Geometric and Algebraic Topological Meth

Geometric and Rlgebrctic Topologicol Methods in Quantum Mechanics

Geometric one) fllgebroic Topologicol Methods in Quantum Mechanics

Giovanni Giachetta & Luigi Mangiarotti University of Camerino, Italy

Gennadi Sardanashvily Moscow State University, Russia

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GEOMETRIC AND ALGEBRAIC TOPOLOGICAL METHODS IN QUANTUM MECHANICS Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 891-256-129-3

ISBN 981-256-129-3

Printed in Singapore.

Preface

Contemporary quantum mechanics meets an explosion of different types of quantization. Some of these quantization techniques (geometric quantization, deformation quantization, BRST quantization, noncommutative geometry, quantum groups, etc.) call into play advanced geometry and algebraic topology. These techniques possess the following main peculiarities. • Quantum theory deals with infinite-dimensional manifolds and fibre bundles as a rule. • Geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. • Geometric and algebraic topological methods can lead to nonequivalent quantizations of a classical system corresponding to different values of topological invariants. Geometry and topology are by no means the primary scope of our book, but they provide the most effective contemporary schemes of quantization. At the same time, we present in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems. Our book addresses to a wide audience of theoreticians and mathematicians, and aims to be a guide to advanced geometric and algebraic topological methods in quantum theory. Leading the reader to these frontiers, we hope to show that geometry and topology underlie many ideas in modern quantum physics. The interested reader is referred to extensive Bibliography spanning mostly the last decade. Many references we quote are duplicated in E-print arXiv (http://xxx.lanl.gov). With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. For the sake of convenience, a few relevant mathematical topics are compiled in Appendixes. V

Contents

Preface

v

Introduction

1

1.

2.

Commutative geometry

17

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

17 23 27 31 39 56 59 70 85

....

Classical Hamiltonian systems 2.1 2.2 2.3 2.4 2.5 2.6 2.7

3.

Commutative algebra Differential operators on modules and rings Connections on modules and rings Homology and cohomology of complexes Homology and cohomology of groups and algebras Differential calculus over a commutative ring Sheaf cohomology Local-ringed spaces Algebraic varieties

91

Geometry and cohomology of Poisson manifolds Geometry and cohomology of symplectic foliations Hamiltonian systems Hamiltonian time-dependent mechanics Constrained Hamiltonian systems Geometry and cohomology of Kahler manifolds Appendix. Poisson manifolds and groupoids

Algebraic quantization 3.1

....

91 110 115 136 157 172 189 195

GNS construction I. C*-algebras of quantum systems . . . 195 vii

viii

Geometric and Algebraic Topological Methods in Quantum Mechanics

3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.

5.

209 217 224 229 234 238 249

Geometry of algebraic quantization

257

4.1 4.2 4.3 4.4 4.5 4.6 4.7

257 271 274 278 282 286 290

Banach and Hilbert manifolds Dequantization Berezin's quantization Hilbert and C*-algebra bundles Connections on Hilbert and C*-algebra bundles Example. Instantwise quantization Example. Berry connection

Geometric quantization 5.1 5.2 5.3 5.4 5.5 5.6 5.7

6.

GNS construction II. Locally compact groups Coherent states GNS construction III. Groupoids Example. Algebras of infinite qubit systems GNS construction IV. Unbounded operators Example. Infinite canonical commutation relations . . . . Automorphisms of quantum systems

Leafwize geometric quantization Example. Quantum completely integrable systems Quantization of time-dependent mechanics Example. Non-adiabatic holonomy operators Geometric quantization of constrained systems Example. Quantum relativistic mechanics Geometric quantization of holomorphic manifolds

295 ....

295 306 312 324 332 335 342

Supergeometry

347

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

347 352 358 366 382 385 388 392 401 423 426

Graded tensor calculus Graded differential calculus and connections Geometry of graded manifolds Lagrangian formalism on graded manifolds Lagrangian supermechanics Graded Poisson manifolds Hamiltonian supermechanics BRST complex of constrained systems Appendix. Supermanifolds Appendix. Graded principal bundles Appendix. The Ne'eman-Quillen superconnection

Contents

7.

8.

9.

10.

ix

Deformation quantization

433

7.1 7.2 7.3 7.4 7.5 7.6

433 444 450 459 472 475

Gerstenhaber's deformation of algebras Star-product Fedosov's deformation quantization Kontsevich's deformation quantization Deformation quantization and operads Appendix. Monoidal categories and operads

Non-commutative geometry

483

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

484 486 492 498 503 507 509 512 514 518

Modules over C*-algebras Non-commutative differential calculus Differential operators in non-commutative geometry . . . . Connections in non-commutative geometry Connes' non-commutative geometry Landsman's quantization via groupoids Appendix. if-Theory of Banach algebras Appendix. The Morita equivalence of C*-algebras Appendix. Cyclic cohomology Appendix. KK-Theory

Geometry of quantum groups

523

9.1 9.2 9.3

523 530 535

Quantum groups Differential calculus over Hopf algebras Quantum principal bundles

Appendixes 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

Categories Hopf algebras Groupoids and Lie algebroids Algebraic Morita equivalence Measures on non-compact spaces Fibre bundles I. Geometry and connections Fibre bundles II. Higher and infinite order jets Fibre bundles III. Lagrangian formalism Fibre bundles IV. Hamiltonian formalism Fibre bundles V. Characteristic classes /f-Theory of vector bundles Elliptic complexes and the index theorem

541 541 546 553 565 569 586 611 618 626 633 648 650

x

Geometric and Algebraic Topological Methods in Quantum Mechanics

Bibliography

661

Index

683

Introduction

Geometry of classical mechanics and field theory is mainly differential geometry of finite-dimensional smooth manifolds, fibre bundles and Lie groups. The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein's General Relativity. In 60-70th, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials [120; 284; 442]. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models [142]. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures [341]. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field [230]. In a general setting, differential geometry of smooth fibre bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fibre bundles and their dynamics is phrased in terms of jet manifolds [169]. Autonomous classical mechanics speaks the geometric language of symplectic and Poisson manifolds [l; 279; 426]. Non-relativistic time-dependent mechanics can be formulated as a particular field theory on fibre bundles over R [294]. At the same time, the standard mathematical language of quantum mechanics and perturbative field theory, except gravitation theory, has been long far from geometry. In the last twenty years, the incremental development of new physical ideas in quantum theory (including super- and BRST symmetries, geometric and deformation quantization, topological field the1

