Table Of ContentQUANTUM
GROUP
SYMMETRY
AND
q-TENSOR
ALGEBRAS
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Q U A N T UM
GROUP
SYMMETRY
AND
q-TENSOR
ALGEBRAS
L. C. Biedenharn
Univ. Texas, Austin
M. A. Lohe
Northern Territory Univ., Australia
World Scientific
Singapore■•NewJersey London •Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
First published 1995
Reprinted 1999
QUANTUM GROUP SYMMETRY AND q-TENSOR ALGEBRAS
Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 981-02-2331-5
Printed in Singapore.
Preface
The impact of quantum groups has been immediate and long-lasting, but
although the initial impetus to the subject came from physics in areas such as the
quantum inverse scattering method and statistical mechanical models, the more vig
orous development has more recently been in mathematics. Yet there is the promise
of a much more extensive application of quantum groups to physics in a way that
mimics the many successful applications of group theory in the past, in which the
symmetry of a physical system is extended to a quantum symmetry. For example,
there is the possibility that any Hamiltonian which is invariant under a Lie group may
be generalized to a quantum group invariant, and that there is an extension of the
algebraic methods which lead to a solution of the physical system with a dependence
on an arbitrary deformation parameter q.
The aim of this monograph is to develop and extend to quantum groups the
symmetry techniques familiar from the application of classical groups to models in
physics. Our exposition is intended to be accessible to graduate physics students and
to physicists wishing to gain an introduction to quantum groups. However, we hope
that experts in quantum groups will also find some topics of interest, or perhaps a
different viewpoint that offers some insight into the properties of quantum groups.
We have taken a uniform approach to quantum groups based on the fundamental
concept of a tensor operator. Properties of both the quantum algebra and co-algebra
are developed from a single point of view using tensor operators, which is especially
helpful for an understanding of the noncommuting coordinates of the quantum plane
interpreted as elementary tensor operators. Representations are constructed using a
generalization of the boson calculus in which g-boson operators, later to be interpreted
as tensor operators, play a central role, including the case when q is a root of unity.
After some introductory remarks and definitions in Chapter 1 we investigate rep
resentations of the quantum unitary groups in Chapter 2, beginning with the g-analog
of the angular momentum group. In Chapter 3 we introduce the concept of a tensor
operator and systematically develop the fundamental properties such as the 'multipli
cation' of two tensor operators using the g-analog of the Clebsch-Gordan coefficients
to produce a third tensor operator. Whereas our initial discussion is based on the
definition of a quantum group as the deformation of the classical Lie algebra, in
Chapter 4 we determine properties of the (/-analog of the Lie group, usually defined
v
VI Quantum Group Symmetry and q-Tensor Algebras
in terms of the dual of the universal enveloping algebra, but in our approach con
structed using tensor operators. Then in Chapter 5 we further develop properties of
the representation matrices and their interpretation as tensor operators. In Chapter
6 we analyse the interesting specialization when q is a root of unity, a case which
appears in significant applications to physics. Algebraic induction is a method of
constructing representations of the classical groups which expresses group properties
in terms of the subgroup and which we generalize to quantum groups in Chapter 7.
Finally, we consider several special topics in Chapter 8 which in some way illustrate
the ideas, particularly those of tensor operators, developed in previous chapters.
We regret that our exposition is necessarily limited in scope and that we have not
been able to include details of the more significant applications to physics, such as to
statistical mechanics and conformal field theory. We have, however, provided refer
ences that will direct the reader to these applications. Nevertheless, partly because of
the rapidly expanding literature there are inevitably many omissions here too and we
apologize in advance to those authors whose work has not been accorded due credit.
Several conference proceedings, for example [1, 2, 3, 4, 5, 6], provide windows looking
onto the wide range of activities relating to quantum groups, and we refer to these
for further references. There have also been several recent monographs on quantum
groups, including those by Kassel [7], Chari and Pressley [8] and also Shnider and
Sternberg [9] (which provides a very extensive bibliography), with a mathematical
style and presentation to which we also refer the reader seeking more details than
we have provided. By contrast, although our presentation is less precise, we have
focussed more directly on the concepts relevant to symmetry techniques in physics.
