Contemporary Problems in Mathematical Physics: Proceedings of the Fourth International Workshop, Cotonou, Republic of Benin 5 - 11 November 2005

Contemporary Problems in Mathematical Physics editors

Jan Govaerts ML Korbert Hounkonnou Alfred Z. Msezane

World Scientific

Contemporary Problems in Mathematical Physics

Proceedings of the Fourth International Workshop on

Contemporary Problems in Mathematical Physics Cotonou, Republic of Benin

5-11 November 2005

editors

JanGovaerts Catholic University of Louvain, Belgium

M.NorbertHounkonnou University of Abomey-Calavi, Republic of Benin

Alfred Z.Msezane Clark Atlanta University, USA

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CONTEMPORARY PROBLEMS IN MATHEMATICAL PHYSICS Proceedings of the Fourth International Workshop Copyright © 2006 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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V

Foreword This volume is the Proceedings of the Fourth International Workshop on Contemporary Problems in Mathematical Physics, which was held in Cotonou (Republic of Benin) from November 5 t h to 1 1 t h , 2005. Being the fourth such scientific gathering, following the First, Second and Third Workshops on Contemporary Problems in Mathematical Physics also organised in Cotonou from October 31 s t to November 5 t h , 1999, October 28 t h to November 2 n d , 2001, and November 1 s t to 7 t h , 2003, respectively, these scientific events have now become a scientific tradition in Western Africa, called to take place on this regular basis every second year in the Republic of Benin, and which its participants are eagerly looking forward to. These COPROMAPH conferences, the acronym by which they have now become known, are organised under the auspices of the International Chair in Mathematical Physics and Applications (ICMPA) located in Cotonou, within the University of Abomey-Calavi, Republic of Benin. The Proceedings of each of the COPROMAPH editions are available as a series of volumes published by World Scientific Publishing, Singapore. Scientific meetings such as the COPROMAPH Workshops provide a unique opportunity for mathematical and theoretical physicists of the African continent to meet and initiate stimulating collaborations both within Africa as well as with scientists from other parts of the world also attending these events. In particular, it is hoped that the greatest benefits will go to the younger generations of African scientists, whose responsability it is to build the basis for a scientific and technological tradition and expertise in Africa extending far beyond the new century into the young millenium, capable of addressing with the appropriate and state-of-theart scientific methods and technological tools the challenges specific to the African environment. On the occasion of the International Year of Physics 2005, the First International COPROMAPH School was also organised, prior to the Workshop itself, from October 31 s t to November 4 t h , 2005, in Cotonou. Being addressed to the younger COPROMAPH participants by laying the ground for some of the plenary and parallel sessions scheduled at the Workshop, two full series of graduate level lecture series, one on Quantum Field Theory and its Perturbation Techniques, the other on An Introduction to String and Superstring Theory, were delivered every morning of that week, with tutorial sessions in the afternoons. On three of these afternoons, student seminars also took place following the tutorials, a rare opportunity for the

vi young participating Ph.D. students to present their work in a more informal setting even though in front of an international and African audience. A general public conference on Einstein's contributions to XX th Century Physics, When Energy Becomes Matter, was also organised the afternoon of November 2 n d , 2005, and met with great success indeed. It was particularly encouraging to see a large participation of high school students very attentive and keen to ask a great many questions. We are grateful to Prof. Jan Govaerts (Belgium) for having delivered these lecture series and the accompanying tutorials, as well as the general public conference. Both the School and the Workshop were satellite events to the IUPAP and UNESCO organised World Conference on Physics and Sustainable Development, which took place from October 31 s t to November 2 n d , 2005, in Durban (Republic of South Africa), also within the context of the International Year of Physics 2005. Media from Benin, whether the written press or radio and television networks, showed their steadfast support by giving a wide coverage to all these events and interviewing a large fraction of participants. This was particularly welcome, since the School and the Workshop were marred by the extreme difficulties the ICMPA has been going through over the recent eighteen months, in spite of the strong support it garnered from international funding agencies such as the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) and the UNESCO (Paris, France), ever since its creation in 2001 in the wake of COPROMAPH2. In fact, all COPROMAPH4 participants expressed their unconditional support for the ICMPA through a signed petition addressed to the Head of State, his Excellency the President of the Republic of Benin, calling on the Highest Authorities of the country to pursue their efforts to support the scientific development of Benin, its sub-region in Western Africa, and beyond towards the whole continent, through initiatives such as the ICMPA, which indeed has had already a strong impact as witnessed by the steady progress in the quality of the successive COPROMAPH editions. This success owes no small debt of gratitude to the Highest Authorities of the Republic of Benin for their unconditional support ever since 2001 until very recently. With the ICMPA having now been granted the status of a UNESCO International Chair, we all may hope that all the promises the ICMPA holds in store for the future of mathematical sciences in Africa will blossom into reality in the coming decade.

