Groups and Related Topics: Proceedings of the First Max Born Symposium PDF
Preview Groups and Related Topics: Proceedings of the First Max Born Symposium
Groups and Related Topics MATHEMATICAL PHYSICS STUDIES Editorial Board: H. ARAKI, Kyoto University, Japan M. CAHEN, Universite Libre de Bruxelles, Belgium A. CONNES, IR.E.S., France L. FADDEEV, Steklov Institute ofM athematics, Leningrad, U.S.S.R. B. NAGEL, K.T.H., Stockholm, Sweden R. RACZKA, Institut Badan Jadrowych, Warsaw, Poland A. SALAM, International Centre for Theoretical Physics, Trieste, Italy W. SCHMID, Harvard University, U.SA. J. SIMON, Universite de Dijon, France D. STERNHEIMER, College de France, France I. T.TODOROV, Institute ofN uclear Research, Sofia, Bulgaria J. WOLF, University of California, Berkeley, U.sA. VOLUME 13 Groups and Related Topics Proceedings of the First Max Rorn Symposium edited by R. Gielerak, J. Lukierski z. and Popowicz University ofWrodaw, Institute ofTheoretical Physics, Wrociaw, Poland SPRINGER SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Max Born Syeposlue (1st : 1991 : WOJnowlce Clstle) Groups Ind related toplcs : proceedlngs of the Flrst Max Born Syeposlue I edlted by R. Glelerak, J. Luklerskl, Ind Z. Popowlcz. p. ce. -- (Mltheeatlcal physlCS studles : v. 13) Includes blbllographlcal references and Index. ISBN 978-94-010-5244-3 ISBN 978-94-011-2801-8 (eBook) DOI 10.1007/978-94-011-2801-8 1. Mathee.tlcal physlcs--Congresses. 2. QUlntue groups -Congresses. 3. Geoeetry, Dlfferentlll--Congresses. 1. Glelerak, R. (Roean), 1951- II. LUklerskl, Jerzy. III. PopOW1CZ, Z. IV. Tltle. V. Series. QC19.2.M39 1991 530.1'5--dc20 92-25133 ISBN 978-94-010-5244-3 AH Rights Reserved © 1992 Springer Science+Business Media Dordrecht Origina1ly published by Kluwer Academic Publishers in 1992 No part of the material protected by this copyright notice may be reproduced or uti1ized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Foreword In the seventies and eighties, scientific collaboration between the Theory Section of the Physics Department of Leipzig University and the Institute of Theoretical Physics of the University of Wroolaw was established. This manifested itself, among other things, in the organization of regular, twice-yearly seminars located alternatively in Wrodaw and Leipzig. These Seminars in Theoretical Physics took place 27 times, the last during November 1990. In order to continue the traditions of German-Polish contacts in theoretical physics, we decided to start a new series of Seminars in Theoretical Physics and name them after the outstanding German theoretical physicist Max Born who was born in 1883 in Wrodaw. We hope that these seminars will continue to contribute to better scientific contacts and understanding between German and Polish theoretical physicists. The First Max Born Symposium was held in Wojnowice Castle, 20 km west of Wrodaw, 27 - 29 September 1991. Wojnowice Castle was built in the 16th century by the noble Boner family, in the Renaissance style, and has been recently adapted as a small conference center. The preferred subjects at the Symposium were Quantum Groups and Integrable Models. The Symposium was organized by Doctors R. Gielerak and Z. Popowicz under the scientific supervision of the undersigned. The organizers would like to thank Wroolaw University, and especially the Rector, Prof. J6zef Zi6lkowski, for fmancial support and for accepting the idea of a new Seminar series. We would also like to express our thanks to the authors, who mostly submitted their paper in camera-ready form, and to Kluwer Academic Publishers for their cooperation in publishing the Proceedings of the First Max Born Symposium. Jerzy Lukierski Table of Contents SECI10N I QUANlUM GROUPS 1 1. M. Chaichian and P. Pre~najder Sugawara Construction and the Q-Defonnation of Virasoro Algebra 3 2. B. Drabant, M. Schlieker, W. Weich and B. Zumino Complex Quantum Groups and Their Dual Hopf Algebras 13 3. S.M. Khoroshkin and V.N. Tolstoy Extremal Projector and Universal R-matrix for Quantized Contragredient Lie (super)algebras 23 4. J. Lukierski and A. Nowicki Quantum Defonnations of D = 4 Poincare Algebra 33 5. R. Matthes "Quantum Group" Structure and "Covariant" Differential Calculus on Symmetric Algebras Corresponding to Commutation Factors on Zn 45 6. A. Schirnnacher Remarks on the Use of R-matrices 55 7. E. Sorace Construction of Some Hopf Algebras 67 8. S. Zakrzewski Realifications of Complex Quantum Groups 83 SECI10N II NON COMMUTATIVE DIFFERENTIAL GEOMETRY 