A Quantum Groups Primer

Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.

Shahn Majid (Author)

9780521010412, Cambridge University Press

Paperback, published 4 April 2002

180 pages, 23 b/w illus. 50 exercises
22.8 x 15.2 x 1.7 cm, 0.265 kg

'This monograph is an excellent reference (and often a true 'eye-opener') for researchers working in quantum groups … S. Majid is well-known for his lively and very informative style of writing, and the reviewed book confirms this opinion. Thus the book is very well written, the proofs contain enough details to make them easily readable but still challenging enough to keep students interested … I can full-heartily recommend this work as a basis for a one-term postgraduate course or as an introductory text to all mathematicians who would like to learn quickly the main ideas, techniques and wide range of applications of quantum group theory.' Zentralblatt MATH

This book provides a self-contained introduction to quantum groups as algebraic objects. Based on the author's lecture notes from a Part III pure mathematics course at Cambridge University, it is suitable for use as a textbook for graduate courses in quantum groups or as a supplement to modern courses in advanced algebra. The book assumes a background knowledge of basic algebra and linear algebra. Some familiarity with semisimple Lie algebras would also be helpful. The book is aimed as a primer for mathematicians and takes a modern approach leading into knot theory, braided categories and noncommutative differential geometry. It should also be useful for mathematical physicists.

Preface
1. Coalgebras, bialgebras and Hopf algebras. Uq(b+)
2. Dual pairing. SLq(2). Actions
3. Coactions. Quantum plane A2q
4. Automorphism quantum groups
5. Quasitriangular structures
6. Roots of Unity. uq(sl2)
7. q-Binomials
8. quantum double. Dual-quasitriangular structures
9. Braided categories
10 (Co)module categories. Crossed modules
11. q-Hecke algebras
12. Rigid objects. Dual representations. Quantum dimension
13. Knot invariants
14. Hopf algebras in braided categories
15. Braided differentiation
16. Bosonisation. Inhomogeneous quantum groups
17. Double bosonisation. Diagrammatic construction of uq(sl2)
18. The braided group Uq(n–). Construction of Uq(g)
19. q-Serre relations
20. R-matrix methods
21. Group algebra, Hopf algebra factorisations. Bicrossproducts
22. Lie bialgebras. Lie splittings. Iwasawa decomposition
23. Poisson geometry. Noncommutative bundles. q-Sphere
24. Connections. q-Monopole. Nonuniversal differentials
Problems
Bibliography
Index.

Subject Areas: Applied mathematics [PBW]