Here is a self-contained introduction to quantum groups as algebraic objects. Based on the author's lecture notes for the Part III pure mathematics course at Cambridge University, the book is suitable as a primary text for graduate courses in quantum groups or supplementary reading for modern courses in advanced algebra. The material assumes knowledge of basic and linear algebra. Some familiarity with semisimple Lie algebras would also be helpful. The volume is a primer for mathematicians but it will also be useful for mathematical physicists.
Author(s): Shahn Majid
Edition: 1
Year: 2002
Language: English
Pages: 168
Contents ......Page 7
Preface ......Page 9
1 Coalgebras, bialgebras and Hopf algebras. Uq(b+) ......Page 11
2 Dual pairing. SLq(2). Actions ......Page 19
3 Coactions. Quantum plane A2 ......Page 27
4 Automorphism quantum groups ......Page 33
5 Quasitriangular structures ......Page 39
6 Roots of unity. uq(sl2) ......Page 44
7 q-Binomials ......Page 49
8 Quantum double. Dual-quasitriangular structures ......Page 54
9 Braided categories ......Page 62
10 (Co)module categories. Crossed modules ......Page 68
11 q-Hecke algebras ......Page 74
12 Rigid objects. Dual representations. Quantum dimension ......Page 80
13 Knot invariants ......Page 87
14 Hopf algebras in braided categories. Coaddition on A2 ......Page 94
15 Braided differentiation ......Page 101
16 Bosonisation. Inhomogeneous quantum groups ......Page 108
17 Double bosonisation. Diagrammatic construction of uq(sl2) ......Page 115
18 The braided group Uq(n+). Construction of Uq(g) ......Page 123
19 q-Serre relations ......Page 130
20 R-matrix methods ......Page 136
21 Group, algebra, Hopf algebra factorisations. Bicrossproducts ......Page 142
22 Lie bialgebras. Lie splittings. Iwasawa decomposition ......Page 149
23 Poisson geometry. Noncommutative bundles. q-Sphere ......Page 156
24 Connections. q-Monopole. Nonuniversal differentials ......Page 163
Problems ......Page 169
Bibliography ......Page 176
Index ......Page 177