2

Geometric and Algebraic Topological Methods in Quantum Mechanics

ory, anomalies, non-commutativity, strings and branes) has called into play advanced geometric techniques, based on the deep interplay between algebra, geometry and topology. Let us briefly survey some peculiarities of geometric and algebraic topological methods in quantum mechanics. Let us recall that, in the framework of algebraic quantization, one associates to a classical system a certain (e.g., von Neumann, C*-, canonical commutation or anticommutation relation) algebra whose different representations are studied. Quantization techniques under discussion introduce something new. Namely, they can provide non-equivalent quantizations of a classical system corresponding to different values of some topological and differential invariants. For instance, a symplectic manifold X admits a set of non-equivalent star-products indexed by elements of the cohomology group i?2(X)[[/i]] [206; 340]. Thus, one may associate to a classical system different underlying quantum models. Of course, there is a question whether this ambiguity is of physical or only mathematical nature. Prom the mathematical viewpoint, one may propose that any quantization should be a functor between classical and quantum categories (e.g., some subcategory of Poisson manifolds on the classical side and a subcategory of C*-algebras on the quantum side) [271]. From the physical point of view, dequantization becomes important. There are several examples of sui generis dequantizations. For instance, Berezin's quantization [145] in fact is dequantization. One can also think of well-known Gelfand's map as being dequantization of a commutative C*-algebra A by the algebra of continuous complex functions vanishing at infinity on the spectrum of A. This dequantization has been generalized to non-commutative unital C*-algebras [105; 239]. The concept of the strict C*-algebraic deformation quantization implies an appropriate dequantization when h —> 0 [269; 372]. In Connes' non-commutative geometry, dequantization of the spectral triple in the case of a commutative algebra C°°(X) is performed in order to restart the original differential geometry of a spin manifold X [107; 368]. I. Let us start with familiar differential geometry. There are the following reasons why this geometry contributes to quantum theory. (i) Most of the quantum models comes from quantization of the original

Introduction

3

classical systems and, therefore, inherits their differential geometric properties. First of all, this is the case of canonical quantization which replaces the Poisson bracket {/, / ' } of smooth functions with the bracket [/, /'] of Hermitian operators in a Hilbert space such that Dirac's condition

[f,T] = -ih{fJ!) holds. Let us mention Berezin-Toeplitz quantization [47; 145; 365] and geometric quantization [141; 401; 426; 438] of symplectic, Poisson and Kahler manifolds. (ii) Many quantum systems are considered on a smooth manifold equipped with some background geometry. As a consequence, quantum operators are often represented by differential operators which act in a pre-Hilbert space of smooth functions. A familiar example is the Schrodinger equation. The Kontsevich deformation quantization is based on the quasi-isomorphism of the differential graded Lie algebra of multivector fields (endowed with the Schouten-Nijenhuis bracket and the zero differential) to that of polydifferential operators (provided with the Gerstenhaber bracket and the modified Hochschild differential) [219; 255]. (iii) In some quantum models, differential geometry is called into play as a technical tool. For instance, a suitable [/(l)-principal connection is used in order to construct the operators / in the framework of geometric quantization. Another example is Fedosov's deformation quantization where a symplectic connection plays a similar role [149]. Let us note that this application has stimulated the study of symplectic connections [165]. (iv) Geometric constructions in quantum models often generalize the classical ones, and they are build in a similar way. For example, connections on principal superbundles [21], graded principal bundles [405], and quantum principal bundles [293] are defined by means of the corresponding oneforms in the same manner as connections on smooth principal bundles with structure finite-dimensional Lie groups. II. In quantum models, one deals with infinite-dimensional smooth Banach and Hilbert manifolds and (locally trivial) Hilbert and C*-algebra bundles. The definition of smooth Banach (and Hilbert) manifolds follows that of finite-dimensional smooth manifolds in general, but infinite-dimensional

4

Geometric and Algebraic Topological Methods in Quantum Mechanics

Banach manifolds are not locally compact, and they need not be paracompact [273; 422]. In particular, a Banach manifold admits the differentiable partition of unity if and only if its model space does. It is essential that Hilbert manifolds (but not, e.g., nuclear manifolds) satisfy the inverse function theorem and, therefore, locally trivial Hilbert bundles are denned. However, they need not be bundles with a structure group. (i) Infinite-dimensional Kahler manifolds provide an important example of Hilbert manifolds [327]. In particular, the projective Hilbert space of complex rays in a Hilbert space E is such a Kahler manifold. This is the space the pure states of a C*-algebra A associated to the same irreducible representation n of A in a Hilbert space E [129]. Therefore, it plays a prominent role in many quantum models. For instance, it has been suggested to consider a loop in the projective Hilbert space, instead of a parameter space, in order to describe Berry's phase [7; 43]. We have already mentioned the dequantization procedure which represents a unital C*-algebra by a Poisson algebra of complex smooth functions on a projective Hilbert space [105]. (ii) Sections of a Hilbert bundle over a smooth finite-dimensional manifold X make up a particular locally trivial continuous field of Hilbert spaces in [129]. Conversely, one can think of any locally trivial continuous field of Hilbert spaces or C*-algebras as being the module of sections of a topological fibre bundle. Given a Hilbert space E, let B C B{E) be some C*-algebra of bounded operators in E. The following fact reflects the nonequivalence of Schrodinger and Heisenberg quantum pictures. There is the obstruction to the existence of associated (topological) Hilbert and C*algebra bundles £ —» X and B —+ X with the typical fibres E and B, respectively. Firstly, transition functions of £ define those of B, but the latter need not be continuous, unless B is the algebra of compact operators in E. Secondly, transition functions of B need not give rise to transition functions of £. This obstruction is characterized by the Dixmier-Douady class of B in the Cech cohomology group H3(X, Z). There is the similar obstruction to the [/(l)-extension of structure groups of principal bundles [73; 86]. One also meets the Dixmier-Douady class as the obstruction to a bundle gerbe being trivial [58; 87]. (iii) There is a problem of the definition of a connection on C*-algebra bundles which comes from the fact that a C*-algebra (e.g., any commutative C*-algebra) need not admit non-zero bounded derivations. An unbounded derivation of a C*-algebra A obeying certain conditions is an infinitesimal generator of a strongly (but not uniformly) continuous one-parameter group

