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American Mathematical Society http://dx.doi.org/10.1090/conm/490

Symmetry in Mathematics and Physics V. S. Varadarajan CONTEMPORARY MATHEMATICS

490

Symmetry in Mathematics and Physics

Conference in Honor of V. S. Varadarajan's Birthday January 18–20, 2008 University of California, Los Angeles, California

Donald Babbitt Vyjayanthi Chari Rita Fioresi Editors

American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss

2000 Mathematics Subject Classification. Primary 16Sxx, 17Bxx, 20Gxx, 32Cxx, 51Bxx, 60Fxx, 81Pxx, 81Txx, 81–02. Photo courtesy of Rahul N. Fernandez

Library of Congress Cataloging-in-Publication Data Symmetry in mathematics and physics : conference in honor of V.S. Varadarajan’s 70th birth- day : January 18–20, 2008, University of California, Los Angeles, California / Donald Babbitt, Vyjayanthi Chari, Rita Fioresi, editors. p. cm. — (Contemporary mathematics ; v. 490) Includes bibliographical references. ISBN 978-0-8218-4731-2 (alk. paper) 1. Symmetry (Mathematics)—Congresses. 2. Symmetry (Physics)—Congresses. I. Varadara- jan, V. S. II. Babbitt, Donald G. III. Chari, Vyjayanthi. IV. Fioresi, Rita, 1966– QA174.7.S96S96 2009 510—dc22 2009004434

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents

Preface vii

Probability

The Role of Weak Convergence in Probability Theory S. R. S. Varadhan 3

Foundations of Quantum Mechanics

Statistical models on a simplex Enrico G. Beltrametti 13

Quantum Error Correcting Codes

Quantum Error Correcting Codes and Weyl Commutation Relations K. R. Parthasarathy 29

Finite and Infinite Dimensional Lie Groups and Algebras and their Representations

Ideals in parabolic subalgebras of simple Lie algebras Vyjayanthi Chari, R. J. Dolbin, and T. Ridenour 47 Quantizations of modules of differential operators Charles H. Conley 61 Geometric realization of PRV components and the Littlewood-Richardson cone Ivan Dimitrov and Mike Roth 83 Ka¸c algebras, quantum doubles and planar algebras S. Jijo and V. S. Sunder 97 On a theorem of Ranee Brylinski Bertram Kostant and Nolan Wallach 105 Conjugacy of maximal toral subalgebras of direct limits of loop algebras Hadi Salmasian 133

v vi CONTENTS

A multiplicity formula for tensor products of SL2 modules and an explicit Sp2n to Sp2n−2 × Sp2 branching formula Nolan Wallach and Oded Yacobi 151

Super Lie Groups and Supergeometry

Super G-spaces Luigi Balduzzi, Claudio Carmeli, and Gianni Cassinelli 159 Quotients in Supergeometry Luigi Balduzzi, Claudio Carmeli, and Rita Fioresi 177 Special geometry of D =4, 5 supersymmetry Mar´ıa Antonia Lledo´ and JoseAntonioL´ opez´ 189

Supersymmetry

Symmetric Spaces in Supergravity Sergio Ferrara and Alessio Marrani 203 Symmetry and Superstring Phenomenology Mary K. Gaillard 229 Supersymmetry: A personal view B. Zumino 241 Preface

The articles in this collection mainly grew out of the talks given at a Conference held at UCLA in January 2008, which honored V. S. Varadarajan on his 70th birthday. The main theme of the Conference was symmetry in mathematics and physics. More precisely, the talks at the conference were dedicated to the interplay between geometry, group theory, and fundamental physics. In addition to the speakers there were a number of doctoral and post doctoral fellows including several students of Varadarajan who had worked under him on these topics throughout his career. Varadarajan’s work over the past 50 years represents a broad spectrum of math- ematics but its main emphasis has been on symmetry in mathematics and math- ematical physics, broadly interpreted. Some of his significant achievements are: development of the infinitesimal method in the theory of infinite dimensional repre- sentations of real semi simple Lie algebras; Fourier transform theory in the complex domain on Riemannian symmetric spaces; theory of local moduli for ordinary mero- morphic linear differential equations at an irregular singularity; theory of unitary representations of super Lie groups and the classification of super particles; and more recently, studies on the physics associated to non-archimedean space-time. The relevance of the representation theory of Lie groups and Lie algebras to the physics of elementary particles and fields has been known for a very long time, going back to the famous 1939 paper of E. P. Wigner on the representations of the Poincare group. Since then this link between representation theory and physics has deepened enormously, and includes quantum field theory and conformal field theory. Then something marvelous happened. In the 1970’s the physicists created a new extension of geometry where the underlying manifolds acquired anti-commuting co- ordinates in addition to the usual commuting ones, reflecting the Fermionic struc- ture of matter. This introduced supergeometry and super Lie groups into the mix and made the connection between geometry and physics much richer. Together with his students, he has made many important contributions to this area. It thus seemed appropriate to have a conference at UCLA devoted to some of these themes. The Conference turned out to be very exciting and stimulating because of the contributions of the participants who came from the United States and abroad. Most of the articles in this volume are thus naturally concerned with the above-mentioned themes: representations of finite and infinite dimensional Lie groups and Lie algebras, super Lie groups and supergeometry, which are at the interface of mathematics and fundamental particle physics, and supersymmetry. The discussions on supergeometry and supersymmetry are especially relevant at this time since some of the experiments at the Large Hadron Collider at CERN may help determine whether supersymmetry is a feature of the world of elementary

