2 W/j.

V /k × V /k −−−−→ ; Then by the universal property of 2 V /k for finite-dimensional vector spaces we find ; a linear map fj k : 2 V /k → W/j such that fjk = fjk  ∧V /k . By the definition of ;2 V , we have a commutative diagram ;2 ∧ −−−−→ V V ×V ⏐ ⏐ ⏐ ⏐ ;2 pk ×pk   pk ∧V /k ;2 V /k × V /k −−−−→ V /k. Now we obtain a natural commutative diagram V ×V ⏐ ⏐ b W ⏐

qj ⏐ 

W/j

;2 V ⏐ ⏐ ;2  pk ;2 V /k ⏐ ⏐ id ;2 ←−− −− V /k ∧

−−−−→

fjk

for each j ∈  (W ). Since W ∼ = W (W ) = limj∈ (W ) W/j we get a unique morphism ; b : 2 V → W such that b = b  ∧. This proves the main assertion. ; It remains to show that the span of the v ∧ w is dense in 2 V as v and w range ; through V . But for each j ∈  (V ), the span of the v ∧ w is mapped by pj : 2 V → ;2 V /j onto the span of the vj ∧wj , vj , wj ∈ Vj . This span, however is V /j. Therefore, ; span{v ∧ w : v, w ∈ V } + j = 2 V for all j, and that proves the assertion.

Miscellaneous Observations Involving Weakly Complete Topological Vector Spaces

649

For S ⊆ V we set def

Ann(S) = {w ∈ V : (∀s ∈ S)s ∧ w = 0}, ; where (v, w)  → v ∧ w : V × V → 2 V is the universal symplectic map of Proposition A2.22. Corollary A2.23. Let W be a vector subspace of V . Then ⎧ ⎪ if dim W = 0, ⎨V Ann(W ) = W if dim W = 1, ⎪ ⎩ {0} if dim W ≥ 2. Proof. We begin by reviewing the assertion in the finite-dimensional case.5 Let v = 0 in n V and let v = e1 , . . . , e5 n be a basis of V . Assume v∧x = 0. We write x = m=1 rm ·em n and find 0 = e ∧ x = m=1 rm · e1 ∧ em . Because of the linear independence of the e1 ∧ em , m = 2, . . . , n, we get r2 = · · · = rn = 0, that is, x = r1 · v. If v1 and v2 are linearly independent and 0 = v1 ∧ x = v2 ∧ x, then x = r1 · v1 = s2 · v2 by the preceding and this implies x = 0. This establishes the claim in finite dimensions. Now let V be a weakly complete topological vector space of arbitrary dimension and let 0 = v;∈ V and assume that v ∧ x = 0 for some x ∈ V . Let j ∈  (V ). By the definition of 2 V and ∧ we have (v + j) ∧V /j (x + j) = (v ∧ x) + j = 0. It follows from the preceding discussion of the finite-dimensional case that x ∈ R · v + j as soon as v ∈ / j, as is the case for all sufficiently small j. Thus x ∈ j∈ (V ) (R · v + j) = R · v, since R · v is closed in V . As in the finite-dimensional case, this is enough to secure the assertion in the general case.

Miscellaneous Observations Involving Weakly Complete Topological Vector Spaces We have seen in Chapter 5 that weakly complete vector groups, that is, the additive groups of weakly complete topological vector spaces, play a significant role in the structure theory of abelian pro-Lie groups. For the sake of completeness, we sample some of the insights we have had. Lemma A2.24 (= 5.10). Let G be an abelian pro-Lie group, and assume that comp(G) is compact. Then the following conclusions hold. (i) There is a weakly complete vector group W and a compact abelian group C which is a product of circle groups such that G may be considered as a closed subgroup of W × C such that G ∩ ({0} × C) = comp(G). (ii) G/ comp(G) is embedded as a closed subgroup into the weakly complete vector group W .

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Appendix 2 Weakly Complete Topological Vector Spaces

Remark A2.25 (= 5.24). For a discrete abelian group, the following statements are equivalent: (i) (ii) (iii) (iv)

G is finitely generated free. G is isomorphic to a closed additive subgroup of Rn for some n = 0, 1, 2, . . . . G is isomorphic to a closed additive subgroup of RJ for some set J . G is isomorphic to a closed additive subgroup of a weakly complete topological vector space.

This result provides an alternative proof of the fact that the prodiscrete free abelian group F of infinite rank cannot be discrete. Example A2.26. There is a closed nondiscrete free abelian subgroup of countably infinitely many generators in RR . Proof. See Proposition 5.2. We remark in passing that an abelian Lie group, being isomorphic to a group Rp × Tq × D for some discrete group (see for instance [102], Exercise E5.18 following Proposition 5.54, or Corollary 7.58), is isomorphic to a subgroup of a connected abelian Lie group if and only if it is algebraically generated by a compact subset. In [34] it is shown that the class of abelian topological groups which are isomorphic to a product C × RX × ZY where C is a compact group and X and Y are countable sets is closed under the passage to closed subgroups and to quotient groups. This points out the fact that quotient groups of RX are complete if X is countable. In this sense the example of RR is minimal in the class of weakly complete topological vector spaces, if one momentarily accepts the Continuum Hypothesis. In [116] it is proved that a closed connected subgroup of RX is a closed vector subspace; consequently it is a direct summand algebraically and topologically by A2.11 (i). In particular, the quotient is a weakly complete topological vector space. For more information we refer the reader to Chapter 5 of this book and for additional information concerning the duality theory of closed subgroups of weakly complete topological vector spaces we refer the reader to [102, Propositions 7.31 through 7.35].

