A proof of the Popov conjecture for quivers.
type: Item
instance of: scholarly article
title: A proof of the Popov conjecture for quivers.
zbMATH Open document ID: 1078.14066
DOI: 10.1016/S0021-8693(03)00430-7
author: Geert Van de Weyer
published in: Journal of Algebra
publication date: 2004-05-27T00:00:00Z
review text: Let \(G\) be a connected reductive algebraic group over an algebraically closed field of characteristic zero, and let \(V\) be a finite dimensional algebraic \(G\)-module. Let \(\pi: V\rightarrow V/\!\!/G\) be the corresponding categorical quotient. The Popov Conjecture [formulated in \textit{V.~L.~Popov}, Funct. Anal. Appl. 10, 242--244 (1976; Zbl 0365.20053)] claims that if \(\pi\) is equidimensional, then \(V/\!\!/G\) is an affine space. The author proves this conjecture in the case where \(V\) is the space of \(\alpha\)-dimensional representation of a quiver \(Q\) on which \(G=\text{GL}_\alpha\) acts by base-change (in this case \(V/\!\!/G\) is the space of isomorphism classes of semisimple representations of \(Q\) of dimension \(\alpha\)).
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 2066838
zbMATH Keywords: reductive group
zbMATH Keywords: quotient variety
zbMATH Keywords: equidimensional morphism
Wikidata QID: Q123352761
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
cites work: Representations with a free module of covariants
cites work: Smooth quiver representation spaces.
cites work: Q4774141
cites work: Semisimple Representations of Quivers
cites work: A Proof of the Popov Conjecture for Tori
cites work: Q3686804
cites work: Q4143433
full work available at URL: s0021-8693(03)00430-7
OpenAlex ID: W2015533415
description: scientific article; zbMATH DE number 2066838