vii viii PREFACE particles. A few of the articles are in probability and the foundations of quantum mechanics, areas in which Varadarajan worked early in his career. The scientific organizing committee consisted of S. R. S. Varadhan (Chair), E. Beltrametti, T. J. Enright, S. Ferrara, K. R. Parthasarathy, and N. R. Wallach. The conference was funded by a generous grant from NSF, from a grant from the Goldman-Sachs corporation, and a matching contribution from a private donor. We are very grateful to these sources for their generosity, which made the conference go on in a very smooth manner. The editors wish to express their thanks to many people whose efforts made this conference a success, including all the participants. They wish to thank IPAM for permission to use their facilities for the conference; Professors Christoph Thiele and Robert Steinberg for inaugurating the conference; the department of mathematics at UCLA for providing help at all stages of the conference and for organizing the web site, especially Babette Dalton, Robert Amodeo, and Natasja Saint-Satyr; and Christine Thivierge of the AMS for her editorial assistance in producing this collection. The editors are also grateful to T. Kibble and Imperial College Press for permission to reprint B. Zumino’s paper “Supersymmetry: A Personal View”, which appears by their courtesy in these pages.

Donald Babbitt, UCLA Vyjayanthi Chari, UC Riverside Rita Fioresi, Universit`a di Bologna, Italy Titles in This Series

490 Donald Babbitt, Vyjayanthi Chari, and Rita Fioresi, Editors, Symmetry in mathematics and physics, 2009 489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic Forms and L-functions II. Local aspects, 2009 488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms and L-functions I. Global aspects, 2009 487 Gilles Lachaud, Christophe Ritzenthaler, and Michael A. Tsfasman, Editors, Arithmetic, geometry, cryptography and coding theory, 2009 486 Fr´ed´eric Mynard and Elliott Pearl, Editors, Beyond topology, 2009 485 Idris Assani, Editor, Ergodic theory, 2009 484 Motoko Kotani, Hisashi Naito, and Tatsuya Tate, Editors, Spectral analysis in geometry and number theory, 2009 483 Vyacheslav Futorny, Victor Kac, Iryna Kashuba, and Efim Zelmanov, Editors, Algebras, representations and applications, 2009 482 Kazem Mahdavi and Deborah Koslover, Editors, Advances in quantum computation, 2009 481 Aydın Aytuna, Reinhold Meise, Tosun Terzio˘glu, and Dietmar Vogt, Editors, Functional analysis and complex analysis, 2009 480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and Pramod Kanwar, Editors, Rings, modules and representations, 2008 479 Timothy Y. Chow and Daniel C. Isaksen, Editors, Communicating mathematics, 2008 478 Zongzhu Lin and Jianpan Wang, Editors, Representation theory, 2008 477 Ignacio Luengo, Editor, Recent Trends in Cryptography, 2008 476 Carlos Villegas-Blas, Editor, Fourth summer school in analysis and mathematical physics: Topics in spectral theory and quantum mechanics, 2008 475 Jean-Paul Brasselet, Jos´e Luis Cisneros-Molina, David Massey, Jos´eSeade, and Bernard Teissier, Editors, Singularities II: Geometric and topological aspects, 2008 474 Jean-Paul Brasselet, Jos´e Luis Cisneros-Molina, David Massey, Jos´eSeade, and Bernard Teissier, Editors, Singularities I: Algebraic and analytic aspects, 2008 473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependent Gross-Pitaevskii equations, 2008 472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, Representation Theory of Real Reductive Lie Groups, 2008 471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonal polynomials, 2008 470 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, 2008 469 Keith Burns, Dmitry Dolgopyat, and Yakov Pesin, Editors, Geometric and probabilistic structures in dynamics, 2008 468 Bruce Gilligan and Guy J. Roos, Editors, Symmetries in complex analysis, 2008 467 Alfred G. No¨el, Donald R. King, Gaston M. N’Gu´er´ekata, and Edray H. Goins, Editors, Council for African American researchers in the mathematical sciences: Volume V, 2008 466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems and applications in fluid dynamics, 2008 465 Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi, Editors, Curves and Abelian varieties, 2008 TITLES IN THIS SERIES