Appendix 3

Various Pieces of Information on Semisimple Lie Algebras

In this appendix we record some results on semisimple or reductive finite-dimensional Lie Algebras which we need in the body of the book. The first portion deals with the question whether in a semisimple Lie algebra every element is a bracket, the second with the question whether a maximal compactly embedded abelian subalgebra of a reductive Lie algebra is contained in only one compactly embedded subalgebra. Prerequisites. In this Appendix, in the first part, we shall make use of Kostant’s Convexity Theorem (see TheoremA3.1), and of basic facts of the theory of finite-dimensional real Lie algebras and their complexification for which we refer to standard sources such as Bourbaki [16]–[20]. We shall also refer to a result of Gordon Brown [33]. In the second part we shall refer to standard source books on semisimple Lie algebras.

Brackets in Semisimple Lie Algebras If A and B are subsets of a finite-dimensional real Lie algebra g, then [A, B] traditionally denotes the linear span of all brackets [x, y], x ∈ A, y ∈ B. We therefore write brack(A, B) = {[x, y] | x ∈ A, y ∈ B}. def

The set brack[g] = brack(g, g) is trivially invariant under scalar multiplication. The commutator algebra g = [g, g] is the sum of a minimal number m(g) of summands [g, g] = brack[g] + · · · + brack[g] . )* + (

(∗)

m(g)

From Example 14.21–14.22 we recall that the minimal number of commutators required to represent a member of the commutator subalgebra of a class 2 nilpotent finite-dimensional real Lie algebras is unbounded;    in fact, we constructed a nilpotent Lie algebra nn such that the commutator subgroup of (nn , ∗) of class 2 of dimension n + n2 = n+1 2 contains an element which is not the product of fewer than n−1 4 commutators, and that implies that the commutator algebra of nn contains an element that is not the sum of fewer than n−1 4 brackets. In other words, the number m(nn ) defined in (∗) above ≤ m(n satisfies n−1 n ). The point is that already in the class of nilpotent Lie algebras g 4 of the lowest possible class of nilpotency to make them nonabelian, the number m(g) is unbounded.

652

Appendix 3 Various Pieces of Information on Semisimple Lie Algebras

 This permitted us to argue that the product ∞ n=2 nn is a class 2 nilpotent proLie algebra with a nonclosed commutator subalgebra. From the structure theory of reductive pro-Lie algebras in Chapter 7 (see Theorem 7.27) we know that a reductive  pro-Lie algebra is a product of the form RI × j ∈J sj for suitable index sets I and J and a family sj of simple real Lie algebras. In Theorem 7.27 we showed that the (algebraic) commutator subalgebra of a reductive pro-Lie algebra is closed. In the proof we use a result which we present in this appendix (Corollary A3.5): For a real reductive Lie algebra g we have m(g) ≤ 2. This follows from a consequence of Gordon Brown’s Theorem of 1963 [33] which we shall quote, that every element in a complex semisimple Lie algebra is a bracket. What we shall prove in this appendix is m(g) ≤ 4 which is a rather direct consequence of the main result of this appendix which is more in the spirit of this book: In a semisimple compact real Lie algebra g every element is a bracket.

Brackets in Compact Lie Algebras The Lie algebra so(3) of all 3 × 3 skew symmetric matrices is isomorphic to the euclidian vector space R3 equipped with the vector product. In this vector space every nonzero element u is the vector product u = v × w of two orthogonal vectors v ⊥ w both of which are orthogonal to u. It follows that every element of so(3) is a bracket. The Lie algebra so(3) is an example of a compact Lie algebra. Recall that a real Lie algebra is called compact if it supports an invariant scalar product, equivalently, that it is isomorphic to the Lie algebra of a compact Lie group. (See [102, Proposition 6.2ff., notably, Theorem 6.6].) The example raises the following question: For which real Lie algebras g it is true that g = brack[g], that is, that every element is a bracket? We shall see that this is the case for instance for all semisimple compact Lie algebras. This is an “infinitesimal” analog of the theorem that in a compact connected semisimple Lie group every element is a commutator (Gotô’s Theorem, see for instance [102, Corollary 6.56]). Before we can show that every element of a compact real Lie algebra is a bracket we have to cite a result known as Kostant’s Convexity Theorem (see for instance [84, p. 257, Theorem III.8.4]). On a semisimple compact Lie algebra g we may and will take the invariant inner product on g given by (X|Y ) = − tr ad X ad Y , the negative of the Cartan–Killing form. In the following theorem we shall consider a compact Lie algebra g and let G be the compact group of its inner automorphisms ead g = ead g . We fix an arbitrary Cartan subalgebra t of g, and take the orthogonal projection p : g → t with respect to an invariant scalar product of g. Moreover, we let W be the Weyl group of g with respect to t. Assume that for x ∈ t, the G-orbit of x in g is denoted G · x and the W -orbit of x in t is denoted W · x. The convex hull of a set S in a real vector spaces is denoted conv(S). If S is compact then conv(S) is precompact and thus compact if the vector space is complete,

