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.