Lectures on algebraic quantum groups
type: Item
instance of: scholarly article
title: Lectures on algebraic quantum groups
zbMATH Open document ID: 1027.17010
author: Kenneth A. Brown
author: Kenneth R. Goodearl
published in: Advanced Courses in Mathematics -- CRM Barcelona
publication date: 2002-07-08T00:00:00Z
review text: Quantum groups can be studied from different angles, and many of these deserve the publication of books accounting their main achievements. This book is mainly concerned with the representation theory of quantum algebras from a ring-theoretical point of view. More precisely, two main types of algebras are studied: ``quantized coordinate algebras'' like the quantized coordinate algebra of an algebraic semisimple group, the algebra of functions on quantum linear spaces, or on quantum matrices, and so on; and quantized enveloping algebras of Lie algebras. There are books focussing on quantized coordinate algebras of semisimple algebraic groups; [\textit{L. I. Korogodskij} and \textit{Y. S. Soibelman}, Algebras of functions on quantum groups. Part I. Mathematical Surveys and Monographs, 56. American Mathematical Society, Providence, RI (1998; Zbl 0923.17017)] has a more geometrical and analytical flavor; and [\textit{A. Joseph}, Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin (1995; Zbl 0808.17004)] is in the spirit of algebraic Lie theory. The origin of research in this area is the seminal work of Soibelman, Vaksman and Levendorskii on representations of the quantum algebra of functions on a compact quantum group. The guiding principle is that the representation theory of a quantized algebra is governed by the Poisson geometry of its classical limit, in particular by the foliation by symplectic leaves (Soibelman et al. attribute this idea to Drinfeld). This principle generalizes the celebrated orbit method of Kirillov. It fits perfectly in the case of the ``standard'' compact quantum group, and less accurately in the multiparametric case. An account is given in [Korogodskij and Soibelman, loc. cit.]; see also the references therein. The results of Soibelman et al. inspired Hodges and Levasseur, who stated a conjecture with an analogous claim for the spectrum of the quantized coordinate algebra of an a…
reviewed by: Nicolás Andruskiewitsch
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W35
Mathematics Subject Classification ID: 17-02
zbMATH DE Number: 1764236
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum algebras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1764236