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Mathematical Monographs Translations of MATHEMATICAL ONOGRAPHS M Volume 238 +Õ>ÌÕÊ Õ`i`Ê -ÞiÌÀVÊ >Ã i`Ê°Ê6>Ã> American Mathematical Society Quantum Bounded Symmetric Domains 10.1090/mmono/238 Translations of MATHEMATICAL ONOGRAPHS M Volume 238 Quantum Bounded Symmetric Domains Leonid L. Vaksman Translated by Olga Bershtein and Sergey D. Sinel'shchikov M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O C C I I American Mathematical Society R E E T ΑΓΕΩΜΕ Y M A Providence, Rhode Island F O 8 U 88 NDED 1 EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair) ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) Leonid L. Vaksman OGRANIQENNYE KVANTOVYE SIMMETRIQESKIE PROSTRANSTVA Translated from a Russian manuscript. The present translation was created under license for the American Mathematical Society and is published by permission. Translated from the Russian by Olga Bershtein and Sergey D. Sinelshchikov. 2000 Mathematics Subject Classification. Primary 17B37, 20G42, 81R50; Secondary 22E47, 33D45, 43A85, 46L52. For additional information and updates on this book, visit www.ams.org/bookpages/mmono-238 Library of Congress Cataloging-in-Publication Data Vaksman, L. L. (Leonid L’vovych), 1951–2007. [Kvantovye ogranichennye simmetricheskie oblasti. English] Quantum bounded symmetric domains / Leonid L. Vaksman ; translated by Olga Bershtein and Sergey Sinelshchikov. p. cm. — (Translations of mathematical monographs ; v. 238) Includes bibliographical references and index. ISBN 978-0-8218-4909-5 (alk. paper) 1. Symmetric domains. 2. Representations of quantum groups. 3. Quantum groups. 4. Non- commutative algebras. I. Title. QC20.7.F84V3513 2010 512.55—dc22 2010009598 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. 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 151413121110 Contents Foreword vii Preface ix Chapter 1. Quantum Disc 1 1.1. A caution to pedants 1 1.2. Topology 2 1.3. Symmetry 15 1.4. An invariant integral 37 1.5. Differential calculi 46 1.6. Integral representations 69 1.7. On kernels of intertwining integral operators 97 Chapter 2. Basic Quantum Theory of Bounded Symmetric Domains 107 2.1. Summary on quantum universal enveloping algebras 107 2.2. Summary on algebras of functions on compact quantum groups 125 2.3. Quantum vector spaces and Harish-Chandra modules 137 2.4. Spaces of functions in quantum bounded symmetric domains 165 2.5. The canonical embedding 182 2.6. Covariant differential calculi and invariant differential operators 209 Chapter 3. Conclusion 233 3.1. Boundaries and spherical principal series 233 3.2. The analytic continuation of holomorphic discrete series and the Penrose transform 235 Bibliography 239 Index 253 v Foreword This book presents basics of the theory of quantum bounded symmetric do- mains, which were developed by L. Vaksman and his team. Our main goal in this foreword is to tell a little bit about the author, about the history of this book, and about its structure. Leonid Vaksman (1951–2007) was a talented mathematician who loved teaching mathematics and explaining mathematical ideas to his colleagues. Starting from his undergraduate years, L. Vaksman was interested in many diverse mathematical fields, from mathematical physics to category theory. This is crucial in understand- ing why L. Vaksman was eager to cooperate fruitfully with very different kinds of students and colleagues, being able to suggest the right topics in each case. Since the late 1980s L. Vaksman ran a seminar, first in Rostov (Russia), then in Kharkov (Ukraine). The range of topics discussed at the seminar was very wide and included such areas as the theory of quantum groups and their representa- tions, the theory of ∗-algebras, noncommutative complex and harmonic analysis, special functions, etc. All related methods and tools were extensively explained and used. Among the participants of the seminar were Ya. Soibelman, L. Korogodski, S. Sinel’shchikov, D. Shklyarov, O. Bershtein, Ye. Kolisnyk. Later L. Vaksman started to write down the problems, results, and ideas dis- cussed at the seminar. These notes resulted in a fundamental survey on the theory of quantum bounded symmetric domains, which also included both complete results discussed earlier and a number of unsolved problems, together with related ideas and hints. Currently, this survey is available only in Russian. Simultaneously, L. Vaksman prepared a smaller version of that survey and supplied it