Maximal Compactly Embedded Subalgebras

653

for instance, if it is finite-dimensional. This applies, in particular, to the convex hull of the compact orbit G · x in g, and so conv(G · x) is compact. Theorem A3.1 (Kostant’s Convexity Theorem). If x is an arbitrary element of the Cartan subalgebra t of a compact Lie algebra g, then p(G · x) is the convex hull of the finite subset W · x in t. This is not a trivial result. It is very useful in many respects, and it will serve us well, too. The following is based on a communication of Karl-Hermann Neeb to the authors. Theorem A3.2. Every element of a semisimple compact Lie algebra is a bracket. Proof. The set of fixed points of the linear action of G on g is the center of g; since the Lie algebra g is semisimple, its center is {0}. Thus 0 is in the closed convex hull conv(G · x) of G · x by [102, Theorem 3.36]. Thus 0 = p(0) ∈ p(conv(G · x)) = conv(p(G · x)). Theorem A3.1 applies and yields conv(p(G · x)) = conv(W · x). Therefore 0 ∈ conv(W · x) = p(G · x) again by Theorem A3.1. Thus there is an inner automorphism γ ∈ G such that γ (x) ∈ ker p. Now assume that x0 is a regular element of g contained in t, that is, dim ker ad x0 is minimal. Then t = ker ad x0 and if t+ abbreviates ker p, the orthogonal complement of t in g, then ad x0 |t+ : t+ → t+ is bijective. Therefore ker p = t+ = [x0 , g]. We conclude that x ∈ γ −1 [x0 , g] = [γ −1 (x0 ), g] ⊆ brack[g]. Corollary A3.3. Let g be any semisimple real Lie algebra. Then every element is the sum of at most four brackets, that is g = brack[g] + brack[g] + brack[g] + brack[g]. Proof. Let g be a semisimple real Lie algebra. Then its complexification gC has a compact real form, that is, there is a real compact subalgebra k such that gC = k + i · k. From Theorem A3.2 we have k = brack(k, k) ⊆ brack(gC , gC ). Thus gC = brack[gC ] + brack[gC ]. So for any x ∈ g there are elements b1 , b2 ∈ k such that g = b1 + i · b2 . By Theorem A3.2 there are elements xj , yj , uj , vj ∈ g, j = 1, 2, such that xj + i · yj , uj + i · vj ∈ k and bj = [xj + i · yj , uj + i · vj ], j = 1, 2. Thus b1 = [x1 , u1 ] − [y1 , v1 ] + i · z1 with z1 ∈ g and i · b2 = −[y2 , u2 ] − [x2 , v2 ] + i · z2 with z2 ∈ g. Therefore x = [x1 , u1 ] − [y1 , v1 ] − [x2 , v2 ] − [y2 , u2 ]. Actually, the result in this corollary is not the best possible representing the status of our knowledge. From the result of Brown [33] we have the following consequence: Proposition A3.4. Every element in a semisimple complex Lie algebra is a bracket. By an even simpler calculation than that in the proof of Corollary A3.3 we then obtain Corollary A3.5. Let g be any semisimple real Lie algebra. Then every element of g is the sum of at most two brackets, that is g = brack[g] + brack[g]. Proof. Let x ∈ g ∈ gC . Then there are elements a, b, u, v ∈ g such that x = [a + ib, u + iv] by Proposition A3.4. Thus x = [a, u] − [b, v]. Corollary A3.5 is the result we used in Chapter 7 in the proof of Lemma 7.26.

654

Appendix 3 Various Pieces of Information on Semisimple Lie Algebras

Maximal Compactly Embedded Subalgebras Proposition A3.6. Let g be a finite-dimensional real reductive Lie algebra and k and k1 two maximal compactly embedded subalgebras containing one and the same maximal compactly embedded Cartan subalgebra t. Then k = k1 . Proof. There is a direct reference for this result: See [84, p. 217, Proposition II.5.6, notably Section (i)]. Another pertinent reference is [157, p. 260ff]. It is clear from Theorem 12.15 that every compactly embedded abelian subalgebra t is contained in a maximal compactly embedded subalgebra k. If t is a Cartan subalgebra, then k is unique by the preceding Proposition A3.6. Write G for the group Inn(g) of inner automorphisms of a finite-dimensional Lie algebra g. Definition A3.7. For a subalgebra k of G we set N(k, G) = {ϕ ∈ G : ϕ(k) = k} and call this group the normalizer of k in G. By Theorem 12.27, G operates transitively on the set of maximal compactly embedded subalgebras m, and the normalizer N(m, G) is the isotropy (or stability) group of this action at the point m. It also acts on the set of all maximal compactly embedded abelian subgroups a. Assume a ⊆ m and set A = ead a , and M = ead m . then N(a, M)/A is the Weyl group of m with respect to a. By Corollary 12.31, G acts transitively on the set  of pairs (a, m) where a is a maximal compactly embedded abelian subalgebra contained in the maximal compactly embedded subalgebra m. Thus N(a, G) acts transitively on the set  of all m containing a. Therefore Lemma A3.8. There is a continuous bijection β : N(a, G)/(N (a, G) ∩ N (m, G)) → .



It follows that  is singleton iff N(a, G) ⊆ N (m, G). If this is the case, then N(a, G)0 ⊆ N (m, G)0 which is the case iff n(a, g) ⊆ n(m, g).

