Foundations of Quantum Group Theory

661 Pages·1995·31.839 MB·English

by  Shahn, Majid

Foundations of Quantum Group Theory
Now in paperback, this is a graduate level text for theoretical physicists and
mathematicians which systematically lays out the foundations for the subject of
quantum groups in a clear and accessible way. The topic is developed in a logical
manner with quantum groups (Hopf algebras) treated as mathematical objects
in their own right. After formal definitions and basic theory, the book goes on
to cover such topics as quantum enveloping algebras, matrix quantum groups,
combinatorics, cross products of various kinds, the quantum double, the semi-
classical theory of Poisson-Lie groups, the representation theory, braided groups
and applications to g-deformed physics. Explicit proofs and many examples will
allow the reader quickly to pick up the techniques needed for working in this
exciting new field.

Foundations of
Quantum Group Theory
SHAHN MAJID
University of Cambridge
CAMBRIDGE
UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge, CB2 2RU, UK http://www.cup.cam.ac.uk
40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
Ruiz de Alarcon 13, 28014 Madrid, Spain
© Cambridge University Press 1995
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1995
First paperback edition (with corrections) 2000
Printed in the United Kingdom at the University Press, Cambridge
Typeset by the author [CRC]
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing in Publication data
Majid, Shahn.
Foundations of quantum group theory / Shahn Majid.
p. cm.
ISBN 0-521-46032-8 (he)
1. Group theory. 2. Quantum theory. I. Title.
QC174. 17.G7M25 1995
512'.55-dc20 95-13641 CIP
ISBN 0 521 46032 8 hardback
ISBN 0 521 64868 8 paperback
For my friends

Contents
Introduction xi
1 Definition of Hopf algebras 1
1.1 Algebras 1
1.2 Coalgebras 5
1.3 Bialgebras and Hopf algebras 6
1.4 Duality 10
1.5 Commutative and cocommutative Hopf algebras 12
1.6 Actions and coactions 15
1.6.1 Actions on algebras and coalgebras 17
1.6.2 Coactions 22
1.7 Integrals and *-structures 28
Notes 36
2 Quasitriangular Hopf algebras 38
2.1 Quasitriangular structures 39
2.2 Dual quasitriangular structures 50
2.3 Cocycles and twisting 55
2.4 Quasi-Hopf algebras 65
Notes 70
3 Quantum enveloping algebras 72
3.1 g-Heisenberg algebra 73
3.2 U (sl2) and its real forms 85
q
3.3 U (g) for general Lie algebras 95
q
3.4 Roots of unity 102
Notes 105
4 Matrix quantum groups 108
4.1 Quantum matrices 111
vii
viii Contents
4.2 Quantum determinants and basic examples 128
4.3 Matrix quantum Lie algebras 143
4.4 Vertex models 149
4.5 Quantum linear algebra 153
4.5.1 Bicovariant formulation 153
4.5.2 Covariant formulation 163
4.5.3 Quantum automorphisms and diffeomorphisms 166
Notes 173
5 Quantum random walks and combinatorics 178
5.1 Combinatorial Hopf algebras 180
5.2 Classical random walks using Hopf algebras 191
5.2.1 Brownian motion on the real line 192
5.2.2 Markov processes 197
5.3 Quantum random walks 204
5.4 Input-output symmetry and time-reversal 209
Notes 220
6 Bicrossproduct Hopf algebras 223
6.1 Quantisation on homogeneous spaces 225
6.2 Bicrossproduct models 239
6.3 Extension theory and cocycles 262
6.4 Quantum-gravity and observable-state duality 286
Notes 297
7 Quantum double and double cross products 302
7.1 Definition of D(H) 304
7.2 Double cross product Hopf algebras 314
7.3 Complexification of quantum groups 335
7.4 Cross product structure of quantum doubles 344
Notes 360
8 Lie bialgebras and Poisson brackets 364
8.1 Lie bialgebras and the CYBE 366
8.2 Double Lie bialgebra 383
8.3 Matched pairs of Lie algebras and their exponentiation 395
8.4 Poisson-Lie groups 420
Notes 432
9 Representation theory 436
9.1 Categories, functors and monoidal products 438
9.2 Quasitensor or braided monoidal categories 451
9.3 Duals, quantum dimensions and traces 465
Contents ix
9.4 Reconstruction theorems 489
9.4.1 Reconstruction in vector spaces 490
9.4.2 Braided reconstruction 499
Notes 520
10 Braided groups and g-deformation 527
10.1 Super and anyonic quantum groups 530
10.2 Braided vectors and covectors 540
10.3 Braided matrices and braided linear algebra 555
10.4 Braided differentiation 568
10.5 Examples of braided addition 588
10.5.1 Coaddition on quantum matrices 589
10.5.2 g-Euclidean space 593
10.5.3 g-Minkowski space 599
Notes 606
References 611
Symbols 625
Index 629
2-Index 640