vii

Well over 100 participants from Africa, Europe and North America attended this Fourth Workshop and its First School. COPROMAPH4 presentations included three series of invited plenary lectures that lasted throughout the Workshop every morning, as well as shorter research communications organised in two parallel sessions every day, according to the following general themes. . Theoretical Methods of Modern Classical and Quantum Physics . Coherent states, Wavelets, Functional Analysis and Orthogonal Polynomials A very significant fraction of these contributions is collected in this Proceedings volume, following a strict refereeing process through which not all the submitted manuscripts could be accepted for publication. The Fourth International Workshop on Contemporary Problems in Mathematical Physics was made possible through the generous financial support in different forms of many Institutions, and for the first time ever, even by private donors from the civil society in Benin, some wishing to remain anonymous. The latter is quite a remarkable and significant fact for which the organisers are extremely grateful. Their most special thanks are expressed to all the Sponsors listed in the next pages. In this respect, another special word of appreciation goes to the UNESCO which through its support to COPROMAPH4, has committed itself for continued support in the coming years. Beyond the contributions of the participants, and in particular those of the invited plenary lecturers who are especially thanked for their most inspiring presentations and valuable notes, the efforts of many people, too numerous to be all named explicitly here, helped shape this Fourth Workshop into the successful meeting it turned out to be once again, and yet even more so in spite of all the odds it had been facing particularly with the future of the ICMPA at stake. That they may all find here the expression of our gratitude, especially those whose names are listed hereafter, the fruits of their hard and dedicated work becoming each time more plain for all to witness with each new COPROMAPH edition. It is especially gratifying to see already these fruits present in an always growing number of the younger African scientists presenting their own contributions of great quality. Let us all meet again, together with new collaborators, at the Fifth International Workshop on Contemporary Problems in Mathematical Physics, COPROMAPH5, early November 2007. Jan Govaerts, M. Norbert Hounkonnou and Alfred Z. Msezane

Scientific Advisory Committee

A. Afouda (Benin) S. T. Ali (Canada) J.-P. Antoine (Belgium) A. Antonevich (Byelorussia) A. Banyaga (USA) S. Belmehdi (France) K. Bota (USA) X. Chapuisat (France) J. M. Combes (France) G. Ciccotti (Italy) M. Englis (Czech Republic) J.-P. Gazeau (France) G. A. Goldin (USA) J. Govaerts (Belgium) M. N. Hounkonnou (Benin) R. Kerner (France) J. R. Klauder (USA) W. Koepf (Germany) Th. Lebel (France) W. A. Lester, Jr. (USA) A. Makhoute (Maroc) A. Z. Msezane (USA) S. Mtingwa (USA) R. Murenzi (Rwanda) M. T. Niane (Senegal) A. Odzijewicz (Poland) A. Ronveaux (Belgium) J.-P. Ryckaert (Belgium) B. Torresani (France)

Organising Committee

A. Afouda (Benin) G. Y. H. Avossevou (Benin) E. Azatassou (Benin) E. Baloi'tcha (Benin) J. 0 . Chabi (Benin) X. Chapuisat (France) J. M. Combes (Prance) L. Combettes (France) G. Dupont (Belgium) N. Fonton (Benin) M. Gosset (France) J. Govaerts (Belgium) M. N. Hounkonnou (Benin) Th. Lebel (France) W. A. Lester, Jr. (USA) A. Z. Msezane (USA) B. Olory (Benin) C. Peugeot (France)

Sponsored and Supported by

Agence Universitaire de la Francophonie (AUF, Bureau Afrique de POuest, Dakar (Senegal)) UNESCO The US State Department through the UNESCO International Basic Sciences Programme The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste (Italy) L'Institut des Hautes Etudes Scientifiques (IHES) — The Schlumberger Foundation Programme (France) The Ministry of Finances and Economy (Government of Republic of Benin) L'Institut de Formation Sociale, Economique et Civique (INFOSEC - Cotonou, Republic of Benin) L'Hotel Eldorado (Cotonou, Republic of Benin) L'Hotel du Port (Cotonou, Republic of Benin) CODIAM (Archeveche de Cotonou, Republique du Benin) Benin national individual anonymous sponsors ATC Beko (Republic of Benin) IATA-Interair South Africa International Chair in Mathematical Physics and Applications (ICMPA), Cotonou (Republic of Benin) Cabinet des Experts Associes (CEA, Benin) Mr Florent Gnacadja (Director of the business L'Angulaire, Cotonou, Republic of Benin) Imprimerie GUTENBERG (Cotonou, Republic of Benin) Universite Paris Sud XI (France) Universite catholique de Louvain (Louvain-la-Neuve, Belgium) Center for Theoretical Studies of Physical Systems (CTSPS, Clark Atlanta University, USA)) University of Ouagadougou (Burkina Faso) Golfe FM (private radio channel, Republic of Benin) Golfe TV (private television channel, Republic of Benin) La Nouvelle Tribune (private daily newspaper, Republic of Benin) Office de Radiodiffusion et de Television du Benin (ORTB) (Republic of Benin)