Introduction

5

of automorphisms of A [62]. Therefore, one may introduce a connection on a C*-algebra bundle in terms of parallel transport curves and operators, but not their infinitesimal generators [15]. Moreover, a representation of A does not imply necessarily a unitary representation of its strongly (not uniformly) continuous one-parameter group of automorphisms. In contrast, connections on a Hilbert bundle over a smooth manifold can be defined both as particular first order differential operators on the module of its sections [296] and a parallel displacement along paths lifted from the base [228]. (iv) Instantwise geometric quantization of time-dependent mechanics is phrased in terms of Hilbert bundles over R [174; 401]. Holonomy operators in a Hilbert bundle with a structure finite-dimensional Lie group are well known to describe the non-Abelian geometric phase phenomena [44]. At present, holonomy operators in Hilbert bundles attract special attention in connection with quantum computation and control theory [159; 181; 349]. III. Geometry in quantum systems speaks mainly the algebraic language of rings, modules and sheaves due to the fact that the basic ingredients in the differential calculus and differential geometry on smooth manifolds (except non-linear differential operators) can be restarted in a pure algebraic way. (i) Any smooth real manifold X is homeomorphic to the real spectrum of the M-ring C°°(X) of smooth real functions on X provided with the Gelfand topology [17; 233]. Furthermore, the sheaf Cg of germs of / G C°°(X) on this topological space fixes a unique smooth manifold structure on X such that it is the sheaf of smooth functions on X. The pair (X, C^) exemplifies a local-ringed space. A sheaf SRon a topological space X is said to be a local-ringed space if its stalk 9lx at each point x £ X is a local commutative ring [414], One can associate to any commutative ring A the particular local-ringed space, called an affine scheme, on the spectrum Spec .4. of A endowed with the Zariski topology [421]. Furthermore, one can assign the following algebraic variety to any commutative finitely generated /C-ring A over an algebraically closed field K. Given a ring fC[x] of polynomials with coefficients in /C, let us consider the epimorphism 4> : K\x\ —» A defined by the equalities Diff S(P, Q) are the jet modules J"P of P. Using the first order jet module JlP, one also restarts the notion of a connection on an ^-module P [260; 296]. Such a connection assigns to each derivation r £ dA of a /C-ring .4 a first order P-valued differential operator VT on P obeying the Leibniz rule V r (ap)=r(o)p + aVT(p). For instance, if P is a C°° (X)-module of sections of a smooth vector bundle Y —> X, we come to the familiar notions of a linear differential operator on Y, the jets of sections of Y —» X and a linear connection on Y —» X. Similarly, connections on local-ringed spaces are introduced [296]. In supergeometry, connections on graded modules over a graded commutative ring and graded local-ringed spaces are defined [2l]. In non-commutative geometry, different definitions of a differential operator on modules over a non-commutative ring have been suggested [50; 136; 286]. Roughly speaking, the difficulty lies in the fact that, if d is a derivation of a non-commutative /C-ring A, the product ad, a € A, need not be so. There are also different definitions of a connection on modules over a non-commutative ring [137; 267]. (iv) Let K. be a commutative ring, A a (commutative or noncommutative) /C-ring, and Z(A) the center of A. Derivations of A make up a Lie /C-algebra 5.4. Let us consider the Chevalley-Eilenberg com-

Introduction

7

plex of /C-multilinear morphisms of DA to A, seen as a DA-module [160; 426]. Its subcomplex O*($A, d) of Z (^-multilinear morphisms is a differential graded algebra, called the Chevalley-Eilenberg differential calculus over A. It contains the minimal differential calculus O*A generated by elements da, a € A. If ^4. is the R-ring C°°(X) of smooth real functions on a smooth manifold X, the module QC°°(X) of its derivations is the Lie algebra of vector fields on X and the Chevalley-Eilenberg differential calculus over C°°(X) is exactly the algebra of exterior forms on a manifold X where the Chevalley-Eilenberg coboundary operator d coincides with the exterior differential, i.e., O*(X>C°°(X),d) is the familiar de Rham complex. In a general setting, one therefore can think of elements of the ChevalleyEilenberg differential calculus Ok(QA, d) over an algebra .4 as being differential forms over A. Similarly, the Chevalley-Eilenberg differential calculus over a graded commutative ring is constructed [160]. IV. As was mentioned above, homology and cohomology of spaces and algebraic structures often play a role of sui generis hidden quantization parameters which can characterize non-equivalent quantizations. (i) First of all, let us mention the abstract de Rham theorem [220] and, as its corollary, the homomorphism H*(X,Z)-+H*(X) of the Cech cohomology of a smooth manifold X to the de Rham cohomology of exterior forms on X. For instance, the Chern classes c* € H2l(X, Z) of a [/(n)-principal bundle P —» X are represented by the de Rham cohomology classes of certain characteristic exterior forms V-2.%(FA) on X expressed into the strength two-form FA of a principal connection A on F - » I [142]. The Chern class c-i of a complex line bundle plays a prominent role in many quantization schemes, e.g., geometric quantization. The well-known index theorem establishes the equality of the index of an elliptic operator on a fibre bundle to its topological index expressed in terms of the characteristic forms of the Chern character, Todd and Euler classes. Let us note that the classical index theorem deals with linear elliptic operators on compact manifolds. They are Fredholm operators. In order to generalize the index theorem to non-compact manifolds, one either imposes conditions sufficient to force operators to be the Fredholm ones or

8

Geometric and Algebraic Topological Methods in Quantum Mechanics

considers the operators which are no longer Fredholm, but their index can be interpreted as a real number by some kind of averaging procedure [375]. (ii) Geometric quantization of a symplectic manifold (X, Cl) is affected by the following ambiguity. Firstly, the equivalence classes of admissible connections on a prequantization bundle (whose curvature obeys the prequantization condition R = ifl) are indexed by the set of homomorphisms of the homotopy group TT\(X) of X to U(l) [257; 312]. Secondly, there are non-equivalent bundles of half-forms over X in general and, consequently, the non-equivalent quantization bundles exist [141]. This ambiguity leads to non-equivalent quantizations. (iii) The cohomology analysis gives a rather complete picture of deformation quantization of symplectic manifolds. Let K, be a commutative ring and K.[[h]\ the ring of formal series in a real parameter h. Let us recall that, given an associative (resp. Lie) algebra A over a commutative ring /C, its Gerstenhaber deformation [166] is an associative (resp. Lie) /C[[/i]]-algebra Ah such that Ah/hAh = A. The multiplication in Ah reads oo

a*b = aob+

Y^ hrCr(a, b) r=l

where o is the original associative (resp. Lie) product and Cr are 2-cochains of the Hochschild (resp. Chevalley-Eilenberg) complex of A. The obstruction to the existence of a deformation of A lies in the third Hochschild (resp. Chevalley-Eilenberg) cohomology group. Let A = C°°(X) be the ring of complex smooth functions on a smooth manifold X. One considers its associative deformations Ah where the cochains Cr are bidifferential operators of finite order. The multidifferential cochains make up a subcomplex of the Hochschild complex of A, and its cohomology equals the space of multi-vector fields on X [433]. If B*'*'1 is the Koszul boundary operator. The algebra B is provided with the graded Poisson bracket [,], and there exists an element 9 of B, called the BRST charge, such that [0,0] = 0 and D = [Q,.] = 5 + d up to extra terms of non-zero ghost number is the nilpotent classical BRST operator. The BRST cohomology is defined as the cohomology of this classical BRST operator. The BRST complex has been built for constrained Poisson systems [245] and time-dependent Hamiltonian systems with Lagrangian constraints [295] as an extension of the Koszul-Tate complex of constraints through introduction of ghosts. Quantum BRST cohomology has been studied in the framework of geometric [419] and deformation [49] quantization. V. Contemporary quantum models appeal to a number of new algebraic structures and the associated geometric techniques.