(∗)

Since M is its own normalizer (see [78, p. 208, Theorem 3.3, and p. 252, Theorem 1.1]), we have n(m, g) = m. On the other hand n(a, g) = z(a, g) as follows from the decomposition of g into isotypic decomposition of the action of A on g. (See e.g. [102], Definition 4.21ff.) Therefore (∗) is equivalent to z(a, g) ⊆ m. However, z(a, g) contains all Cartan subalgebras containing a. Therefore (∗) implies that m contains a Cartan subalgebra c containing a. Since a is maximal abelian, we conclude a = c. Thus we have seen Proposition A3.9. If a is a maximal compactly embedded abelian subalgebra of a reductive Lie algebra g, then the following conditions are equivalent: (i) a is a Cartan subalgebra of g. (ii) a is contained in a unique maximal compactly embedded subalgebra of g.

Postscript

655

Postscript We started the Appendix with the most elementary example, namely, so(3). Here everyone knows from elementary linear algebra that every element is a commutator. The example of sl(2, R) is a bit less elementary,  0  but indeed, in sl(2,  element  R) every  is a bracket as well. For a proof set h = 01 −1 and p + = 00 01 , p − = 01 00 . Then def

def

h = [p+ , p− ] and the element k = p + + p − satisfies [h, k] = u = p + − p − . Also [h, p± ] = ±2 ·p ± . Since every element of sl(2, R) is conjugate to either h, or u, or p + (see for instance [84, p. 108, Proposition II.3.6]), every element of sl(2, R) is a bracket. So whenever an element of a Lie algebra is contained in a subalgebra isomorphic to either so(3) or sl(2, R), then it is a bracket. The fact that every element of sl(n, K) for a field K of characteristic zero is a commutator was shown by Shoda in 1937 [182]. G. Brown proves in 1963 [33] that each element is a commutator in any semisimple Lie algebra over any field of characteristic zero that is splittable (déployable in the sense of Bourbaki [19, p. 75]; in fact his result reaches further as far as the field is concerned [33]. Theorem A3.2 originates from correspondence between K. H. Hofmann and K.-H. Neeb [111]. We present it here because it is of independent interest and does not fall into any of the categories covered by Brown’s Theorem. It seems to be unknown whether in an arbitrary real semisimple Lie algebra every element is a bracket. The second portion of this Appendix provides some information which illuminates the role of a maximal compactly embedded abelian subalgebra a of a reductive Lie algebra and those maximal compactly embedded subalgebras m containing it. Only if a is in fact a Cartan subalgebra is it true that m is uniquely determined.

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List of Symbols

A, 115 [A, B], 420 AbTopGr, 119 Ad(g), 131 ad x, 624 α(G), 104 A(h), 366 A(h, G), 366 A(J ) , 150 Ann(S), 649 0BA : A → B, 169 B 1 (K, n), 488 C(N), 463, 488 Cofin, 631 Cofin, 634 comm(g, h), 115 comp(G), 17, 217 CompGr, 72 comp(L)(G), 217 ConnGr, 72 Der(E), 624 D(G), 405 Eeff , 280 E(G), 121 E(G), 121 E1 ⊥ E2 , 494 Ess , 280 expG , 110 E0 , 280 Fin, 631, 634 f (ϕ), 307 G , 420 Ga , 121  13 G, gad , 20

[g, g], 282 [g, g], 282 GN , 91 G0 , 2, 3, 115, 121 g0 (h), 49, 320 H 1 (K, n), 488 H (V ), 573, 601 h(V ), 573  (L), 138 Inn(g), 352, 359 j ∈J

Ej , 277

KZ(G), 399 LCGr, 72

Lcoad , 272 L(G), 110  272 L, LieAlg, 123 LieAlgGr, 119 LieGr, 119 LieProGr, 135 lim F = x, 88 limj ∈J Gj , 78 L , 272 MaxK(G), 530 M(G), 195 M ∼ N, 195 N, 73 , 91 N Ncored (G), 38, 447 n ⊕δ h, 264 N (G), 38, 447 n(g), 23, 303 N (G), 148, 164 N(H, G), 31, 373

668

List of Symbols

n(h, g), 31, 373 nilcore(G), 579 nilcore(g), 576 N π H , 15 ν(G), 61, 585 ν(g), 586 νMN : G/N → G/M, 91 νN : GN → G/N , 91 P (G), 32, 255 PN , 91 proLieAlg, 138 proLieAlgGr, 140 proLieComGr, 140 proLieGr, 2 proSimpConLieGr, 255 protoLie, 205 1 p ∞ · Z/Z, 101, 590 Q, 79, 588 Q(G), 57, 534 qN : G → G/N , 91 Qp , 589 R, 73 ran f , 69 RefTopGr, 272, 273, 637 R(G), 38, 432

r(g), 23, 291 Scal, 119 S ⊥ , 272

T (G), 15 Top, 119 TopGr, 2, 71, 119 Top∗ , 119 Tp , 80, 589 , 270 V V  , 270 V 0 (h), 316 x · ω, 271 Z, 72, 79, 588 Zp , 588 Z(p ∞ ), 100, 590 z(g), 282 ζ (G), 104 Z(G), 197 Z(H, G), 30, 371 Z 1 (K, n), 488 1 ZN ∩H (H, N ), 463 Z2 (G), 453