For their invaluable help in different forms in the successful organisation of this scientific event, special thanks are also expressed to Professor K. Dramana, for his uncompromising support of the COPROMAPH Workshops as well as the ICMPA Dr Paulin Hounsounon-Tolin Mrs Lidwine A. B. Hounkonnou Mrs Ginette I. Baloi'tcha Mr Lenard Azandegbe Mr Laurent Metognon Mr Atchike Coffi (Director of the Central Administration Automobile Park, Ministry of Finances and Economy, Benin) Mr Bienvenu Olory Mr Gerard Akindes Mr Vincent Cakpo Mr Jean Djessou

Xlll

Contents

Plenary Lectures Representation Theory in Classical and Quantum Physics J.-P. Antoine Lectures on the Gauge Theory/Gravity Correspondence R. de Mello Koch Introductory and Fundamental Mathematical Aspects of Supersymmetry S. J. Gates, Jr.

3

65

120

Parallel Sessions Group I. Theoretical Methods of Modern Classical and Quantum Physics Bosonization of the Schwinger Model by Noncommutative Chiral Bosons J. Ben Geloun, J. Govaerts and M. N. Hounkonnou Bol Loops as a New Approach in Physics Th. B. Bouetou

173

181

Random Phase Approximation with Exchange for the Inner-Shell Electron Photoionization Z. Chen and A. Z. Msezane

198

A New Analytical Approach to the Atmosphere Characterization by a Backscattered LIDAR Signal G. Debiais, F. K. Guedje and M. N. Hounkonnou

212

xiv Nonabelian Global Chiral Symmetry Realisation in the Two-Dimensional N Flavour Massless Schwinger Model L. Gouba, J. Govaerts and M. N. Hounkonnou

230

Superconductivity and Electric Fields: A Relativistic Extension of BCS Superconductivity J. Govaerts and D. Berirand

237

Analytical Solutions of a Generalized Nonlinear Reaction-Diffusion Equation M. Kabir Mahaman and M. N. Hounkonnou

255

The Algebraic Structure of a Generalized Coupled Dispersionless System K. V. Kuetche, T. C. Kofane and B. T. Bouetou

262

A Density Functional Theory Study of the Adsorption of CH3 on the Pt(100) and N i ( l l l ) Surfaces P. S. Moussounda, B. M'Passi-Mabiala, M. F. Haroun, P. Legate, G. Rakotovelo, A. Rakotomahevitra and C. Demangeat

282

The Magnetic Structure of FeMn Layers Across a Cu Spacer B. M'Passi-Mabiala, B. R. Malonda-Boungou, L. Mouketo and C. Demangeat

295

Generation of Matrices with Specified Eigenvalues H. V. Mweene

304

The Potential Group Method for Sturm-Liouville Equations K. Sodoga, M. N. Hounkonnou and G. Debiais

315

Group II. Coherent States, Wavelets, Functional Analysis and Orthogonal Polynomials The Beta-Geometric Model Applied to Fecundability in a Sample of Married Women D. B. Adekanmbi and T. A. Bamiduro

325

XV

On the Existence and Uniqueness of Solutions to the Thermal Filtration Model F. B. Agusto and 0. M. Bamigbola

333

Basic Set of Polynomials: A General Overview A. Anjorin and M. N. Hounkonnou

338

Wavelets and Wavelet Frames on the 2-Sphere J.-P. Antoine

344

On Compact Elements of Banach Algebras U. N. Bassey

363

Application of the Adomian Decomposition Method to Solve the Duffing Equation and Comparison with the Perturbation Method G. Bissanga

372

Hermite Interpolation Polynomial of Several Real Variables J. Dzoumba, D. Moukoko and Y. Nkasa

378

Braid Index with Up to Ten Crossings O. A. Fadipe-Joseph

387

Hazard Rate Prediction in Life Time Data Analysis K. E. Gneyou

397

Laguerre-Freud Equations for the Recurrence Coefficients of Some Discrete Semi-Classical Orthogonal Polynomials of Class Two C. Hounga, M. N. Hounkonnou and A. Ronveaux