10

Geometric and Algebraic Topological Methods in Quantum Mechanics

(i) For instance, SUSY models deal with graded manifolds and different types of supermanifolds, namely, H°°-, G°°-, GH°°-, G-supermanifolds over (finite) Grassmann algebras, R°°- and .R-supermanifolds over ArensMichael algebras of Grassmann origin and the corresponding types of DeWitt supermanifolds [21; 22; 69]. Their geometries are phrased in terms of graded local-ringed spaces. Let us note that one usually considers supervector bundles over G-supermanifolds. Firstly, the category of these supervector bundles is equivalent to the category of locally free sheaves of finite rank (in contrast, e.g., with Gff°°-supermanifolds). Secondly, derivations of the structure sheaf of a G-supermanifold constitute a locally free sheaf (this is not the case, e.g., of G°°-supermanifolds). Moreover, this sheaf is again a structure sheaf of some G-superbundle (in contrast with graded manifolds). At the same time, most of the quantum models uses graded manifolds. They are not supermanifolds, though there is the correspondence between graded manifolds and DeWitt if00-supermanifolds. By virtue of the well-known Batchelor theorem, the structure ring of any graded manifold with a body manifold Z is isomorphic to the graded ring AE of sections of some exterior bundle AE* —> Z. In physical models, this isomorphism holds fixed from the beginning as a rule and, in fact, by geometry of a graded manifold is meant the geometry of the graded ring AE- For instance, the familiar differential calculus in graded exterior forms is the graded Chevalley-Eilenberg differential calculus over such a ring. (ii) Non-commutative geometry is mainly developed as a generalization of the calculus in commutative rings of smooth functions [107; 194; 267]. In a general setting, any non-commutative /C-ring A over a commutative ring /C can be called into play. One can consider the above mentioned Chevalley-Eilenberg differential calculus O*A over A, differential operators and connections on A-modules (but not their jets). If the derivation K.module T)A is a finite projective module with respect to the center of A, one can treat the triple (A, (3A, O*A) as a non-commutative space. For instance, this is the case of the matrix geometry, where A is the algebra of finite matrices, and of the quantum phase space, where A is a finitedimensional algebra of canonical commutation relations. Non-commutative field theory also can be treated in this manner [133; 359], though the bracket of space coordinates [xfl,x1/} = ie^

in this theory is also restarted from Moyal's star-product xM * xv [99; 133].

Introduction

11

A different linear coordinate product [x»,xl'] = i ©° be an Abelian group bundle over the unit space ©° of a groupoid ©. The pair (©,21) together with a homomorphism © —> Iso2l is called the ©-module bundle. One can associate to any ©-module bundle a cochain complex C*(©,21). Let 21 be a 6-module bundle in groups [/(I). The key point is that, similarly to the case of a locally compact group [129], one can associate a C*-algebra C*(0,u) to any locally compact groupoid © provided with a Haar system by means of the choice of a two-cocycle a G C2(©,21) [367]. The algebras C*(<S,a) and C*(©, (xy,x,y) is a Lagrangian submanifold of the symplectic manifold (© x ©x 6 , f l e f i 6 f i ) [27]. A Poisson manifold P is called integrable if there exists a symplectic groupoid ©(P) over P. It is unique up to an isomorphism. Integrable Poisson manifolds subject to a certain class morphisms (isomorphism classes of regular dual pairs) make up a suitable category Poisson [270]. Since the groupoid ©(P) is I- and /-simple connected, one considers the category LG of Lie groupoids possessing this property. Any Lie groupoid yields an associated Lie algebroid A(<S) which is the restriction to ©° of the vertical tangent bundle of the fibration r : © -> ©° [287]. The key point is that, similarly to the dual of a Lie algebra, the dual A*( C*(©)

13

Introduction

is a functor from the category LPoisson to the above mentioned category of C*-algebras [271]. It is a desired functorial quantization. This functor is equivariant under the Morita equivalence of Poisson manifolds in LG [444] and that of C*-algebras [371]. Furthermore, the functorial quantization A*( C*(6) is amplified into the above mentioned strict quantization of C*(->= u(da) £ H

which obey the relations (bu)(a) — bu(a),

u(ba) = u(b)a + (ub)(a).

Another problem of geometry of Hopf algebras is the notion of a quantum principal bundle [75; 82; 293]. In the case of Lie groups, there are two equivalent definitions of a smooth principal bundle, which is both a set of trivial bundles glued together by means of transition functions and a bundle provided with the canonical action of a structure group on the right. In the case of quantum groups, these two notions of a principal bundle are not matched, unless the base is a smooth manifold [139; 355]. • The first definition of a quantum principal bundle repeats the classical one and makes use of the notion of a trivial quantum bundle, a covering of a quantum space (e.g., by a family of non-intersecting closed ideals), and its reconstruction from local pieces [76] which however is not always possible [81]. • The second definition of a quantum principal bundle is algebraic [74; 293]. Let H be a Hopf algebra and V a right K-comodule algebra with respect to the coaction /3 : V —» V H. Let : (3(p) = p ® l }

M = {p£V

be its invariant subalgebra. The triple (P,H,f3) is called a quantum principal bundle if the map ver : V ® V 9 (p ® q) ^ p/3{q) £ ? ® W M

M

is a linear isomorphisms. This condition, called the Hopf-Galois condition, is a key point of this algebraic definition of a quantum principal bundle. By some reasons, one can think of it as being a sui generis local trivialization. (v) Finally, one of the main point of Tamarkin's proof of the formality theorem in deformation quantization is that, for any algebra A over a field of characteristic zero, its Hochschild cochain complex and its Hochschild cohomology are algebras over the same operad [219; 411]. This observation has been the starting point of 'operad renaissance' [253; 297]. Monoidal categories provide numerous examples of algebras for

Introduction

15

operads. Furthermore, homotopy monoidal categories lead to the notion of a homotopy monoidal algebra for an operad. In a general setting, one considers homotopy algebras and weakened algebraic structures where, e.g., a product operation is associative up to homotopy [276]. Their well-known examples are A^-spaces and Aoo-algebras [403]. At the same time, the formality theorem is also applied to quantization of several algebraic geometric structures such as algebraic varieties [255; 450].

Chapter 1

Commutative geometry

In comparison with classical mechanics and field theory phrased in terms of smooth finite-dimensional manifolds, quantum theory speaks the algebraic language adapted to describing systems of infinite degrees of freedom. Geometric techniques are involved in quantum theory due to the fact that the differential calculus over an arbitrary ring can be denned. Their relation to the familiar differential geometry of smooth manifolds is based on the fact that any manifold can be characterized in full by a certain algebraic construction and, furthermore, there is the categorial equivalence between the vector bundles over a smooth manifold and the finite projective modules over the ring of smooth real functions on this manifold.