Index

abelian Lie algebra, 624 abelian sequence of ideals, 285 abelian sequence of normal subgroups, 420 absorbing, 630, 631 addition on L(G), 115 adjoint module, 20, 270 morphism, 131 representation, 131, 270 Adjunction Theorem for L, 127 Ado’s Theorem for Pro-Lie Algebras, 332 affine subspace, 226, 643 Alexander Subbasis Theorem, 197 algebra Banach, 109 pro-Lie, 138 algebraic dual, 632, 634 algebraically locally compact L-module, 45, 494, 561 almost connected, 2, 7, 101, 195, 218, 478 metrizable, 203 prosolvable, 486 Alternative Open Mapping Theorem for Pro-Lie Groups, 556 analytic morphism of pro-Lie groups, 360 semisimple group of bounded type, 455 semisimple group of unbounded type, 455 subgroup, 29, 360, 416 annihilator, 272, 281, 636, 638, 640 Annihilator Mechanism, 638, 640 antiisomorphism, 273, 275, 640 approximately nilpotent, 308 arbitrarily small co-Lie subgroups,

148, 160 associated representation, 331 atom, 454 of bounded type, 455 of unbounded type, 455 automorphic action, 222 automorphism, 462, 464 inner, 356 special, 24 Axiom of Choice, 633 back adjunction, 12, 127, 257 Baire Category Theorem, 85 Baire space, 85, 497 balanced, 630, 631, 636 Banach algebra, 109 bicommutant, 372 Bipolar Lemma, 636, 637 Bohr compactification, 104 Bohr compactification morphism, 104 bonding maps, 78 bounded type, 41, 455 bracket multiplication on L(G), 115 Campbell–Baker–Hausdorff–Dynkin series, 336 Campbell–Hausdorff multiplication, 626 Campbell–Hausdorff neighborhood, 107, 626 canonical decomposition, 132 Cartan pair, 322 Cartan subalgebra, 49, 322 category, 633 complete, 69 discrete, 65 has equalizers, 67 has finite limits, 67 has intersections of coherent retracts, 68 has limits, 67 has products, 67

670

Index

has projective limits, 67 has pullbacks, 67 of compact groups, 72 of connected groups, 72 of diagrams, 64 of locally compact groups, 72 of topological groups, 119 of topological vector spaces, 633 small, 64 Cauchy filter basis, 94, 111 Cauchy net, 95 center, 282 Center-free Embedding Lemma, 394 centralizer, 30, 371 centrally supplemented, 58, 541 Characterisation Theorem for Local Compactness, 555 for Potentially Compact Connected Pro-Lie Groups, 522 for Strong Local Connectivity, 181 of Reductive Pro-Lie Groups, 452 of Semisimple and Reductive Connected Pro-Lie Groups, 435 of Simply Connected Pronilpotent Pro-Lie Groups, 484 of Simply Connected Prosolvable Pro-Lie Groups, 487 closed commutator algebra, 282 closed subcategory, 69 Closed Subgroup Theorem, 8 Closed Subgroup Theorem for Pro-Lie Groups, 158 closeness relation, 195 Closure Theorem for Semisimple Analytic Subgroups, 439 co-Lie subgroup, 148 coadjoint module, 21, 272 coboundary, 488 cocycle, 71, 462–464, 488 coequalizer, 276 cofactor, 463 cofinal, 81 cofinal function, 74, 81

cofinal functor, 74 cofinality, 74 Cofinality Lemma, 75, 83, 86 Cofinality Lemma Revisited, 81 coherent coretractions, 68 cohomology, 488 cokernel, 169 colimit, 276 colimit cone, 276 commutant, 30, 371 commutator, 115 subalgebra, 282 subgroup, 420 Commutator Series Theorem for Pro-Lie Groups, 429 Commutator Subgroup Theorem for Dense Analytic Subgroups, 382 compact, 652 compact L-module, 45, 494, 561 Compact Generation Theorem for Abelian Pro-Lie Groups, 236 compact Lie algebra, 652 compact open topology, 270, 629, 632, 634, 636–638 compact-free, 217 compactly embedded, 47, 54, 500, 521, 561 generated, 218, 643 simple, 56, 483, 530 topologically generated, 218 compactly simple, 402 complemented, 571 complete, 2, 95 lattice, 273, 275, 640 complete category, 69 Complete Groups and Projective Limits Theorem, 96 completeness of locally compact groups, 95 Completeness Theorem for Pro-Lie Groups, 158 for Projective Limits of Lie Groups, 136 for Topological Groups with pro-Lie

Index

Algebras, 141 for Topological Groups with Pro-Lie Component, 142 of the Category of Groups with Lie algebras, 124 completion, 104 concomitant Dynkin algebra, 109 cone, 64 conilsubmodule, 302 Conjugacy Theorem of Cartan Subalgebras, 329 Connectivity Theorem for Maximal Compact Subgroups, 547 constant functor, 64 constant morphism, 169 Continuity Theorem for Functors, 77 continuous G-module, 497 continuous L-module, 269 continuous action, 222 continuous cross section, 340 continuous functor, 77 contravariant functor, 273, 637 convergence of a filter basis, 88 convex, 631, 636 coproduct, 276 coquotient, 170 coreductive radical, 38, 283, 303, 447 coretraction, 67 Correspondence Theorem for Subalgebras and Subgroups, 375 cotorus subgroup, 242 counit, 12 countably nilpotent, 36, 287, 421 countably solvable, 22, 35, 285, 420 countably topologically nilpotent, 36, 287, 423 countably topologically solvable, 22, 36, 286, 422 crossed homomorphism, 71 defining morphism, 360 derivation, 308, 624