464 Gestur Olafsson,´ Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms, geometry, and wavelets, 2008 463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational arithmetic geometry, 2008 462 Giuseppe Dito, Hugo Garc´ıa-Compe´an, Ernesto Lupercio, and Francisco J. Turrubiates, Editors, Non-commutative geometry in mathematics and physics, 2008 461 Gary L. Mullen, Daniel Panario, and Igor Shparlinski, Editors, Finite fields and applications, 2008 460 Megumi Harada, Yael Karshon, Mikiya Masuda, and Taras Panov, Editors, Toric topology, 2008 459 Marcelo J. Saia and Jos´e Seade, Editors, Real and complex singularities, 2008 458 Jinho Baik, Thomas Kriecherbauer, Luen-Chau Li, Kenneth D. T-R McLaughlin, and Carlos Tomei, Editors, Integrable systems and random matrices, 2008 457 Tewodros Amdeberhan and Victor H. Moll, Editors, Tapas in experimental mathematics, 2008 456 S. K. Jain and S. Parvathi, Editors, Noncommutative rings, group rings, diagram algebras and their applications, 2008 455 Mark Agranovsky, Daoud Bshouty, Lavi Karp, Simeon Reich, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems III, 2008 454 Rita A. Hibschweiler and Thomas H. MacGregor, Editors, Banach spaces of analytic functions, 2008 453 Jacob E. Goodman, J´anos Pach, and Richard Pollack, Editors, Surveys on Discrete and Computational Geometry–Twenty Years Later, 2008 452 Matthias Beck, Christian Haase, Bruce Reznick, Mich`ele Vergne, Volkmar Welker, and Ruriko Yoshida, Editors, Integer points in polyhedra, 2008 451 David R. Larson, Peter Massopust, Zuhair Nashed, Minh Chuong Nguyen, Manos Papadakis, and Ahmed Zayed, Editors, Frames and operator theory in analysis and signal processing, 2008 450 Giuseppe Dito, Jiang-Hua Lu, Yoshiaki Maeda, and Alan Weinstein, Editors, Poisson geometry in mathematics and physics, 2008 449 Robert S. Doran, Calvin C. Moore, and Robert J. Zimmer, Editors, Group representations, ergodic theory, and mathematical physics: A tribute to George W. Mackey, 2007 448 Alberto Corso, Juan Migliore, and Claudia Polini, Editors, Algebra, geometry and their interactions, 2007 447 Fran¸cois Germinet and Peter Hislop, Editors, Adventures in mathematical physics, 2007 446 Henri Berestycki, Michiel Bertsch, Felix E. Browder, Louis Nirenberg, Lambertus A. Peletier, and Laurent V´eron, Editors, Perspectives in Nonlinear Partial Differential Equations, 2007 445 Laura De Carli and Mario Milman, Editors, Interpolation Theory and Applications, 2007 444 Joseph Rosenblatt, Alexander Stokolos, and Ahmed I. Zayed, Editors, Topics in harmonic analysis and ergodic theory, 2007

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. The articles in this volume mainly grew out of talks given at a Conference held at UCLA in January 2008, which honored V. S. Varadarajan on his 70th birthday. The main theme of the Conference was symmetry in mathematics and physics, areas of mathematics and mathematical physics in which Varadarajan has made significant contributions during the past 50 years. Very early in his career he also worked and made significant contributions in the areas of probability and the foundations of quantum mechanics. Topics covered by the articles in this volume are probability, quantum mechanics, symmetry (broadly interpreted in mathematics and physics), finite and infinite dimen- sional Lie groups and Lie algebras and their representations, super Lie groups and supergeometry (relatively new but active and important fields at the interface between mathematics and physics), and supersymmetry. The latter topic takes on a special impor- tance since one of the first experiments at the Large Hadron Collider at CERN will be a test of whether supersymmetry exists in the world of elementary particles. A reprint of an exposition of supersymmetry by one of its founders, B. Zumino, appears in this volume.

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