Quantum Groups: A Path to Current Algebra

ISBN-13: 9780521695244 / Angielski / Miękka / 2007 / 160 str.

Ross Street; Michael Murray; Phil Broadbridge

Opis

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.

Recenzje

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

Spis treści

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.