671

descending nil-sequence, 296 diagram, 64 dimension, 554 direct product, 464 direct sum, 277, 313 directed set, 66 discrete category, 65 disjoint, 162, 620 divisible group, 404 domain of definition, 627 dual algebraic, 632, 634 topological, 270, 632, 634 dual module, 272 Duality Theorem for L-Modules, 274 for Real Vector Spaces, 272, 637 Dynkin algebra, 107, 624 Dynkin Algebra Fundamental Theorem, 626 effective submodule, 280 elementwise compact, 217 embedding, 94 embedding morphism, 94 Embedding Preservation Theorem in Lie’s Third Theorem, 263 epic-monic factorisation, 169 equalizer, 65, 66, 112 Equivalence Theorem for Nilpotency, 297 for Nilpotency of Connected Pro-Lie Groups, 444 for Prosimple and Simple Connectedness, 344 for Solvability, 295 for Solvability of Connected Pro-Lie Groups, 427 of the Definitions of Pro-Lie Group, 161 equivariant, 341 Existence Theorem for Cartan Subalgebras, 50 for Manifold Factors, 555

672

Index

for Maximal Compactly Embedded Subalgebras, 48 of Cartan Subalgebras, 325 Explicit Structure of Weakly Complete Topological Vector Spaces, 638 exponential, 340 strictly, 340 exponential function, 110, 625 of a group, 110 Exponential Generation Theorem for Simply Connected Pro-Lie Groups, 350 exponentially generated, 360 subgroup, 29 extended radical, 57, 534 extremely rich, 571 faithful, 331 faithful module, 331 fibered product, 65 filter basis, 89 convergence of, 89 Filter Basis Theorem for Closed Affine Subspaces, 643 for Closed Linear Subspaces, 641 filter of identity neighborhoods, 630 finest locally convex vector space topology, 273, 633, 637, 639 finite intersection property, 197 finite limit, 66 finite-dimensional, 396 finite-dimensional pro-Lie group, 32 finite-dimensional topological vector space, 630 First Axiom of Countability, 114 First Fundamental Lemma, 152 fixed point-free, 394 Fréchet space, 497 front adjunction, 12, 127 full closed subcategory, 102 full subcategory, 2, 69, 119, 637 full subcategory of C-objects, 119 fully characteristic, 405, 523 function

equivariant, 341 exponential, 110 functor, 273, 637 cofinal, 74 constant, 64 continuous, 77 contravariant, 273, 637 left adjoint, 126 procontinuous, 77 Fundamental Lemma First, 152 Second, 154 Third, 156 Fundamental Theorem on Projective Limits, 88 generating Lie algebra, 115, 121 G-module, 497 group compactly simple, 56 having duality, 238, 594 Lie, 107 of p-adic integers, 588 of p-adic rationals, 589 prodiscrete, 15, 149 protodiscrete, 149 with pro-Lie algebra, 140 Group Theoretical Characterisation Theorem of Procompact Pro-Lie Algebras, 517 Hahn–Banach Theorem, 636 Hausdorff series, 336 Hilbert L-module, 45, 494 idealizer, 31, 373 idempotent, 264 identity arc component, 121 identity component, 115, 121 inductive, 278 injective, 227 injective limit, 276 injective object, 227 injectivity, 464

Index

inner automorphism, 50, 131, 352, 356, 359 inner automorphism group, 357, 359 inner derivation, 356, 624 intersection, 66 inverse limit, 78 inversion, 627 involutive, 294 vector subspace, 294 isomorphy of local groups, 627 isomorphy of partial groups, 627 Jacobi identity, 624 jointly continuous topological G-module, 46 jointly topological G-module, 497 k-space, 83 kernel, 71, 169, 331 kernel of a module, 331 Largest Compact Normal Subgroup of a Pro-Lie Group, 530 Largest Compactly Embedded Connected Normal Subgroup of a Pro-Lie Group, 527 Largest Compactly Embedded Ideal of a Pro-Lie Algebra, 515 lattice, 273, 275, 640 left adjoint functor, 126 Levi summand, 293 Levi–Mal’cev Theorem for Compact Connected Groups, 518 for Pro-Lie Algebras: Conjugacy, 310 for pro-Lie Algebras: Existence, 294 for Reductive Pro-Lie Algebras, 284 Lie algebra, 624 abelian, 624 compact, 652 generating, 121 of a topological group, 115 stable, 407 Lie bracket, 624

673

Lie group, 107 Lie projective component, 140 Lie subalgebra, 109 Lie’s Third Fundamental Theorem for Pro-Lie Algebras, 252 Lie’s Third Theorem, Functorial Version, 253 limit, 64, 66 inverse, 78 projective, 65 strict projective, 86 strong projective, 86 limit cone, 64 limit diagram, 64 Limit Existence Theorem, 70 limit maps, 78 Limit Theorem via Equalizers and Products, 69 linear, 643 linear functional, 634, 637 linear Lie group, 1, 109 linear morphism, 629 linearly compact, 643 L-module, 19, 269 local group, 627 associated with G and U , 627 associated with L and B, 628 local one parameter subgroup, 110 locally arcwise simply connected, 344 locally compact subgroup, 630 locally convex, 631 locally finite-dimensional L-module, 45, 273 locally precompact, 630 loopwise simply connected, 250 Main Corollary on Maximal Compact Connected Subgroups, 525 Main Theorem on Maximal Compactly Embedded Subgroups: Existence and Conjugacy, 525 manifold factor, 554 manifold rank, 60, 555 map

