Quantum groups and non-commutative geometry. (English) Zbl 0724.17006

Manin, Yu. I.

Montréal: Université de Montréal, Centre de Recherches Mathématiques (CRM). 91 p. $ 12.00 (1988).

This book may well be used as an introduction to quantum groups – or some class of noncommutative, noncocommutative Hopf algebras. The starting point of the author is the Hopf algebra of “noncommutative functions” on a quantum group, which is in some sense predual to the Hopf algebras considered by V. Drinfel’d using deformations of enveloping algebras. The author generalizes the original concept of quantum group and its Hopf algebra of noncommutative functions to the concept of “quadratic Hopf algebras” defined by arbitrary quadratic relations in a finitely generated tensor algebra. The definition makes sense and seems to be natural in view of the fact, that the commutation relation between some elements of quantum matrix groups are more complicated, than the integrated Heisenberg commutation relation satisfied in linear quantum spaces. The considerations of the author are based on category theory arguments as well as on direct computations. There are several instructive examples included. The methods are purely algebraic in the spirit of algebraic geometry, no differential calculus on quantum groups or quantum spaces is discussed. The relation to the Yang-Baxter equation is pointed out and put in a category-theoretic framework.
The content is given as follows: Introduction, 1. The quantum group \(\mathrm{GL}(2)\), 2. Bialgebras and Hopf algebras, 3. Quadratic algebras as quantum linear spaces, 4. Quantum matrix spaces I. Categorical viewpoint, 5. Quantum matrix spaces II. Coordinate approach, 6. Adding missing relations, 7. From semigroups to groups, 8. Frobenius algebras and the quantum determinant, 9. Koszul complexes and growth rate of quadratic algebras, 10. Hopf *-algebras and compact matrix pseudogroups, 11. Yang-Baxter equations, 12. Algebras in tensor categories and Yang-Baxter functors, 13. Some open problems, Bibliography.

Reviewer: Helmut Boseck (Greifswald)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
17-02	Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
16T05	Hopf algebras and their applications
16T10	Bialgebras
16T20	Ring-theoretic aspects of quantum groups
16T25	Yang-Baxter equations
14A22	Noncommutative algebraic geometry
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

quantum groups; quadratic Hopf algebras; compact matrix pseudogroups; Yang-Baxter equations; tensor categories