1.1

Commutative algebra

In this Section, the relevant basics on modules over commutative algebras is summarized [272; 288]. An algebra A is an additive group which is additionally provided with distributive multiplication. All algebras throughout the book are associative, unless they are Lie algebras. A ring is a unital algebra, i.e., it contains a unit element 1. Unless otherwise stated, we assume that 1 ^ 0 , i.e., a ring does not reduce to the zero element. One says that A is a division algebra if it has no a divisor of zero, i.e., ab = 0, a,b £ A, implies either a — 0 or b = 0. Non-zero elements of a ring form a multiplicative monoid. If this multiplicative monoid is a multiplicative group, one says that the ring has a multiplicative inverse. A ring A has a multiplicative inverse if and only if it is a division algebra. A field is a commutative ring whose non-zero elements make up a multiplicative group. A subset I of an algebra A is called a left (resp. right) ideal if it is a 17

18

Geometric and Algebraic Topological Methods in Quantum Mechanics

subgroup of the additive group A and ab e l (resp. 6a € 1) for all a E A, b e l If J is both a left and right ideal, it is called a two-sided ideal. An ideal is a subalgebra, but a proper ideal (i.e., 1 ^ A) of a ring is not a subring because it does not contain a unit element. Let A be a commutative ring. Of course, its ideals are two-sided. Its proper ideal is said to be maximal if it does not belong to another proper ideal. A commutative ring A is called local if it has a unique maximal ideal. This ideal consists of all non-invertible elements, of A. A proper twosided ideal I of a commutative ring is called prime if db £ 1 implies either a £ J or b £ 1. Any maximal two-sided ideal is prime. Given a two-sided ideal 1 c A, the additive factor group A/1 is an algebra, called the factor algebra. If A is a ring, then A/1 is so. If J is a prime ideal, the factor ring A/1 has no divisor of zero, and it is a field if J is a maximal ideal. Remark 1.1.1. We will refer to the following particular construction in the sequel. Let K be a commutative ring and S its multiplicative subset which, by definition, is a monoid with respect to multiplication in K. Let us say that two pairs (a,s) and (a',s'), a,a' £ /C, s,s' £ S, are equivalent if there exists an element s" £ S such that s"{s'a - so,') = 0. We abbreviate with a/s the equivalence classes of (a, s). The set S~1IC of these equivalence classes is a ring with respect to the operations s/a + s'/a' := (s'a + sa')/(ss'), (a/s) • (a'/s') := (aa')/(ss'). There is a homomorphism $s : K 3^ a/1 £ S^IC

(1.1.1)

such that any element of $s(S) is invertible in S^1^. If a ring K has no divisor of zero and S does not contain a zero element, then $ s (1.1.1) is a monomorphism. In particular, if 5 is the set of non-zero elements of K-, the ring S~1fC is a field, called the field of quotients of the fraction field of /C. If K. is field, its fraction field coincides with K. • Given an algebra A, an additive group P is said to be a left (resp. right) A-module if it is provided with distributive multiplication A x P —> P by elements of A such that (ab)p = a(bp) (resp. (ab)p = b(ap)) for all a, b € A and p £ P. If A is a ring, one additionally assumes that lp = p = pi for

19

Chapter 1 Commutative Geometry

all p € P. Left and right module structures are usually written by means of left and right multiplications (a, p) H-» ap and (a, p) >—> pa, respectively. If P is both a left module over an algebra A and a right module over an algebra A', it is called an (A — .4')-bimodule (an .4-bimodule if A — .4'). If A is a commutative algebra, an (.4 — .4)-bimodule P is said to be commutative if ap = pa for all a € A and p £ P. Any left or right module over a commutative algebra A can be brought into a commutative bimodule. Therefore, unless otherwise stated, any module over a commutative algebra A is called an .4-module (see Section 8.1). A module over a field is called a vector space. If an algebra A is a module over a ring K., it is said to be a IC-algebra. Any algebra can be seen as a Z-algebra. Remark 1.1.2. Any AC-algebra A can be extended to a unital algebra A by the adjunction of the identity 1 to A. The algebra A, called the unital extension of A, is defined as the direct sum of ^-modules K © A provided with the multiplication (Ai,ai)(A2,a2) = (AiA2,Aia2 + A 2 ai+aia 2 ),

Ai,A 2 £/C,

ai,a2eA

Elements of A can be written as (A, a) = Al + a, A € /C, a G A. Let us note that, if A is a unital algebra, the identity 1^ in A fails to be that in A. In this case, the algebra A is isomorphic to the product of A and the algebra K,(l — 1A)D In this Chapter (except Sections 1.5C), all associative algebras are assumed to be commutative, unless they are graded. The following are standard constructions of new modules from old ones. • The direct sum Pi © P 2 of ,4-modules Pi and P 2 is the additive group Pi x P 2 provided with the .4-module structure a(Pi,P2) = (api,ap2),

Pi,2 G P l i 2 ,

a & A.

Let {Pi} ie / be a set of modules. Their direct sum ©P* consists of elements (..., pi,...) of the Cartesian product n Pi s u c n that pi ^ 0 at most for a finite number of indices i € I. • The tensor product P ® Q of ^-modules P and Q is an additive group which is generated by elements p® q, p € P, q 6 Q, obeying the relations {p + p') ® q = P ® q + p' ® q, p ® (q + q') = p®q+p®q'', p e P, q € Q, a € A, pa (8) q = p aq,

20

Geometric and Algebraic Topological Methods in Quantum Mechanics

(see Remark 10.4.1), and it is provided with the .4-module structure a(p ® q) = (ap) ®q = p® (qa) = (p q)a.

If the ring A is treated as an .4-module, the tensor product A ®^ Q is canonically isomorphic to Q via the assignment A ®A

QBa®qaq£Q.

• Given a submodule Q of an .4-module P, the quotient P/Q of the additive group P with respect to its subgroup Q is also provided with an «4-module structure. It is called a factor module. • The set Horn ,4 (P, Q) of .4-linear morphisms of an .4-module P to an .4-module Q is naturally an .4-module. The .4-module P* = Horn ^(P, A) is called the dual of an .4-module P. There is a natural monomorphism P-» P**. An .4-module P is called free if it has a basis, i.e., a linearly independent subset I C P spanning P such that each element of P has a unique expression as a linear combination of elements of / with a finite number of non-zero coefficients from an algebra A. Any vector space is free. Any module is isomorphic to a quotient of a free module. A module is said to be finitely generated (or of finite rank) if it is a quotient of a free module with a finite basis. One says that a module P is protective if it is a direct summand of a free module, i.e., there exists a module Q such that P®Q is a free module. A module P is projective if and only if P = pS where 5 is a free module and p is a projector of S, i.e., p 2 = p. If P is a projective module of finite rank over a ring, then its dual P* is so, and P** is isomorphic to P. THEOREM

1.1.1. Any projective module over a local ring is free.

•

Now we focus on exact sequences, direct and inverse limits of modules [288; 303]. A composition of module morphisms P - U Q -^->T is said to be exact at Q if Ker j = Im i. A composition of module morphisms

O^P -UQ -Ur^o

(1.1.2)

21

Chapter 1 Commutative Geometry

is called a short exact sequence if it is exact at all the terms P, Q, and T. This condition implies that: (i) i is a monomorphism, (ii) Ker j = Imi, and (iii) j is an epimorphism onto the quotient T — Q/P. THEOREM 1.1.2. Given an exact sequence of modules (1.1.2) and another ,4-module R, the sequence of modules

0->EomA{T,R) i^RomA{Q,R)

^Eom{P,R)

