This is a graduate-level text that systematically develops the foundations of the subject. Quantum groups (i.e. Hopf algebras) are treated as mathematical objects in their own right; basic properties and theorems are proven in detail from this standpoint, including the results underlying key applications. After formal definitions and the 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 semiclassical theory of Poisson-Lie groups, the representation theory, braided groups and applications to q-deformed physics. The explicit proofs and many worked examples and exercises will allow readers to quickly pick up the techniques needed for working in this exciting new field.
Author(s): Shahn, Majid
Edition: 1
Publisher: Cambridge University Press
Year: 1995
Language: English
Pages: 627
Front Cover
Front Matter
Dedication
Contents
Introduction
1 Definition of Hopf algebras
1.1 Algebras
1.2 Coalgebras
1.3 Bialgebras and Hopf algebras
1.4 Duality
1.5 Commutative and cocommutative Hopf algebras
1.6 Actions and coactions
1.6.1 Actions on algebras and coalgebras
1.6.2 Coactions
1.7 Integrals and *-structures
Notes for Chapter 1
2 Quasitriangular Hopf algebras
2.1 Quasitriangular structures
2.2 Dual quasitriangular structures
2.3 Cocycles and twisting
2.4 Quasi-Hopf algebras
Notes for Chapter 2
3 Quantum enveloping algebras
3.1 q-Heisenberg algebra
3.2 Uq(sl2) and its real forms
3.3 Uq(g) for general Lie algebras
3.4 Roots of unity
Notes for Chapter 3
4 Matrix quantum groups
4.1 Quantum matrices
4.2 Quantum determinants and basic examples
4.3 Matrix quantum Lie algebras
4.4 Vertex models
4.5 Quantum linear algebra
4.5.1 Bicovariant formulation
4.5.2 Covariant formulation
4.5.3 Quantum automorphisms and diffeomorphisms
Notes for Chapter 4
5 Quantum random walks and combinatorics
5.1 Combinatorial Hopf algebras
5.2 Classical random walks using Hopf algebras
5.2.1 Brownian motion on the real line
5.2.2 Markov processes
5.3 Quantum random walks
5.4 Input-output symmetry and time-reversal
Notes for Chapter 5
6 Bicrossproduct Hopf algebras
6.1 Quantisation on homogeneous spaces
6.2 Bicrossproduct models
6.3 Extension theory and cocycles
6.4 Quantum-gravity and observable-state duality
Notes for Chapter 6
7 Quantum double and double cross products
7.1 Definition of D{H)
7.2 Double cross product Hopf algebras
7.3 Complexification of quantum groups
7.4 Cross product structure of quantum doubles
Notes for Chapter 7
8 Lie bialgebras and Poisson brackets
8.1 Lie bialgebras and the CYBE
8.2 Double Lie bialgebra
8.3 Matched pairs of Lie algebras and their exponentiation
8.4 Poisson-Lie groups
Notes for Chapter 8
9 Representation theory
9.1 Categories, functors and monoidal products
9.2 Quasitensor or braided monoidal categories
9.3 Duals, quantum dimensions and traces
9.4 Reconstruction theorems
Notes for Chapter 9
10 Braided groups and q-deformation
10.1 Super and anyonic quantum groups
10.2 Braided vectors and covectors
10.3 Braided matrices and braided linear algebra
10.4 Braided differentiation
10.5 Examples of braided addition
10.5.1 Coaddition on quantum matrices
10.5.2 q-Euclidean space
10.5.3 q-Minkowski space
Notes for Chapter 10
References
Symbols
Index
2-Index
