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 the latest 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 -- Index.