(1.1.3)

is exact at the first and second terms, i.e., j * is a monomorphism, but i* HI need not be an epimorphism. One says that the exact sequence (1.1.2) is split if there exists a monomorphism s :T —> Q such that j o s = IdT or, equivalently,

Q = i{P) ® s{T) ^P®T. The exact sequence (1.1.2) is always split if T is a projective module. A directed set 7 is a set with an order relation < which satisfies the following three conditions: (i) i < i, for all i € I; (ii) if i < j and j < k, then i < k; (iii) for any i,j € / , there exists k £ I such that i < k and j < k. It may happen that i ^ j , but i < j and j < i simultaneously. A family of modules {P{\i^i (over the same algebra), indexed by a directed set / , is called a direct system if, for any pair i < j , there exists a morphism r* : Pi —> Pj such that r\ = I d P i ,

r) or{ = ri,

i<j
P^ such that r ^ = r£, o rj for all i < j . The module P^ consists of elements of the direct sum ©Pj modulo the identification of elements of Pi with their images in Pj for all i < j . An example of a direct system is a direct sequence Po —»Pi ^ • • • P / M . . . ,

J = N.

(1.1.4)

It should be noted that direct limits also exist in the categories of commutative algebras and rings, but not in categories whose objects are non-Abelian groups.

22

Geometric and Algebraic Topological Methods in Quantum Mechanics

THEOREM 1.1.3. Direct limits commute with direct sums and tensor products of modules. Namely, let {Pi} and {Qi} be two direct systems of modules over the same algebra which are indexed by the same directed set / , and let P^ and Q^ be their direct limits. Then the direct limits of the direct systems {Pi © Qi} and {Pt Qi} are P^ © Qoo and Poo ® Qoo, respectively. •

A morphism of a direct system {Pi,rlj}i to a direct system {Qi>,plj,}i> consists of an order preserving map / : / — » / ' and morphisms Fj : Pj —> Qf(i) which obey the compatibility conditions 0S^)oFi = Fior). If PQQ and Qoo are limits of these direct systems, there exists a unique morphism F^ : P ^ —> Qoo such that

p£>oFi=F00ori0. Moreover, direct limits preserve monomorphisms and epimorphisms. To be precise, if all Ft : Pi —> Q/(») axe monomorphisms or epimorphisms, so is $00 : Poo —* Qoo- As a consequence, the following holds. THEOREM

1.1.4. Let short exact sequences 0-^Pi

^Qi

^Ti^O

(1.1.5)

for all i £ I define a short exact sequence of direct systems of modules {P,}/, {Qi}i, and {Tj}/ which are indexed by the same directed set / . Then there exists a short exact sequence of their direct limits O^Poo ^ Q o o ^ T o o ^ O .

(1.1.6)

• In particular, the direct limit of factor modules Qi/Pi is the factor module Qoo/Poo- By virtue of Theorem 1.1.3, if all the exact sequences (1.1.5) are split, the exact sequence (1.1.6) is well. Example 1.1.3. Let P be an ^-module. We denote P®k =®P. Let us consider the direct system of ^-modules with respect to monomorphisms A -^(A®P)

— > - - - ( ^ © P © - - - © P 0 f e ) —>••• •

Chapter 1 Commutative Geometry

23

Its direct limit ®P = A® P ••• ®P®k®---

(1.1.7)

is an N-graded ,4-algebra with respect to the tensor product . It is called the tensor algebra of a module P. Its quotient with respect to the ideal generated by elements pp'+p'' ®p, p,p' e P, is an N-graded commutative algebra, called the exterior algebra of a module P. • We restrict our consideration of inverse systems of modules to inverse sequences

P° «— P1
Pi such that 7if° = irj o itf for all i < j . It consists of elements (... ,p*,...), pl € Pl, of the Cartesian product f] Pl such that p1 = K{ (p3) for all i < j . THEOREM 1.1.5. Inductive limits preserve monomorphisms, but not epimorphisms. If a sequence

Q-^Pi

^Q{

-51*T*,

ieN,

of inverse systems of modules {P1}, {Q1} and {T1} is exact, so is the sequence of the inductive limits poo

/f»oo

0_>p°° *-^>Q°° ?—>T°°.

n In contrast with direct limits, the inductive ones exist in the category of groups which are not necessarily commutative. Example 1.1.4. Let {Pi} be a direct sequence of modules. Given another module Q, the modules Hom(Pj, Q can be endowed with the two different Amodule structures (afc)(p) := a$(p),

($ • o)(p) := $(ap),

o£i,

p£?.

(1.2.1)

For the sake of convenience, we will refer to the second one as the .A*-module structure. Let us put <Sa$ := a$ - $ • a,

(1-2.2)

a £ A.

1.2.1. An element A Q which obeys the Leibniz rule u(ab) =u(a)b + au(b),

a,b G A.

(1.2.10)

It should be emphasized that this derivation rule differs from that (6.2.3) of graded derivations. A Q-valued derivation u of A is called inner if there O exists an element q £ Q such that u(a) = qa — aq.

26

Geometric and Algebraic Topological Methods in Quantum Mechanics

If Q = A, the derivation module $A of A is also a Lie algebra over the ring K. with respect to the Lie bracket [ti,!i'] = i i o t i ' - u ' o t i ,

u, u'£ A.

(1.2.11)

Accordingly, the decomposition (1.2.9) takes the form (1.2.12)

DiSi(A) = A®dA.

An s-order differential operator on a module P is represented by a zero order differential operator on the module of s-order jets of P as follows. Given an .A-module P, let us consider the tensor product A ®JC P of ^-modules A and P . We put 5b(a®p) := {ba)®p-a®(bp),

p € P,

a,b £ A.

(1.2.13)

Let us denote by /xfc+1 the submodule of A®K. P generated by elements of the type S*0

O-.-OSbk(a®p).

The k-order jet module Jk{P) of a module P is defined as the quotient of the ^-module A JC P by /i fe+1 . We denote its elements c ®k p. In particular, the first order jet module JX{P) consists of elements c®\p modulo the relations Sa o 5b(l ®ip) = ab®xp-b

i (op) - a ®i (bp) + 1 ®i (abp) = 0. (1.2.14)

The /C-module Jk (P) is endowed with the A- and .A*-module structures b(a ®k p) := ba ®k p,

b» (a®kP) •= a®k(bp).

(1.2.15)

There exists the module morphism Jk :P3p^l®kPe

Jk(P)

(1.2.16)

of the .A-module P to the ,4*-module Jk(P) such that Jk{P), seen as an ,4-module, is generated by elements Jkp, p 6 P. Due to the natural monomorphisms /j,r —> fis for all r > s, there are .4-inodule epimorphisms of jet modules 7rj+1 : Ji+1(P)

- • J*(P).

(1-2.17)

In particular, Trl:J1(P)Ba®ip^ap£P.

(1.2.18)

27

Chapter 1 Commutative Geometry

The above mentioned relation between differential operators on modules and jets of modules is stated by the following theorem [261]. THEOREM 1.2.3. Any Q-valued differential operator A of order k on an ,4-module P factorizes uniquely

A : P A j f e ( P ) —+Q through the morphism Jk (1.2.16) and some ,4-module homomorphism fA :

Jk(P) -> Q.