674

Index

bonding, 78 limit, 78 perfect, 83 proper, 83, 86 symplectic, 647 Maximal Compactly Embedded Subalgebras: Conjugacy, 511 Maximal Compactly Embedded Subalgebras: Existence, 504 maximal connected abelian subgroup, 467 maximally compactly embedded subalgebra, 454 maximal zero submodule, 280 metabelian, 328, 339, 598 metrizability of topological groups, 86 minimal analytic subgroup, 368, 416 module, 19, 269, 497 morphism defined by h ⊆ g, 366 Morphism Duality Theorem for Weakly Complete Topological Vector Spaces, 640 morphism of G-modules, 497 morphism of L-modules, 20, 269 multiplication, 627 natural isomorphism, 64 natural transformation, 64 neighborhood filter, 88 niladapted, 331 niladapted module, 331 nilcore, 61, 576, 579, 585 nildimension, 585 nilmodule, 301 nilpotent, 36, 287, 421 nilpotent module, 301 nilpotent of class 2, 380, 599 nilradical, 23, 38, 61, 303, 447 nilsequence of ideals, 285 nilsequence of normal subgroups, 421 no compact central subgroups, 402 no small subgroups, 118 normalizer, 31, 321, 373, 462, 654 null morphism, 169

null object, 169 ω-nilmodule, 301 ω-nilpotent module, 301 one parameter subgroup, 110 one-dimensional topological vector space, 630 Open Mapping Theorem, 42, 85, 408 orthogonal subspaces, 494 orthogonal sum, 494 outer automorphism group, 357 outer derivation algebra, 356 p-adic integers, 79, 588 p-adic rationals, 79, 589 p-adic solenoid, 80, 589 p-adic topology, 589 partial group, 627 partial one parameter subgroup, 110 partially ordered set, 66 passing to intersections of coherent retracts, 69 perfect, 83, 282, 285, 420 Poincaré group, 32, 259, 352, 400 compactified, 400 point of adherence, 451 pointed category, 169 polar, 636 Polish space, 233, 644 potentially compact, 54, 521 Prüfer group, 405 pre-Hilbert L-module, 45, 493 pre-Hilbert module, 561 precompact, 630–633 Preservation Theorem for Semidirect Products, 265 primary components, 405 procompact, 45, 494 procompact L-module, 561 procompact pro-Lie algebra, 47, 500 procontinuous functor, 77 procountable, 209 prodiscrete, 15, 149 product, 65

Index

Product Theorem via Projective Limits and Finite Products, 67 profinite, 2 profinite-dimensional, 316 profinite-dimensional Lie algebra, 138 profinite-dimensional L-module, 20, 45, 273 pro-Heisenberg algebra, 573 pro-Heisenberg group, 573, 601 projection, 65 projective (abelian group), 242 projective limit, 65, 634 projective limit associated with (G, N ), 91 projective limit in C, 66 projective limit of Lie groups, 135, 161 projective system, 66, 82 projective system D, 6 projective system associated with (G, N ), 91 pro-Lie algebra, viii, 6, 138 procompact, 47, 500 reductive, 20, 277 semisimple, 20, 282 pro-Lie group, vii, 2, 149, 164 finite-dimensional, 32 prosimply connected, 11, 249 simply connected, 13 pro-Lie group: Definition A, 160 pro-Lie group: Definition B, 161 pro-Lie group: Definition C, 161 Pro-Lie Group Theorem, 157 Pro-Lie Groups with Pronilpotent Lie Algebra, 337 pronilpotent, 37, 289, 424 pro-open, 208 Pro-Open Mapping Theorem for Pro-Lie Groups, 209 proper, vii, 83, 86 prosimply connected pro-Lie group, 11, 249 prosolvable, 22, 37, 289, 424 proto-Lie group, 149, 407

675

protodiscrete, 149 pullback, 65, 66 Pullback Theorem via Products and Equalizers, 66 pure, 540 pushout, 276 quotient morphism, 169 Quotient Preservation Theorem in Lie’s Third Theorem, 261 Quotient Theorem for Pro-Lie Groups, 170 Quotient Theorem for Pro-Lie Groups Revisited, 202 radical, 23, 38, 291, 432 coreductive, 38, 283, 303, 447 extended, 57, 534 solvable, 23, 38, 291 range, 69 reductive, 38, 277, 282, 434 reductive module, 20, 277 reductive pro-Lie algebra, 20, 277 reductive proto-Lie group, 472, 613 Reflection Theorem to Simply Connected Pro-Lie Groups, 255 reflexive, 238, 272, 594, 636 topological vector space, 273, 637 relative injective for E , 227 relatively compact element, 217 relatively compact one parameter subgroup, 217 relatively open, 208 Resolution Theorem for Connected Pro-Lie Groups, 538 Resolution Theorem of Abelian Pro-Lie Groups, 244 restricted product, 150 retract, 67, 340 retraction, 67, 340 Retraction Theorem for Full Closed Subcategories, 102 Retraction Theorem for Subcategories of TopGr, 103 rich, 571