We gratefully acknowledge and thank our many colleagues for their support, both
direct and indirect over many years, who helped us to formulate and develop our
investigation, in particular (but not only) R. Askey, V. Dobrev, H.-D. Doebner, D.
Flath, B. Gruber, J. Louck, M. Tarlini, J. Towber, P. Truini, and also Professor
M. Nomura for his interest in the project. We thank Brenda Gage for her efficient
secretarial help, and especially our wives Sarah and Thilagam for their encouragement
and support.
L. C. Biedenharn and M. A. Lohe, May 1995.
NOTATION: We use standard notations, but in particular note that IR+ denotes
the set of positive real numbers and Cx denotes the nonzero complex numbers. The
symbol = is used in order to emphasize that the equation constitutes a definition.
Where necessary we denote g-analog functions and operators with a suffix q (for
example, ^-integers are denoted [n] , generators J|) which we omit for convenience
q
when confusion with the q = 1 case is unlikely.
Equations are numbered, where necessary, consecutively within each chapter to
gether with items such as theorems, lemmas, remarks and examples; the first digit
specifies the chapter in which this item appears.
Contents
Preface v
1 Origins of Quantum Groups 1
1.1 Quantum Inverse Scattering Method 2
1.2 Applications of Quantum Groups 5
1.3 Special Functions and Quantum Groups 7
1.4 Definition of Quantum Group 11
2 Representations of Unitary Quantum Groups 15
2.1 The Prototype for Quantum Groups: W,(su(2)) 16
2.1.1 Co-Algebra Structure 17
2.2 Irreducible Unitary Representations of W,(su(2)) 19
2.3 The Jordan Map and Unitary Symmetry 24
2.4 The ^-Generalization of the Boson Calculus 26
2.4.1 Realizations of g-Boson Operators 29
2.4.2 The g-Boson Realization of U (su(2)) Unitary Irreps 33
g
2.4.3 Realization on a Projective Space 35
2.4.4 Mixed Symmetry States and Irreps of W (u(2)) 40
g
2.5 Irreducible Unitary Representations of U (u(n)) 43
q
2.5.1 The g-Boson Construction for U (u(n)) 49
q
2.6 Appendix: Gel'fand-Weyl States and Young Frames 50
2.7 Appendix: Properties of ^-Numbers 55
2.7.1 Symmetries and Identities of q-Numbers 57
2.7.2 The g-Binomial Theorem 60
2.8 Appendix: g-Calculus and g-Functions 63
2.8.1 g-Derivation and Integration 63
2.8.2 The q-Exponential Function 64
2.8.3 Basic Hypergeometric Functions 67
vn
viii Quantum Group Symmetry and q-Tensor Algebras
3 Tensor Operators in Quantum Groups 71
3.1 Introduction 71
3.2 Classical Theory of Tensor Operators 73
3.2.1 The Classification Problem for Tensor Operators 76
3.2.2 Operator Patterns and the Characteristic Null Space 77
3.3 Tensor Operators in Quantum Groups 81
3.4 The Algebra of q-Tensor Operators 85
3.4.1 Ug(su(2)) q-Tensor Operators and Coupling Coefficients . . .. 86
3.4.2 Examples of g-Tensor Operators in W,(su(2)) 89
3.4.3 W,(u(n)) g-Tensor Operators 93
3.5 g-Wigner-Clebsch-Gordan Coefficients 94
3.5.1 Special Cases of q-Wigner-Clebsch-Gordan Coefficients . . .. 96
3.5.2 Symmetries of q-Wigner-Clebsch-Gordan Coefficients 99
3.6 q-6j and g-Racah Coefficients 102
3.6.1 Asymptotic limit of the q-6j symbol 105
3.7 The Pattern Calculus and Elementary Tensor Operators 107
