Quantum deformation of classical groups
type: Item
instance of: scholarly article
title: Quantum deformation of classical groups
zbMATH Open document ID: 0806.16040
author: Takahiro Hayashi
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-02-15T00:00:00Z
review text: Let \(A(G)\) be the coordinate algebra of a Lie group \(G\). A quantum deformation \(A(G_ q)\) of \(A(G)\) is a one-parameter family of Hopf algebras whose representation theories are the same as those of \(A(G)\). The author constructs the quantum deformations of \(A(O(N))\), \(A(SO(N))\) and \(A(Sp(N))\) by using Jimbo's solutions of the Yang-Baxter equation and determines their Peter-Weyl decompositions. For this purpose, he investigates the so-called quantum matrix bialgebras, and defines Hopf algebras \(A(G_ q)\) as quotients of corresponding quantum matrix bialgebras. Also, he studies group-like elements of these quantum matrix bialgebras and gives a new realization of the universal \(R\)-matrix.
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 81U20
zbMATH DE Number: 548922
zbMATH Keywords: coordinate algebra
zbMATH Keywords: Lie group
zbMATH Keywords: quantum deformation
zbMATH Keywords: one-parameter family of Hopf algebras
zbMATH Keywords: Yang-Baxter equation
zbMATH Keywords: Peter-Weyl decompositions
zbMATH Keywords: quantum matrix bialgebras
zbMATH Keywords: group-like elements
zbMATH Keywords: universal \(R\)-matrix
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195168856
OpenAlex ID: W2037182182
cites work: The diamond lemma for ring theory
cites work: On algebras which are connected with the semisimple continuous groups
cites work: Q3202270
cites work: Polynomial representations of \(GL_n\)
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Quantum multilinear algebra
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Yang-Baxter equation and representation theory. I
cites work: Quantum deformations of certain simple modules over enveloping algebras
cites work: Hopf algebras and vector symmetries
cites work: QUASITRIANGULAR HOPF ALGEBRAS AND YANG-BAXTER EQUATIONS
cites work: Some remarks on Koszul algebras and quantum groups
cites work: Q5336354
cites work: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra
cites work: Analogues de la forme de Killing et du théorème d'Harish-Chandra pour les groupes quantiques
cites work: Matric bialgebras and quantum groups
cites work: Finite dimensional representations of quantum groups
cites work: Q4435456
cites work: Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups
DOI: 10.2977/PRIMS/1195168856
description: scientific article; zbMATH DE number 548922