Quantum Groups, Quantum Categories and Quantum Field Theory
This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects a
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
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Jtirg Frohlich
Quantum Groups, Quantum Categories and Quantum Field Theory
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Jurg Frohlich Theoretische Physik ETH - Honggerberg CH-8093 ZUrich, Switzerland Thomas Kerler Department of Mathematics Harvard University Cambridge, MA 02138, USA
Mathematics Subject Classification (1991): 00-02, 13, 18D I 0, 18D99, 15A36, 16D60, 16W30, 16W20, 18E, 20F36, 20K01, 20Ll7,46L, 46M, 81R05, 81R50, 8IT05, 8IT40
ISBN 3-540-56623-6Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56623-6Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper
Contents
1 Introduction and Survey of Results 2 Local Quantum Theory with Braid Group Statistics Some Aspects of Low-Dimensional, Local Quantum Field Theory
17
2.2
Generalities Concerning Algebraic Field Theory . . . . . . . . . . . . . ..
24
2.3
Statistics and Fusion of Intertwiners; Statistical Dimensions
32
2.4
Unitary Representations of the Braid Groups Derived from Local Quantum
Superselection Sectors and the Structure of Fusion Rule Algebras 3.1
41
45
Definition of and General Relations in Fusion Rule Algebras, and their Appearance in Local Quantum Field Theories . . . . . . . . . . . .
46
3.2
Structure Theory for Fusion Rule Algebras . . . . . . . . . . . . . . . . ..
51
3.3
Grading Reduction with Automorphisms and Normality Constraints in Fusion Rule Algebras
3.4
4
17
2.1
Theory; Markov Traces. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
3
1
"
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Fusionrules with a Generator of Dimension not Greater than Two
Hopf Algebras and Quantum Groups at Roots of Unity
v
72
102
5
Representation Theory of
119
5.1
119
Highest Weight Representations of
122
5.2 The Irreducible and Unitary Representations of
6
126
5.3
Decomposition of Tensor Product Representations . . ..
5.4
Fusion Rules, and q-Dimensions: Selecting a List of Physical Representations 135
Path Representations of the Braid Groups for Quantum Groups at
Roots of Unity
141
6.1
Quotients of Representation Categories :
141
6.2
Braid Group Representations and Fusion Equations ..
152