412

A Study of a Viscoelastic Contact Problem O. P. Layeni and A. P. Akinola

420

List of Participants

427

Plenary Lectures

3

R E P R E S E N T A T I O N THEORY IN CLASSICAL A N D Q U A N T U M PHYSICS J.-P. ANTOINE Institut

de Physique Theorique, Universite catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium E-mail: AntoineQfyma.ucl.ac.be

We review the basic notions of group theory, in particular Lie groups and Lie algebras, and of representations of the latter. Then we examine briefly their occurrence in classical physics for the description of invariance properties of physical systems and the concomitant conservation laws resulting from Noether's theorem. In the last section, finally, we give an overview of the applications of group representation theory in quantum physics, with special emphasis on the proper mathematical description of symmetry properties, both in quantum mechanics and in quantum field theory.

Organization of the Lectures 1. Motivation: why group theory? Why representations? 2. Mathematical tools I: Group theory 2.1. Basic notions: a quick reminder 2.2. Lie groups and Lie algebras 2.3. Simple and semisimple Lie algebras 2.4. Integration on a locally compact group 3. Mathematical tools II: Representations 3.1. Basic notions 3.2. Irreducibility of representations 3.3. Schur's lemma and its generalizations 3.4. Representations of Lie algebras 3.5. Representations of compact groups 3.6. Square integrable representations 4. Classical Physics 4.1. Conservation laws in classical mechanics 4.2. Hamiltonian mechanics: representations with respect to Poisson brackets

4

4.3. Classical field theory: Symmetries and Noether's theorem 5. Quantum Physics 5.1. Symmetries in Quantum Mechanics 5.2. Symmetries in Quantum Field Theory 5.3. Coherent states 5.4. Applications in quantum physics 1. Motivation Before starting this review, there are two basic questions we ought to answer: Why do we need group theory in physics? Why do we need representations of groups and algebras? As for the first question, it is, of course, closely linked to the notion of symmetry. Symmetry, or the lack of it, has always fascinated Man, from the ornaments on Mycenian jewelry, several centuries B.C., to the most contemporary developments in high energy physics (string theory) or condensed matter physics (quasicrystals). And group theory is the mathematical language for describing symmetries, which really developed in the 19th century. 1 Then, a quick look at the various instances where it appears reveals that the dominant themes are classification and simplification. Let us give a few examples. . Crystallography: this was the first occurrence of group theory in physics. Originally at least, the goal was to list all allowed types of crystals (there are 230 of them in 3 dimensions, so that a classification principle was indeed indispensable). The complete classification of crystal into 32 crystal classes was achieved by listing the 32 point groups (finite groups). Combining the latter with lattice translations then leads to the 230 space groups. Their complete listing is due to E. V. Fedorov (1885) and A. Schonflies (1891). . Spectroscopy: already in the 1930s, the use of the rotation group SO(3), under the impetus of Wigner, led to considerable progress in unraveling atomic spectra. This was not easily accepted, however, as is manifested in the term Gruppenpest qualifying the grouptheoretical approach! . Describing symmetry or invariance properties of physical systems: here dynamics enters, but geometry remains an essential tool (remember, F. Klein defines geometry as the invariance under a certain group of transformations; for instance, Euclidean geometry is the study of properties invariant under Euclidean transformations, namely, rotations and translations).

5

. Tensorial properties of physical quantities, a fundamental tool in relativity, are defined in terms of a certain transformation group (rotations, Lorentz). . Elementary particles: since the 1960s, the proliferation of new particles (which by the way are far from being elementary!) also demanded a systematization, that group theory indeed provided. . Classifying interactions between particles in terms of their invariance groups: this is now a purely dynamical aspect. . Understanding conservation laws: by Noether's theorem, invariance of a Lagrangian implies existence of conservation laws (more about this in Sec. 4.3). As for the second question, why do we need representations of groups and algebras, the answers differ in the classical and in the quantum cases. . In classical (Hamiltonian) physics, states are functions on phase space and the dynamics is described by the Poisson algebra, defining canonical transformations. Thus generators of symmetry groups obey a Poisson algebra analogous to the Lie algebra g of the symmetry group G, i.e., a representation of g. . In quantum physics, states are rays in a Hilbert space S) (from the superposition principle) and observables are linear operators acting on Sj. Thus, symmetry operations must be realized by operators acting on S) and possessing the same properties as the corresponding group G, i.e., a linear representation of G; the corresponding infinitesimal generators are described by a representation of the Lie algebra g.

