Relative invariants for representations of finite dimensional algebras
type: Item
instance of: scholarly article
title: Relative invariants for representations of finite dimensional algebras
zbMATH Open document ID: 1031.16014
published in: Manuscripta Mathematica
publication date: 2002-09-17T00:00:00Z
review text: The author considers a finite dimensional basic algebra \(\Lambda\) over a field \(K\) of characteristic 0. For a fixed dimension vector \(\beta\) one defines the representation space \(R(\Lambda,\beta)\) and a group \(\text{GL}(\beta)\) such that the \(\text{GL}(\beta)\)-orbits in \(R(\Lambda,\beta)\) correspond to the isomorphy classes of \(\beta\)-dimensional modules. A relative invariant is a polynomial function on \(R(\Lambda,\beta)\) on which \(\text{GL}(\beta)\) acts by a character. Using a construction of \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 385-395 (1991; Zbl 0779.16005)] one can associate relative invariants to finite dimensional \(\Lambda\)-modules if \(\Lambda\) is a path algebra of a quiver. \textit{J. Weyman} and the reviewer [J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)] showed that Schofield's relative invariants span the space of relative invariants for quivers without oriented cycles. \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)] generalized this result to arbitrary quivers. In this paper, the author generalizes Schofield's construction to arbitrary finite dimensional basic algebras. The relative invariants obtained this way will again span the space of relative invariants. To get algebra generators of the algebra of relative invariants it suffices to take the relative invariants corresponding to indecomposable representations. A similar result was obtained by \textit{J. Weyman} and the reviewer [J. Algebra 258, No. 1, 216-227 (2002; Zbl 1048.16005)] in the case where \(\Lambda\) is the quotient of a finite dimensional path algebra.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16W22
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 16R30
zbMATH DE Number: 1801621
zbMATH Keywords: quivers
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: relative invariants
zbMATH Keywords: path algebras
zbMATH Keywords: indecomposable representations
author: Matyas Domokos
reviewed by: Harm Derksen
MaRDI profile type: Publication
full work available at URL: s002290200258
OpenAlex ID: W1967945316
DOI: 10.1007/S002290200258
description: scientific article; zbMATH DE number 1801621