The book is devoted to algebraic aspects of the theory of quantum groups. The attention is concentrated on two main objects – the quantized enveloping algebras Uq(G) and quantize coordinate rings Oq(G), which are dual to each other. The first introductory part of the book contains a systematic construction of Uq(G) and Oq(G) and a description of their basic properties for the case when G is a classical simple Lie group and q is not a root of unity. It also includes a summary of basic facts on semisimple Lie algebras and appendices describing various algebraic tools needed in the book. Then the book splits into two parts, which are to a certain extent independent. The first one is devoted to the generic case, while the other treats the cases when q is a root of unity. In the generic case, discussion concentrates mainly on a study of prime and primitive ideals in Oq(G), in an analogy to the case of the coordinate ring of an affine algebraic variety. In the case of roots of unity, the structure and representations of Oє(G) are described. The book offers a very nice introduction to the subject, containing many illuminating comments and remarks, a lot of exercises and also a formulation of open directions for research from this algebraic point of view.

Reviewer:

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