101 1. A. Borowiec, W. Marcinek and Z. Oziewicz On Multigraded Differential Calculus 103 2. R. Coquereaux Yang Mills Fields and Symmetry Breaking: From Lie Super-Algebras to Non Commutative Geometry 115 3. J. Rembielifiski Differential and Integral Calculus on the Quantum C-Plane 129 viii SECTION ill INTEGRABLE SYSTEMS 141 1. S. Albeverio and J. Schafer Rigorous Approach to Abelian Chern-Simons Theory 143 2. R. Flume The Confonnal Block Structure of Perturbation Theory in Two Dimensions 153 3. B. Fuchssteiner An Alternative Dynamical Description of Quantum Systems 165 4. L. Hlavaty On the Solutions of the Yang-Baxter Equations 179 5. M. Karowski, W. Muller and R. Schrader State Sum Invariants of Compact 3-Manifolds with Boundary and 6j-symbols 189 SECTION IV MISCELLANEOUS 197 1. K. Fredenhagen Product of States 199 2. K.-E. Hellwig Quantum Measurements and Infonnation Theory 211 3. J . .t.opuszaflski A Comment on a 3-Dimensional Euclidean Supersymmetry 223 4. B. Schroer Chiral Nets and Modular Methods 233 5. M. Salmhofer and E. Seiler Chiral Symmetry Breaking - Rigorous Results 247 6. V.A. Soroka, D.P. Sorokin, V.I. Tkach and D.V. Volkov On a Twistor Shift in Particle and String Dynamics 259 7. A. Uhlmann The Metric of Bures and the Geometric Phase 267 SECI10N I QUANTUM GROUPS SUGAWARA CONSTRUCTION AND THE Q-DEFORMATION OF VIRASORO ALGEBRA M. Chaichian Department of High Energy, Physics, University of Helsinki Siltavuorenpenger 20 C, SF-00170 Helsinki, Finland. and P. Presnajder Department of Theoretical Physics, Comenius University Mlynska dolina F2, CS-84215 Bratislava, Czechoslovakia December 20, 1991 Abstract The q-deformed Virasoro algebra is obtained using the annihilation and cre ation operators of the q-deformed infinite Heisenberg algebra, which has the Hopf structure. The generators of the q-deformed Virasoro algebra are expressed as a Sugawara construction in terms of normal ordered binomials in these annihilation and.creation operators, and become double-indexed as the reminder of a degeneracy removal. The obtained q-deformed Virasoro algebra with central extension reduces to the standard one in the non-deformed limit. 1. Introduction For many years great attention has been paid to the Virasoro algebra, the underlying symmetry of the string theory [11. The Virasoro algebra is the central e~tension of the Lie algebra of conformal transformations of a complex plane. The generation of the latter are given in terms of annihilation and creation operators satisfying the communication relation (1) in the following way 3 R. Gielerak tit aI. (tIds.), Groups and RtI/Qted Topics, 3-12. C 1992 K1IIWt!r AcaMmlc I'ublishers. 4 M. CHAICHIAN AND P. PRESNAJDER Lm = (a+)m+la (2) They sat.isfy the Lie algebra relations IL", Lm] = (m - n)Lm+" (3) During the last few years a growing interest has appeared in the study of quan tum groups and algebras 12,3]. In particular, a great deal of attention has been paid to q-deformations of centreless Virasoro algebra (4-7] and to its central .extension [8- 10]. Several of t,bese investigations (8,9] are based on the use of q-deformed harmonic oscillators satisfying the relations Ill] Ia ,a +] ,= q- N , IN, aJ = -a, [N.a+J = a+, (4) = where lA, BI, AB - qAB, and q is the deCormation parameter. The q-deformed Virasoro generators are expressed by the following generalization of eq. (3): Lm = q-N(a+)m+~a, (5) and they satisfy the following, algebraic relations (6) where IA,B)Cf,..,,) = qlAB - q2BA, and [x) = (q'" - q-"')/(q - q-l). Furthermore, the problem of iheir central extensions has been investigated by using the q-deformed gener alizations of the Jacobi identity (8-10). However, until now these constructions have not been shown to have a Hopf algebra structure [12-14). In this letter we shall follow an approach based on the Sugawara construction in the string theory, leading to a Virasoro algebra with central extension (see for example, 11)). The generators Lm are represented by an infinite set of bosonic operators a", with n-integer, satisfying the infinite Heisenberg algebra H(oo), defined by the relations = (am, an) mIl c5m+",o, = [H,am) 0, (7) with H as a central element. The formula Cor Lm reads as follows 1 +00 Lm = 2 E : aA:a,,: c5"+",m, (8) 'tn=-oo where the colons denote the normal ordering. Owing to qua.ntization, the operators Lm satisfy the (centrally-extended) Virasoro relations (see ref. (1)): [L", LmJ = (m - n)H L,,+m + 112112(n3 - n)c5n+m,o. (9)
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