BGP-reflection functors and Lusztig's symmetries: a Ringel-Hall algebra approach to quantum groups
type: Item
instance of: scholarly article
title: BGP-reflection functors and Lusztig's symmetries: a Ringel-Hall algebra approach to quantum groups
zbMATH Open document ID: 0999.17026
author: Jie Xiao
author: Shilin Yang
published in: Journal of Algebra
publication date: 2002-12-03T00:00:00Z
review text: This paper contributes a very interesting Ringel-Hall algebra approach to quantum groups through BGP-reflection functors and Lusztig's symmetries. In this paper, the authors deduce Lusztig's symmetries \(T_{i,1}''\), \(i\in I\), by applying the Bernstein-Gelfand-Ponomarev reflection functors to the Drinfeld doubles of Ringel-Hall algebras, according to the canonical isomorphism between the Ringel-Hall algebras (composition algebras) and the quantum groups. The fundamental properties of \(T_{i,1}''\) including the following are obtained conceptually, that is, (1) \(T_{i,1}''\), \(i\in I\), induce automorphisms of the quantum groups \(U_q(g)\) on the integrable modules; (2) \(T_{i,1}''\), \(i\in I\), satisfy the braid group relations. This extends and completes the results of \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Algebra 221, 135--160 (1999; Zbl 0955.16016)].
reviewed by: Fang Li
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16T20
zbMATH DE Number: 1658814
zbMATH Keywords: Ringel-Hall algebra
zbMATH Keywords: quantum groups
zbMATH Keywords: BGP-reflection functors
zbMATH Keywords: Lusztig's symmetries
zbMATH Keywords: braid group relations
MaRDI profile type: Publication
full work available at URL: jabr.2001.8762
OpenAlex ID: W1997540855
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DOI: 10.1006/JABR.2001.8762
description: scientific article; zbMATH DE number 1658814