with a prelim- inary chapter explaining his approach to the main topics of the book. After his death this text was prepared for publication by O. Bershtein, Ye. Kolisnyk, and S. Sinel’shchikov. This is the book you are looking at now. The core of this book is the exposition of basic results on quantum bounded symmetric domains. Some of these results have already been published as articles and/or as preprints, some others are presented here for the first time. Regretably, it was impossible to include all the nice results in this shorter version. On the other hand, the large introductory Chapter 1 is devoted to the simplest quantum bounded symmetric domain, namely, the quantum disc. The idea of this chapter is to explain the problems, results, and methods of the book in the simplest case (which is already quite nontrivial). We are grateful to H. P. Jakobsen for helpful discussions on some parts of the book and his warm hospitality during our visit to Copenhagen in September 2008, vii viii FOREWORD where the translation work started. We also acknowledge the dedicated work of Ye. Kolisnyk in preparing this book for publication. O. Bershtein, S. Sinel’shchikov Kharkov, 2009 Preface This book presents a study of quantum analogs of bounded symmetric domains. The latter domains are steady subjects of attraction for specialists in geometry, algebra, and analysis, basically as sources of exactly solvable problems of complex analysis, noncommutative harmonic analysis, and classical mathematical physics. The simplest bounded symmetric domain is the unit disc D = {z ∈ C||z| < 1}. Its quantum analog was introduced by S. Klimek and A. Lesniewski [173]. We find it reasonable to start with Chapter 1 on the quantum disc. Our motive here is to avoid distracting the reader by algebraic details, but instead to produce an outline of the problems of noncommutative complex and harmonic analysis in which we are interested. This chapter can be used as background material to a seminar for university students in mathematics. The problems discussed in Chapter 1 admit a reformulation (many of them, even a solution) in a much more general context, namely within the framework of quantum theory of bounded symmetric domains [289, 290], whose basics are expounded in Chapter 2. This demonstrates an interplay between the theory of quantum groups and noncommutative complex analysis. A more detailed view of Chapters 1 and 2 is given in the table of contents. The author’s intention for subsequent chapters was to present results of O. Bershtein, Ye. Kolisnyk, D. Proskurin, D. Shklyarov, S. Sinel’shchikov, A. Stolin, L. Turowska, L. Vaksman, and G. Zhang, [33, 36, 34, 308, 258, 284, 320, 279, 318, 281, 287], together with unpublished results on quantum bounded symmetric domains (some of those are already present in xxx.lanl.gov, see, e.g., [317], or will appear therein in the nearest future). Regretably, this plan is not accessible due to some nonmathematical reasons. Instead, we are going to form a Web page dedicated to quantum bounded symmetric domains, which will contain a draft of the conjectured full version of the book. It is expected to be twice as large as the present volume; it will, in particular, contain a discussion on unsolved problems. I am deeply grateful to my students O. Bershtein, Ye. Kolisnyk, L. Korogodski, D. Shklyarov, and coauthors Ya. Soibelman and A. Stolin. Also, my special thanks to H. Jakobsen, A. Klimyk, E. Koelink, S. Kolb, Yu. Samoilenko, K. Schm¨udgen, L. Turowska, and G. Zhang for numerous helpful discussions on the results of this book. A special role in my life was played by Vladimir Drinfeld. In the mid-1980s he taught me the basics of the theory of quantum groups and helped me to return to mathematics after an involuntary break I had to take from it for many years. Here is some historical background. In the late 1970s, a study of exactly solvable problems of statistical mechanics and quantum field theory led L. Faddeev and his team to the creation of the quantum method of inverse scattering problems [293, ix xPREFACE 294, 305]. They introduced the quantum Yang-Baxter equation and associated its solutions to series of exactly solvable problems of mathematical physics. Their work [206] presents the solutions of this equation which were known before 1980; the authors mention that “its deep relations to the mathematical fields like group theory and algebraic geometry are coming into the picture”.
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