Quantum Groups
A Path to Current Algebra

Quantum Groups: A Path to Current Algebra presents algebraic concepts and techniques.

Ross Street (Author)

9780521695244, Cambridge University Press

Paperback, published 18 January 2007

160 pages, 26 b/w illus. 25 exercises
22.9 x 15.2 x 0.9 cm, 0.24 kg

"The book is very well written [and] it is quite concise."
E.J. Taft, Mathematical Reviews

Algebra has moved well beyond the topics discussed in standard undergraduate texts on 'modern algebra'. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an 'algebra'. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a 'coalgebra'. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term 'quantum group', along with revolutionary new examples, was launched by Drinfel'd in 1986.

Introduction
1. Revision of basic structures
2. Duality between geometry and algebra
3. The quantum general linear group
4. Modules and tensor products
5. Cauchy modules
6. Algebras
7. Coalgebras and bialgebras
8. Dual coalgebras of algebras
9. Hopf algebras
10. Representations of quantum groups
11. Tensor categories
12. Internal homs and duals
13. Tensor functors and Yang-Baxter operators
14. A tortile Yang-Baxter operator for each finite-dimensional vector space
15. Monoids in tensor categories
16. Tannaka duality
17. Adjoining an antipode to a bialgebra
18. The quantum general linear group again
19. Solutions to exercises
References
Index.

Subject Areas: Physics [PH], Mathematical foundations [PBC]