3.7.1 The Pattern Calculus Rules for Elementary q-Tensor Operators 109
3.7.2 A Conceptual Derivation of the Pattern Calculus Rules . . .. 113
4 The Dual Algebra and the Factor Group 115
4.1 Introduction 115
4.2 Matrix Quantum Groups 117
4.2.1 The n-Dimensional Matrix Quantum Groups 120
4.2.2 Noncommuting g-Coordinates and the Quantum Plane . . .. 121
4.3 The Classical Unitary Factor Groups 124
4.3.1 The (7(2) Factor Group and the Rotation Matrices 132
4.4 Extension to the Quantum Factor Algebra 134
4.4.1 Basis Polynomials in an Irrep of the Quantum Factor Algebra 139
4.4.2 Derivation of q-WCG Coefficients 141
4.5 Commutation Rules for Elements of the Quantum Matrix 144
4.5.1 Generalization to the Quantum Hyperplane 148
4.6 A g-Boson Realization of Noncommuting Elements 148
4.7 Irreps of the Matrix Quantum Group 153
4.7.1 Fractional Linear Transformations 154
Contents ix
5 Quantum Rotation Matrices 157
5.1 Fundamental Properties of the Quantum Rotation Matrices 158
5.1.1 Special Cases 161
5.2 Generating Function 162
5.2.1 Symmetries of the Quantum Rotation Matrix 164
5.3 Tensor Operator Properties of the Quantum Rotation Matrices . . .. 165
5.4 The Wigner Product Law 166
6 Quantum Groups at Roots of Unity 169
6.1 The Special Linear Quantum Group for q a Root of Unity 170
6.1.1 Invariants of li(si(2)) at Roots of Unity 173
q
6.1.2 Irreducible Nilpotent Representations of U (su(2)) 175
g
6.2 Irreducible Cyclic Representations of U (sl(2)) 178
q
6.2.1 Unitary Cyclic Representations of U (su(2)) 181
q
6.2.2 Factorized Matrix Elements 184
6.2.3 Analytic Extension of U(2) Representations 188
6.3 g-Boson Operator Construction of Representations 189
6.3.1 Cyclic Representations for Even p 195
6.4 Hermitean Adjoints of g-Boson Operators 196
6.5 Cyclic g-Boson Operators in a Fock Space 200
6.5.1 Unitary Cyclic Representations in a Fock Space 203
6.6 Cyclic Representations in a Space of Polynomials 204
6.7 Algebraic Induction at Roots of Unity 206
7 Algebraic Induction of Quantum Group Representations 209
7.1 Introduction and Summary 209
7.2 The Algebraic Borel-Weil Construction 210
7.3 Algebraic Induction for the Classical Group 1/(2) 212
7.4 Algebraic Induction for the Quantum Group U (u(2)) 215
q
7.5 The Algebraic Induction Construction for the Classical Unitary Groups 218
7.6 Extension of Algebraic Induction to the Unitary Quantum Groups . . 223
7.6.1 The Isomorphism of Quantum Group Algebras 229
7.6.2 An Alternative Form for the Induced Irrep Vectors 234
7.7 Algebraic Induction for £/(3) and its Quantum Extension 236
7.7.1 Explicit Induced Vectors for U{3) 236
X Quantum Group Symmetry and q-Tensor Algebras
7.7.2 Algebraic Induction for W,(u(3)) 242
7.7.3 Explicit Induced Vectors for W,(u(3)) 245
7.7.4 Basic Hypergeometric Functions and Watson's Formula . . .. 247
7.8 Appendix: The Construction of Tensor Operators in the Classical Uni
tary Groups U(n) 250
8 Special Topics 253
8.1 The g-Harmonic Oscillator 254
8.1.1 g-Coherent States 255
8.2 Physical Interpretation of Noncommuting Coordinates 257
8.3 Group Invariance of the Canonical Commutation Relations 260
8.3.1 Weyl-Ordered Polynomials and the Symplecton 261
8.3.2 The g-Symplecton 267
Bibliography 275
Index 290
