The isomorphism of Hall algebras
type: Item
instance of: scholarly article
title: The isomorphism of Hall algebras
zbMATH Open document ID: 0836.16009
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: This article is a valuable contribution to the theory of Hall algebras, introduced and first studied by \textit{C. M. Ringel} [see: Topics in Algebra, Pt. 1, Banach Cent. Publ. 26, Part I, 433-447 (1990; Zbl 0778.16004), Adv. Math. 84, 137-178 (1990; Zbl 0799.16013) and Invent. Math. 101, 583-591 (1990; Zbl 0735.16009)]. The Hall algebra, \({\mathcal H}(R)\) of a finitary ring \(R\) has a basis formed by isoclasses of finite modules and a multiplication that, in a sense, reflects the number of modules with given two-story filtrations. The composition algebra of \(R\) is the subalgebra \({\mathcal C}(R)\) determined by finite simple \(R\)-modules. The aim of the paper is to give necessary/sufficient conditions for the isomorphism of Hall or composition algebras associated to rings \(R\), \(R'\). One main result is that, for representation directed algebras over finite fields, \(R\), \(R'\), \({\mathcal H}(R)\cong{\mathcal H}(R')\) if and only if there exists a bijection between the isoclasses of simple modules that preserves endomorphism rings and the first and second extension groups of each pair of simples.
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 614249
zbMATH Keywords: Hall algebras
zbMATH Keywords: isoclasses of finite modules
zbMATH Keywords: simple modules
zbMATH Keywords: composition algebras
zbMATH Keywords: representation directed algebras
zbMATH Keywords: endomorphism rings
zbMATH Keywords: extension groups
author: Jin Yun Guo
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
full work available at URL: jabr.1994.1139
OpenAlex ID: W1995526787
DOI: 10.1006/JABR.1994.1139
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description: scientific article; zbMATH DE number 614249