CONTEMPORARY MATHEMATICS 490 -ÞiÌÀÞÊÊ >Ì i>ÌVÃÊ>`Ê* ÞÃVà viÀiViÊÊÀÊvÊ6°Ê-°Ê6>À>`>À>>¿ÃÊ ÀÌ `>Þ >Õ>ÀÞÊ£nqÓä]ÊÓään 1ÛiÀÃÌÞÊvÊ >vÀ>]ÊÃÊ}iiÃ]Ê >vÀ> >`Ê >LLÌÌ 6Þ>Þ>Ì Ê >À ,Ì>ÊÀià `ÌÀà 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,