2. Mathematical Tools I: Group Theory Since it is important to have precise notions in mind before tackling applications, we shall devote the next two (long) sections to the essential mathematical tools we will need, basic group theory and representations, respectively. Most proofs are omitted, so that this text is not really a crash course in group theory. No prior knowledge is assumed, although the pace might appear fast to the complete debutant. In that case, recourse to a standard textbook may be advised, such as those of Bacry,2 Barut-Raczka, 3 or Gilmore. 4

6

2.1.

Basic notions:

a quick

reminder

Definition 2.1. A group is a set G equipped with an internal composition law, called product, (g,g') >-> gg', such that (i) the product is associative: 31(5253) = (5152)33, for all 31,52,53 £ G; (ii) there exists a (unique) neutral element e £ G such that eg = ge = g, for all g € G; (iii) every element g S G has an inverse g~l 6 G such that gg~l = 9~l9 = e. The group is abelian or commutative if 3132 = 3231, for all 31,32 £ G. Equivalently, all commutators are trivial, that is, gig-ig^g^1 = e> f° r a u 3i,32 S G. Examples: . Z 2 = {1,-1}. Vi = {I 2 , R, - I 2 , - i ? } , where i? = d i a g ( - l , 1). . Z 4 = {1, i, - 1 , — i}, cyclic group of order 4; this group is different from V4 (consider the square of each element)!

•s°(2)={(:™rc»;)'°^ gH is a (canonical) homomorphism of G onto G/H and Ker a = H. (ii) Conversely, let a : G —> G' be a homomorphism. Then G/Ker a ~ Im a. In other words, H is an invariant subgroup of G if and only if it is the kernel of some surjective homomorphism a : G —> G' such that G' ~ G/H. Example: Take again the group 0(2), with its subgroups SO(2) and {l,R}: . SO(2) G, Lg(h) = gh. The corresponding differential is a map between tangent bundles: (dLg)h : ThG -»• TghG. A vector field X on G is left-invariant G, (dLg)hX(h) = X(gh). The central result is the following.

if dLgX

= X, i.e., at h £

Theorem 2.22. The vector space g of left-invariant vector fields on a Lie group G is isomorphic to the tangent space at the neutral element: g ~ TeG. Examples: . SO(3): the generators are J i , J2, J3, where Jk generates the rotations around the x^-axis; they obey the commutation relations [Ji, h] = ih, [J2,h] = iJ\, [Jz,J\] = 1J2,

&

[Jk, Ji] = iektmJm,

where eum is the totally antisymmetric unit tensor. . SU(2): the generators £ 1 , 2 2 , ^ 3 obey the commutation relations

These are identical to those of SO(3)! In other words, the two Lie algebras so(3) and su(2) are isomorphic (but not the groups SO(3) and SU(2), as we will see below). Note that Sj = \&j, the Pauli matrices, with commutation relations [o~k, g : (X, Y) >-> [X, Y] satisfying the Jacobi identity: [aX + pY, Z] = a[X, Y] + /3[Y, Z], for allX, Y, Z 6 g, a,/3 € K, [X,Y] = -[Y,X],

forallX,y

eg,

[X, [Y, Z]} + [Y, [Z, X] + [Z, [X, Y]] = 0, for all

X,Y,Zeg.

Theorem 2.24. (Ado) Every real or complex finite dimensional Lie algebra is isomorphic to a matrix Lie algebra, where [X, Y] = XY — YX.

18 As in the case of groups, comparing Lie algebras requires a proper notion of homomorphism. The natural definition reads as follows. Definition 2.25. Let g,g' be two Lie algebras over K. A map tp : g —I g1 is a Lie algebra homomorphism if: (i) V is linear: Si simple and [fli.fl,-] = 0 if i ^ j .

iei

Example: so(4) = su(2) ©su(2) & SO(4) ~ [SU(2) x SU(2)]/Z 2 Thus, in order to classify the semisimple Lie algebras, it suffices to classify all the simple Lie algebras. This has been done by Cartan in his thesis (1894), another masterpiece of mathematics. We will now sketch this fundamental result. Cartan's method of classification consists in choosing a standard basis in the Lie algebra and translating its properties in graphical terms, which results in the so-called root diagram. Then the latter can be completely classified. To that effect, one considers first a complex Lie algebra g (i.e., with complex parameters) and one solves the eigenvalue equation aAA{X) = aX,

i.e.,

[A,X] =aX, a € C.

(9)

Then the results may be summarized as follows. Let g be a semisimple Lie algebra and A € g an element with a maximal number of distinct eigenvalues. Then: (1) 0 is the only degenerate eigenvalue; its multiplicity / is called the rank of g. (2) Choose I independent eigenvectors associated to 0, Hi, i = 1,2,...,/: adA(ffi) = [/!,#*] = 0. Since [.4,^4] = 0, A = Y^ici^i a n d one can choose A = Hi. The abelian subalgebra f) of g generated by {Hi, i = 1,2,...,/} is called a Cartan subalgebra.

