Ringel-Hall algebras and Lusztig's symmetries
type: Item
instance of: scholarly article
title: Ringel-Hall algebras and Lusztig's symmetries
zbMATH Open document ID: 1017.16008
DOI: 10.1016/S0021-8693(02)00123-0
author: Bangming Deng
author: Jie Xiao
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The authors of the paper under review have shown [in J. Algebra 251, No. 1, 110-149 (2002; Zbl 1010.16011)] that the Drinfeld double \({\mathcal D}(\Lambda)\) of the Ringel-Hall algebra of a finite-dimensional hereditary algebra \(\Lambda\) augmented by a torus algebra is isomorphic to the quantized enveloping algebra of a generalized Kac-Moody algebra in the sense of \textit{R. Borcherds} [J. Algebra 115, No. 2, 501-512 (1988; Zbl 0644.17010)]. Moreover, the action of Lusztig's symmetries is extended to \({\mathcal D}(\Lambda)\). \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Algebra 221, No. 1, 135-160 (1999; Zbl 0955.16016)] as well as \textit{J. Xiao} and \textit{S. Yang} [J. Algebra 241, No. 1, 204-246 (2001; Zbl 0999.17026)] have used Bernstein-Gelfand-Ponomarev reflection functors in order to derive the fundamental properties of Lusztig's symmetries on the Drinfeld double of the composition subalgebra of a Ringel-Hall algebra of \(\Lambda\) augmented by a torus algebra. In the paper under review the authors prove that the Bernstein-Gelfand-Ponomarev reflection functors considered as operators on \({\mathcal D}(\Lambda)\) as well as operators on integrable highest weight \({\mathcal D}(\Lambda)\)-modules coincide with Lusztig's symmetries up to a canonical isomorphism and satisfy the braid group relations. This answers a question of \textit{B. Sevenhant} and \textit{M. Van den Bergh} [op. cit.].
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1868010
zbMATH Keywords: Hopf algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: integrable highest weight modules
zbMATH Keywords: Drinfeld doubles
zbMATH Keywords: Ringel-Hall algebras
zbMATH Keywords: finite-dimensional hereditary algebras
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: generalized Kac-Moody algebras
zbMATH Keywords: Lusztig symmetries
zbMATH Keywords: Bernstein-Gelfand-Ponomarev reflection functors
zbMATH Keywords: braid group relations
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00123-0
OpenAlex ID: W1965017853
description: scientific article; zbMATH DE number 1868010