On the double of the Hall algebra of a quiver
type: Item
instance of: scholarly article
title: On the double of the Hall algebra of a quiver
zbMATH Open document ID: 0955.16016
author: Bert Sevenhant
author: Michel Van den Bergh
published in: Journal of Algebra
publication date: 2000-11-29T00:00:00Z
review text: Let \(Q\) be a quiver and \(k\) be a finite field. Then Ringel defined the twisted Hall algebra \(H(Q)\) of \(Q\), and Green proved that \(H(Q)\) is a Hopf algebra in a certain twisted sense. Furthermore, one introduced certain Cartan elements to make \(H(Q)\) into a real Hopf algebra. In the paper under review the authors give a double construction of this Hopf algebra, and denote the double by \(U(Q)\). The main interest of the paper is the following: if two quivers \(Q\) and \(Q'\) have the same underlying graph, how are the two doubles related to each other? The authors show that one can use finitely many times the usual reflection functor associated to a source vertex to define an isomorphism between \(U(Q)\) and \(U(Q')\). Furthermore, the authors define an automorphism \(\widehat t_i\) of \(U(Q)\) for each vertex \(i\) in \(Q\) and show that the \(\widehat t_i\) restrict to the automorphism of the quantum enveloping algebra, which is Lusztig's braided group action. In fact, the authors prove that the \(\widehat t_i\) satisfy the braid relations for Dynkin cases on the whole of \(U(Q)\) instead of just for the quantum enveloping algebra. They conjecture also that this is true for all cases. Perhaps with the knowledge of representation theory of algebras and a careful analysis of the paper under review one might solve this conjecture.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1384442
zbMATH Keywords: twisted Hall algebras
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Fourier transforms
zbMATH Keywords: quivers
zbMATH Keywords: reflection functors
zbMATH Keywords: quantum enveloping algebras
zbMATH Keywords: doubles
reviewed by: Changchang Xi
MaRDI profile type: Publication
full work available at URL: jabr.1999.7958
OpenAlex ID: W2048435474
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DOI: 10.1006/JABR.1999.7958
description: scientific article; zbMATH DE number 1384442