23

(3) Let Ea be an eigenvector of Hi associated to the (nondegenerate) eigenvalue a i : [H\,Ea] = ot\Ea. Using the Jacobi identity, one gets [Hi,Ea] = atEa,

i=

l,2,...,I.

Therefore, all the Hi are diagonalized simultaneously. Denote by ai,a2,-..,oei the eigenvalues of the common eigenvector Ea. The vector a = {a;}i=i,...,z £ Rl is called a root vector and the set of all roots is called the root diagram of g. Thus the root diagram is a set of vectors in tf, where I is the rank of g. In the standard (Cartan) basis {Hi, i = 1, 2 , . . . , I; Ea} of 0 (complexified), the commutation relations take the form: [Hi,Hj]=0,

i,j =

1,2,...,I,

Na Ea

if

[Hi,Ea]=aiEa, [Ea,E_a}=aiHi, \E 1

°"

(10) a

3 is a

E ] = i P +P> ( + z) nonzero root, 0l \0, if {a + P) is not a root.

The set of roots has the following properties: . ka is a root if and only if k = ±1. . The set of all roots is invariant with respect to the reflection in the hyperplane perpendicular to the pair ±a. . All these reflections generate a finite group W, called the Weyl group. Example 1: so (3), with rank 1 and commutation relations [J3,J±] = ± ^ ± ,

[J+,J-] = Jz,

where J± = —=( Jx ± iJ2). V2

Root diagram:

J_ M

J3

J+

•

•

Fig. 1. The root diagram of SU(2).

24

Example 2: su(3) = A2, of rank 2 The root diagram consists of six nonzero roots, the tips of which draw a regular hexagon. The Weyl group is isomorphic to the 6-element permutation group 53, generated by reflections with respect to the 3 lines orthogonal to roots (dashed lines). 11

013 = ai + a-i

Fig. 2.

The root diagram of SU(3) (from Ref. 11).

The root diagram has the following further properties: (i) For any pair of roots a, j3: • 2(a/3) . . . . . the ratio —.—r- is a positive integer (aa) . (5 — - — — a is a root (reflection of /? with respect to the line (aa) perpendicular to ± a ) (ii) If 6ap is the angle between the roots a, ft, one has M) 2 mn , m, n e N + . cos2 0ap = (aa){p(i) 4 As a consequence, only a few values are allowed for the angles 8ap and the ratio of lengths, as follows: 6a0 : 30° 45° 60° 90° 120° 135° 150° 180° Ratio of lengths : -\/3 \/2 1^ arbitrary 1 v 2 V3 1^

25

Classification of simple Lie algebras. Dynkin diagrams One can define an order relation on root systems: . a = (ai,02, • • • ,Q/) is positive if the first nonzero component is positive (lexicographic order), . /?>aif/3-a>0. A root is said to be simple if it is positive and cannot be decomposed into the sum of two positive roots. Then: . If g has rank I, there exist / linearly independent simple roots (i.e., simple roots are a basis of M'), . If a,/3 are simple roots, the angle 6ap takes only the values 90°, 120°, 135°, 150°. These facts are encoded graphically in terms of the so-called Dynkin diagrams. The principles are the following: . Each simple root is represented by a small circle. . The number of links between two circles is 0,1,2 or 3 whenever the angle between the corresponding roots is 90°, 120°, 135° or 150°, respectively. Example: Three simple roots r i , r 2 , r 3 with angles (?i,r2) —120°, (r2,r3) = 135°,(r 1 ,r 3 ) = 90°: n r2 r3 O 0==0 . Closed loops are forbidden:

O-

-b

6

O

. Each circle can support at most three links, so that a connection like the following one is forbidden: O

Q

. Simple roots can have two different lengths only; one uses white disks for the short ones, black disks for the long ones.

26

The result of Cartan's analysis is that there exists four infinite series of simple complex Lie algebras, corresponding to classical groups, plus five exceptional algebras (with no associated classical group). The four infinite series are Ai (I ^ 1),-B; {I ^ 2),C; (I > 3) and A (I ^ 4), where I denotes the rank of the algebra and the restrictions on I guarantee that all algebras are different. Indeed, there are some isomorphisms for the lower ranks: Ai ~BX ~CUB2~C2,A3~D3,D2~A1(BA1.