•

The proof is based on the fact that the morphism Jk (1.2.16) is a fc-order i7 (P)-valued differential operator on P. Let us denote fc

J: P3pi->l®p£A®P. Then, for any f e Horn .4 (.4 ® P,Q), we obtain 6btfoJ)(p) = f(6b(l®p)). The correspondence A >—> fA defines an .A-module isomorphism DiBa(P,Q)= Horn A(J'(P),Q). 1.3

(1.2.19)

Connections on modules and rings

We employ the jets of modules in previous Section in order to introduce connections on modules and commutative rings [296]. Let us consider the jet modules Js = JS{A) of the ring A itself. In particular, the first order jet module J1 consists of the elements a <S>\ b, a,b £ A, subject to the relations ab ! 1 - b i a - a <S>i b + 1 ®i (ab) = 0. The A- and ^.'-module structures (1.2.15) on J1 read c(a ! b) := (ca) i 6,

c • (a <E>i b) := a ®i (cb) = (a ® i b)c.

Besides the monomorphism J1 : A^a^->l®la£Jl (\.2.1(^), there is the .4-module monomorphism ii : A 3 a h^> a ®! 1 £ Jx.

(1.3.1)

28

Geometric and Algebraic Topological Methods in Quantum Mechanics

With these monomorphisms, we have the canonical A-module splitting (1.3.2)

J1=ii(A)®O1, oJ x (6) = a i b = ab ®x 1 + a(l ®i b - b i 1),

where the Amodule O1 is generated by the elements 1 ®i & — &(g>i 1 for all 6 € A Let us consider the corresponding Amodule epimorphism /i1:J13l®i6i-+ligi1&-fc®1lGC1

(1.3.3)

and the composition (1.3.4)

d1 = h1oJ1:Aab^l®lb-b®1l€O1,

which is a /C-module morphism. This is a C^-valued derivation of the Jt-ring A which obeys the Leibniz rule d1 (ab) = 1 ®i ab - ab ®i 1 + a ®i b - a ! b = a d ^ + ( d 1 ^ . It follows from the relation (1.3.1) that adx6 = (d1b)a for all a, 6 € A Thus, seen as an ,4-module, O1 is generated by the elements dla for all o e A Let O1* = Horn .4 ( 0 \ .4) be the dual of the A-module O1. In view of the splittings (1.2.12) and (1.3.2), the isomorphism (1.2.19) reduces to the duality relation (1.3.5)

DA=OU, QA 9 u 4>u e O1*,

(f>u(d1a) := u(a),

a€A

(1.3.6)

In a more direct way (see Proposition 8.2.1 below), the isomorphism (1.3.5) is derived from the facts that C 1 is generated by elements dla, a E A, and that (dla) is a derivation of A for any £ O1*. However, the morphism oi

_>

oi.*

=

0 ^»

need not be an isomorphism. Let us define the modules Ok, k = 2,..., as the exterior products of the Amodule O1. There are the higher degree generalizations hk

:ji(Ok-1)^Ok,

dk = hk o J 1 : Ok-1 -+ Ok

(1.3.7)

of the morphisms (1.3.3) and (1.3.4). The operators (1.3.7) are nilpotent, i.e., dk o dk~l = 0. They form the cochain complex 0-»/C -^A

^ O

1

^...Qkd{a®1b)®p.

(1.3.9)

Then the isomorphism (1.3.2) leads to the splitting J\P)

= {A®OX)®P

= {A®P)®

{O1 ® P),

(1.3.10)

a (g>i bp O1 ® P,

V(p) = l ® i p - r ( p ) .

(1.3.15)

Though this complementary morphism in fact is a covariant differential on the module P, it is traditionally called a connection on a module. It satisfies the Leibniz rule V(ap) =d1ap + aV(p),

(1.3.16)

30

Geometric and Algebraic Topological Methods in Quantum Mechanics

i.e., V is an (O1 ® P)-valued first order differential operator on P. Thus, we come to the equivalent definition of a connection [260]. DEFINITION 1.3.2. A connection on an ^.-module P is a /C-module mor• phism V (1.3.15) which obeys the Leibniz rule (1.3.16). The morphism V (1.3.15) can be extended naturally to the morphism V : O1 ® P -> O2 P.

Then we have the morphism iJ = V2 :P^O2®P,

(1.3.17)

called the curvature of the connection V on a module P. In view of the isomorphism (1.3.5), any connection in Definition 1.3.2 determines a connection in the following sense. DEFINITION 1.3.3. A connection on an A-module P is an ^-module morphism ViA3ut-^VueDifii(P,P)

(1.3.18)

such that the first order differential operators V u obey the Leibniz rule V u (ap) = u(a)p + aVu(p),

a € A,

p £ P.

(1.3.19)

• 1

Definitions 1.3.2 and 1.3.3 are equivalent if O = DA*. The curvature of the connection (1.3.18) is defined as a zero order differential operator R{u, u') = [V«, Vu.) - V[UiU/]

(1.3.20)

on the module P for all u, u' € DA. Let P be a commutative .A-ring and DP the derivation module of P as a AT-ring. Definition 1.3.3 is modified as follows. DEFINITION 1.3.4. A connection on an ,4-ring P is an .4-module morphism 5 i 9 u H V u £ DP,

(1.3.21)

which is a connection on P as an ^-module, i.e., obeys the Leinbniz rule • (1.3.19).

31

Chapter 1 Commutative Geometry

Two such connections V u and V^ differ from each other in a derivation of the .A-ring P, i.e., which vanishes on A C P. The curvature of the connection (1.3.21) is given by the formula (1.3.20). 1.4

Homology and cohomology of complexes

This Section summarizes the relevant basics on homology and cohomology of complexes of modules over a commutative ring [288; 303]. Let K. be a commutative ring. A sequence 0^B0 £-2?! o are trivial. A chain complex 5» is acyclic if there exists a homotopy operator h. This is defined as a set of module morphisms hp:Bp^Bp+1,

peN,

such that h p _! odp + dp+l o h p = IdB p ,

p&N+.

It follows that, if dpbp = 0, then bp = dp+i(hpbp), and Hp>0(Bt,) = 0.

32

Geometric and Algebraic Topological Methods in Quantum Mechanics

A chain complex (1.4.1) is said to be a chain resolution of a module B if it is acyclic and H0(B*) = B. This complex defines the exact sequence 0 SI

(1.4.3)

is denned as a family of degree-preserving /C-module homomorphisms 7p

: i? p -> s ; ,

peN,

which commute with the boundary operators, i.e., 9

P+I°7P+I

=lp°dp+i.

Chapter 1 Commutative Geometry

33

It follows that if bp £ Bp is a cycle or a boundary, then 7P(6P) G B'p is well. Therefore, the chain morphism of complexes (1.4.3) yields the induced homomorphism of their homology groups

[ 7 ], : H.(B.)