676

Index

right adjoint, 76 S-group, 175, 591 Sandwich Theorem, 40, 435 Sandwich Theorem for Abelian Pro-Lie Groups, 245 scalar morphism, 114 scalar multiplication, 111, 112 second center, 453 Second Fundamental Lemma, 154 Second Isomorphism Theorem for Pro-Lie Groups, 413 semidirect cofactor, 42, 71, 488 semidirect factor, 42, 71 semidirect product, 71, 462, 463 semidirect sum, 264, 313 semireflexive, 238, 594, 636 semisimple, 20, 38, 277, 282, 434 semisimple Lie algebra of bounded type, 455 semisimple Lie algebra of unbounded type, 455 semisimple module, 20, 277 semisimple pro-Lie algebra, 20 semisimple radical, 280 separable, 233, 644 separable space, 233, 644 separation of points, 634, 637 σ -compact, 84, 233, 644 simple module, 20, 277 simply connected, 11, 249, 343, 344, 355, 629 simply connected pro-Lie groups category of, 13, 256, 343, 344 small category, 64 small subgroups, 118 solvable, 22, 35, 285, 420 solvable radical, 23, 291 solvable radical of the group G, 38, 432 special automorphism, 309, 359 splits, 273, 275, 640 Splitting a Maximal Compact Subgroup topologically, 551 Splitting Theorem

for Reductive Lie Groups, 466 for Simply Connected Prosolvable Groups with Compact Quotients: Conjugacy, 489 for Simply Connected Prosolvable Groups with Compact Quotients: Existence, 476 for Vector Group with Compact Quotients: Conjugacy, 489 for Vector Group with Compact Quotients: Existence, 474 splitting theorem, 42, 490 Strict Exactness Theorem for , 259 Strict Exactness Theorem for L, 188 strict morphism, 188 strict projective limit, 86 strict projective system, 86 strictly exact, 188 strictly exponential, 340 strong operator topology, 132, 270, 498 strong projective limit, 86 strong projective system, 86 strongly locally connected, 177 Structure Theorem for Algebraically Locally Compact Modules, 495 for Almost Connected Pro-Lie Groups, 485 for Groups with Finite Dimensional Nilcore, 581 for Procompact Lie Algebras, 501 for Procompact Modules, 497 for Prosimply Connected Pro-Lie Groups, 343 for Prosolvable Pro-Lie Groups, 478 of Pro-Lie Algebras with Finite Dimensional Nilcore, 577 of Reductive L-Modules, 281 of Semisimple Compact Connected Groups, 518 of Semisimple Locally Finite Dimensional Modules, 277 of Semisimple Pro-Lie Algebras, 283

Index

on Simply Connected Pro-Lie Groups with prosolvable Lie algebra, 342 subbasis for the set of closed sets, 196 subgroup algebraically generated, 115 submodule, 273 Subspace Theorem for Weakly Complete Topological Vector Spaces, 639 Supplement for the Minimal Analytic Subgroup Generated by the Full Lie Algebra, 535 supplemented, 571 Sylow groups, 405 symplectic map, 647 tangent bundle of a pro-Lie group, 15 terminating, 285, 420 terminating nil-sequence of ideals, 296 Theorem Closed Subgroup for Projective Limits, 96 on Closed Connected Normal Subgroups of Simply Connected Pro-Lie Groups, 349 on Completeness of TopGr, 72 on the Compact Group Associated with a Procompact Module, 498 on the Descending Central Series of Pro-Lie Groups, 446 Weak Pro-Open Mapping for Pro-Lie Groups, 171 Third Fundamental Lemma, 156 topological G-module, 497 L-module, 20, 270 dimension, 235, 554 dual, 270, 632, 634 group with a Lie algebra, 115 Lie algebra, 123 topological group having enough compact sets, 239 having no small subgroups, 118

677

Topological Product Decomposition Theorem for Connected Pro-Lie Groups, 553 topologically compactly generated, 643 nilpotent, 36, 287, 423 perfect, 286, 422 solvable, 22, 36, 286, 422 topological vector space, 629, 630 finite-dimensional, 630 locally convex, 631 one-dimensional, 630 reflexive, 273 topology compact open, 636 generated by a subbasis, 197 of pointwise convergence, 132, 270, 634, 637, 638 of uniform convergence, 634 of uniform convergence on compact subsets, 270 toral homomorphic image, 242 transfinite nilmodule, 301 transfinitely nilpotent, 36, 287, 421 nilpotent module, 301 solvable, 22, 35, 285, 420 topologically nilpotent, 36, 287, 423 topologically solvable, 22, 36, 286, 422 Trotter formulae, 626 unbounded type, 455 uniqueness of limits, 65 unit, 12 universal covering morphism, 259 universal group, 259 universal morphism, 259, 352 vector group complement, 222 Vector Group Splitting Theorem for Abelian Pro-Lie Groups, 230 for Connected Abelian Pro-Lie Groups, 221 vector space manifold factor, 554

678

Index

very rich, 571 weak ∗-topology, 634 weak topology, 633, 634 weakly complete, 83, 138, 273, 633, 634, 637, 639 topological vector space, 246, 633 vector group, 212, 220, 246 vector space, 16

vector subgroup, 246 Weil’s Lemma for Pro-Lie Groups, 216 well-complemented, 571 Z-topology, 197 zero-dimensional Bohr compactification, 104 zero-dimensional Bohr compactification morphism, 104, 400