(11)

The five exceptional algebras are denoted G2,-F4, £?6,-E7 and Eg. All the simple Lie algebras are listed in Table 2.3, together with their Dynkin diagrams. The next step is to list the real forms of the simple Lie algebras and the corresponding Lie groups. Let go be a real Lie algebra. Its complexification is the complex Lie algebra fl = 0Q consisting of all elements of the form X + iY, X,Y £ 00) the bracket being extended by linearity. Conversely, a real form of g is a real Lie algebra 0i such that g is isomorphic to the complexification of gi. Of course, a complex Lie algebra may have several nonisomorphic real forms. There is one fundamental restriction, however. Theorem 2.31. Every semisimple complex Lie algebra has a real form which is compact. Starting from this compact real form, one may now obtain a noncompact one, as follows. The tool is the notion of involutive automorphism of the compact Lie algebra g, that is, an automorphism a of g such that er2 = I. Such a map a has eigenvalues ± 1 and it splits g into eigensubspaces: g = 6®p, where the eigenspace t corresponds to the eigenvalue + 1 , i.e., it is the set of fixed points of a. The commutation relations of g are the following: [1,6] C 6,

(thus 6 is a subalgebra)

[e,p] = p, [p,p]Ct. Then the Lie algebra g* = t ©p*, where p* = ip, is another real form of g c , and it is noncompact (this construction is called Weyl's unitary trick). The commutation relations of g* read: [t, 6] C E,

[e,p*] = p*. [p*,p*]c-e.

(6 is still a subalgebra)

27

Thus classifying the real forms amounts to classify the involutive automorphisms, and again the result was obtained by Cartan. For instance, the complex Lie algebra B2 has three different real forms, namely, so(5), which is the compact one, so(4,1) and so(3,2), which are noncompact. All this extends to the corresponding Lie groups. In the example of B2, one gets SO(5), which is compact, SO(4,l), the de Sitter group, and SO(3,2), the Anti-de Sitter group, which are both noncompact. For each nonexceptional Lie algebra, we list in Table 2.3 the real forms of the corresponding classical Lie groups. For each of them, there is one compact form (SU(/+1), SO(2/+l), Sp(2Z), SO(2/)) and several noncompact forms (for I ^ 3). The isomorphisms of the low-rank Lie algebras (11) in turn entail local isomorphisms for the corresponding groups. For instance, SU(2) ~ SO(3) ~ Sp(2) or SU(4) ~ SO(6) locally (but not always globally). 2.4. Integration

on a locally compact

group

The main advantage of locally compact groups is that they allow a theory of integration, which will prove crucial in a number of situations. Let G be a locally compact group, in particular a Lie group. Then one defines: . A left invariant measure on G, that is, a measure /i£ on G which satisfies the following relation for any ^x-integrable function / : / f(9ogWL(g)

= [ f{g)dtxL{g),

JG

for all g0 G G,

(12)

JG

or, equivalently, dHL{galg)

=duL(g),

or IJ,i(gQ1E) =

HL{E),

for every Borel set E of G.

. A right invariant measure fiR on G: f f(99o)d/jR(g)

= f f(g)dm(g),

JG

(13)

JG

dmigQo1)

= d(J,R(g),

fi^Eg^1)

=

HR(E),

for every Borel set E of G.

Then the fundamental result of Haar is the following. Theorem 2.32. Up to normalization, every locally compact group possesses a unique left invariant measure \IL and a unique right invariant measure fiR. These two measures, called Haar measures, are equivalent.

Abstract Lie algebra

Ai (I > 1) Bi (I ^ 2)

Real form of corresponding Lie group

SXJ(l + l)orS\J(p,q),p

+q=l+ l

S0(2Z + 1) or SO(p, q), p + q = 21 + 1

Q (I ^ 3)

Sp(2l)orSp(p,q),p

+ q = 2l

Di (I > 4)

SO{2l) oiSO(p,q),p

+ q = 2l

FA G2 E6 E7

Table 2.3. Dynkin diagrams for the simple Lie

29

.RemarJcs: (1) Two measures are called equivalent if they have the same sets of measure zero. (2) If /i is a left invariant measure, then ju, image of fi by the homeomorphism g H-> g~l, is a right invariant measure and vice-versa:

= ,

1(g) = fig"1)-

Since \IL ~ HR, there exists a continuous function A : G ->• E + , called the modular function, such that: diAL{g) = A(g)dnR(g).

(14)

The modular function has the following properties: (1) A(g) > 0, (2) A(e) = 1,

for all g € G,

(3) A(gi)A{g2)

= A{gig2),

for all gi, g2 £ G.

In other words, A is a character of the group G. One has also: dm(g) = A{g~x) dfiL{g) = duUgg') = A{g')diiL{g).

dnL{g~l),

The group G is said to be unimodular if A(g) = 1, for all g G G, i.e., HL - HRExamples of unimodular groups: . . . . .

