Quantum Groups

Quantum Groups : A Path to Current Algebra

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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.show more

Product details

  • Paperback | 160 pages
  • 152 x 226 x 10mm | 240.4g
  • Cambridge, United Kingdom
  • English
  • 26 b/w illus. 25 exercises
  • 0521695244
  • 9780521695244
  • 1,530,761

Review quote

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

Table of contents

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.show more

About Ross Street

Ross Street is a Professor of Mathematics and Director of the Centre of Australian Category Theory at Macquarie University. He is also a Fellow of the Australian Academy of Science.show more