-» H.(B't),

[7]([6]) := [7(6)],

(1-4.4)

where [b] denotes the homology class of 6 € B». Let 7 , 7 ' : £» —> B't be two different chain morphisms of the same chain complexes. By a chain homotopy h is meant a family of /C-module homomorphisms

hp:Bp^

B'p+1,

peN,

of degree +1 such that

d'p+i °hP + hp-i o dp = 7 P - 7p.

If a chain homotopy exists, the chain morphisms 7 and 7' are called homotopic. The difference 7 — 7' of homotopic chain morphisms sends cycles onto boundaries, i.e., these morphisms induce the same homomorphisms [7], and [7']* (1.4.4) of homology groups. In particular, a chain morphism 7 (1.4.3) is said to be a homotopy equivalence if there exists a chain morphism £ : B'% —» B* such that the compositions £ ° 7 and 70£ are homotopic to the identity morphisms of the chain complexes B* and B'^, respectively. Chain complexes connected by a homotopy equivalence are called homotopic. Their homology groups are isomorphic. Let us consider a short sequence of chain complexes

0 -» C, - ^ B . -$-> F . -> 0,

(1.4.5)

34

Geometric and Algebraic Topological Methods in Quantum Mechanics

represented by the commutative diagram 0

0

I

I

dp C p - i B2 be two different cochain morphisms. By their cochain homotopy h is meant a family of homomorphisms

hP:Bl^Bl~\

peN+,

such that $P-I ohP + hp+16p = 7 P - ip. If a cochain homotopy exists, the cochain morphisms 7 and 7' are called homotopic. Homotopic cochain morphisms 7 and 7' induce the same homomorphisms [7]* and [7']* (1.4.11) of cohomology groups. One says that the cochain morphism 7 (1.4.10) is a homotopy equivalence if there exists a cochain morphism £ : B2 —* B\ such that the compositions C ° 7 and 7 o £ axe homotopic to the identity morphisms of the complexes B* and B2, respectively. Complexes connected by a homotopy equivalence are called homotopic, and their cohomology groups are isomorphic. Let us consider a short sequence of complexes 0 -^ C* ^UB* -^-> F* -> 0,

(1.4.12)

38

Geometric and Algebraic Topological Methods in Quantum Mechanics

represented by the commutative diagram 0

0

.._ U J._... Tp

...

y

I

BP

7p+l

BP+1

- ^

CP I ...

»

I

Cp+i

pp i f U

> •• •

I >...

pp+i

I

I

0

0

It is said to be exact if all columns of this diagram are exact, i.e., 7 is a cochain monomorphism and £ is a cochain epimorphism onto the quotient F* = B*/C*. The following assertions are similar to Theorems 1.4.1 and 1.4.2. 1.4.3. The short exact sequence of complexes (1.4.12) yields the long exact sequence of their cohomology groups THEOREM

0->H°(C*)

l

^H°(B*)

[

^H°(F*)

—>HP(C*) ^ > F P ( B * )

^H\C*)

1

^>HP(F*)

—>•••

^HV+1{C*)

(1.4.13) —••••.

• THEOREM

1.4.4. A direct sequence of complexes B*

_ » B J —»• • • J3J 7 ^B* k + 1 -*..•

(1.4.14)

admits a direct limit B^ which is a complex whose cohomology ,ff*(.B£o) is a direct limit of the direct sequence of cohomology groups H*(B*0)

-*H*{Bl)

-^...H*(B*k)b^>]H*(B*k+1)

—....

This statement is also true for a direct system of complexes indexed by an arbitrary directed set. •

39

Chapter 1 Commutative Geometry

1.5

Homology and cohomology of groups and algebras

Subsections: A. Homology and cohomology of groups, 39; B. The Koszul complex, 44; C. Hochschild cohomology, 49; D. Chevalley-Eilenberg cohomology, 53. We briefly sketch homology and cohomology of some algebraic systems needed in the sequel. These are homology and cohomology of groups, homology of the Koszul complex, Hochschild cohomology, ChevalleyEilenberg cohomology.

A. Homology and cohomology of groups Homology and cohomology of groups demonstrate the standard techniques of constructing homology and cohomology of algebraic systems [288]. Given a set Z, one can introduce a chain complex as follows. Let Zk be a A;+l

free Z-module whose basis is the Cartesian product x Z. In particular, Zo is a free Z-module whose basis is Z. Let us define Z-linear homomorphisms d0 : Zo 3mi(zl0) i-> Ylmi

eZ

'

m

ȣz.

(1.5.1)

i

3k+1

: Zk+1 -+ Z*,

ke N, fe+1

dk+i(z0, • • •, z/M-i) = ^2(~'i-y(zo,---,Zj,...,zk+i), j=o

(1.5.2)

where the caret "denotes omission. It is readily observed that dkodk+\ = 0 for all k € N. Thus, we obtain the chain complex 0 0(G,A) = 0. For instance, Hk>o{G,Z)=0, where Z is treated as a trivial G-module. We refer the reader to [288] for cohomology of cyclic and • free groups. By cohomology of a group G with coefficients in a G-module A is also meant the cohomology HQ(G, A) of the subcomplex QQ of the complex Q* whose fc-cochains are .A-valued functions of k arguments from G which vanish whenever one of the arguments is equal to 1. It is easily verified

that skgk c gk+1.

Let us show that the cohomology group HQ(G,A) classifies the extensions of the group G by an additive group A. Such an extension is defined as a sequence (1.5.14)

O-*A^W-^G->1

of group homomorphisms, where i is a monomorphism onto a normal subgroup of W and TT is an epimorphism onto the factor group G = W/A, i.e., Imi = 7r~ 1 (l). By analogy with sequences of additive groups, the sequence (1.5.14) is said to be exact. For the sake of simplicity, we will identify A with its image i(A) c W, and write the group operation in W in the additive form. Any extension (1.5.14) yields a homomorphism of G to the group of automorphisms of A as follows. Let w(g) be representatives in W of elements g £ G. Then any element w G W is uniquely written in the form w = aw + w(g), aw € A. Let us consider the automorphism <j)g : a H-> ga = w(g) + a — w(g),

a G A.

(1.5.15)

Certainly, this automorphism depends only on an element g G G, but not on its representative in W. The association g >—> g defines a desired homomorphism 0:G->AutA

(1.5.16)

This homomorphism <j> makes A to a G-module denoted by A^. Conversely, the homomorphism <j> (1.5.16) corresponds to some extension (1.5.14) of the group G. Among these extensions, there is the semidirect product W = Ax^G with the group operation (a, g) + (a', g') = (a + ga', gg'),

ga' = g{a').

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Chapter 1 Commutative Geometry

THEOREM 1.5.1. There is one-to-one correspondence between the classes of isomorphic extensions (1.5.14) of a group G by an Abelian group A associated to the same homomorphism (1.5.16) and the elements of the • cohomology group H^(G,A^) [288].

Outline of proof. Given representatives w(gi) and w(g2) of elements