Lectures on Algebraic Quantum Groups

In September 2000, at the Centre de Recerca Matematica in Barcelona, we pre­ sented a 30-hour Advanced Course on Algebraic Quantum Groups. This textbook consists of an expanded set of lectures on algebraic aspects of quantum groups, concentrating particularly on quantized coordinate rings of algebraic groups and spaces and on quantized enveloping algebras of semisimple Lie algebras. The approach, a mixture of introductory textbook, lecture notes and overview survey, is designed to allow access by graduate students and by researchers new to the areas, as well as by experts, and to provide a basis for further study of the subject. Thus, large parts of the material are developed in full textbook style, with many examples and numerous exercises; other portions are discussed with sketches of proofs, while still other material is quoted without proof. Much associated material is outlined in the appendices. Among the topics covered in the book are a discussion of the nature of the prime spectrum of a "generic" quantum algebra and details of how the Hopf algebra structure of the algebra and the Poisson algebra structure of the centre carry important consequences for quantized algebras when the quantum parameter is a root of unity. The book is structured in three parts: one introductory part with many examples plus background material; one concentrating on generic quantized coordinate rings; and one dealing with quantized algebras at roots of unity. Many examples and exercises at the end of each chapter are provided. The book serves also as a survey book for researchers and contains open problems and questions.