Abelian groups Compact groups Simple and semisimple groups Inhomogeneous groups: E(3), 7^(1,3), ... Discrete groups.

Examples of nonunimodular groups: . The affine group of E: {(&, a) : b e K, a £ E, a ^ 0}, i.e., K xi 1» . The ox + 6 subgroup of the affine group (a > 0), i.e., E x E+ . The similitude group SIM(n) = R" x (K+ x SO(n)). The Haar measures provide an easy criterion for compactness of the group: G is compact if and only if vol G < oo, where vol G = / d(iL(g) = / JG

JG

dfj,R(g).

30

Examples: . SO(n),SU(n) are compact; . SO(p,q),SXJ(p,q),Wl,P(1,3)

are noncompact.

Note: A similar discussion may be done for measures on homogeneous spaces X = G/H, but there is an essential difference. Indeed, a homogeneous space does not always admit an invariant measure, some (known) criteria have to be satisfied. However, it always admits a quasi-invariant measure, i.e., a measure equivalent (but not equal) to its translates. 3. Mathematical Tools II: Representations 3.1. Basic

notions

Definition 3.1. A linear representation of a group G in a vector space V is a homomorphism T : G —> GL(V), where GL(T^) denotes the set of all invertible linear operators on V: Tigito) = T(9l)T(g2), 1

l

It follows that Tig- ) = T(g)~ denned as the dimension of V.

for all gug2

£ G.

and T(e) = I. The dimension of T is

The most useful case is that where V is a Hilbert space ft and the operators T(g) are bounded. Then T is a homomorphism of G into GL(Sj), the set of bounded operators with bounded inverse. If G is a Lie group, T is called strongly continuous if \\(T(g) - I)0|| -> 0, when g ->• e, for all £ fj.

(15)

Definition 3.2. When Sj is a Hilbert space, the representation T is called unitary if T(^) is a unitary operator for every g £ G, i.e., T(g~x) = (T(g)f\T(g)h)

= (/I/!), for all*? £ G, for all,/,/* £ Sj.

Exampies: . G = SO(2), fj = L2(S1)

P W / 1 M = /(V> + ) = T(gg') = T(g'g) = T(g')T(g), that is, T(g) £ V. By Corollary 3.11, T(g) = X(g)I with \X(g)\ = 1 , i.e., dim T = 1. •

34

Example: G= S0(2) 9(f) =

cos if — sin f GS0(2), sirup cos y

g(27r)=g(0)=I.

Every UIR of S0(2) is of the form Tk(g(ip)) = eikv , k G Z. Indeed, one has T{g(if)) = A(p)I, since S0(2) is abelian. Thus T(g(ipi))T{g(tp2))

= T(g(fl)g(f2))

= T{g{fX +

whose only continuous solutions are X(f) = elkv, k G R. Finally, the condition T(g(2n)) = eik2v = I implies that k G Z. This is equivalent to the theory of Fourier series] Indeed, an arbitrary function / G L2^1) may be expanded in a Fourier series: 00

/(¥>)= £

c,e^,

(21)

k= — 00

so that 00

L2(51)=

0

J5fc,

d i m % = l-

(22)

k= — 00

The regular representation of SO(2) acts in L 2 (5 X ) and reads: 00

[ULW>)f](=

k=~oo

J2

ckTk(g(i>))eik\

k= — 00

i.e., (22) corresponds to the decomposition of UL into 1-dimensional UIRs: 00

UL= 0 Tt. & = —00

Lemma 3.13. (Schur's lemma: general) Let U\ be a UIR in Sji, U2 a unitary representation in $)2, and T : $)i —> ^2 a bounded operator that intertwines U\ and 1/2- Then either T = 0, orT is a multiple of an isometry, i.e., there exists a constant A > 0 such that

Wml^XMWl,, for ail 4 eft.

35

Proof: Prom the hypotheses, we have T'TU^g)

= T*U2{g)T = U^g)T'T,

for alls £ G.

Then by Schur's classical lemma 3.9, either T = 0 or T*T = XI.

D

Lemma 3.14. (Schur's lemma: extended) Let U be a UIR of G in the Hilbert space S) and U' a unitary representation in $j'. Let T : Sj —> $j' be a closed linear operator with dense, U-invariant domain D(T), that intertwines U and U' • Then either T = 0, or T is a multiple of an isometry (hence bounded). 3.4. Representations

of Lie

algebras

Throughout this section, we assume that T is a finite dimensional representation of G. Consider first the case of a one-parameter group, namely, SO(2): g((p) = e~l