A characterization of algebras generated by idempotents
type: Item
instance of: scholarly article
title: A characterization of algebras generated by idempotents
zbMATH Open document ID: 1482.16029
author: Wei Hu
author: Zhankui Xiao
published in: Journal of Pure and Applied Algebra
publication date: 2021-06-09T00:00:00Z
review text: This paper is devoted to finding a method to determine whether an algebra is generated by idempotents. The main result of the paper is as follow. Theorem. Let \(A\) be a finite dimensional \(F\)-algebra. Then \(A\) is generated by idempotents if and only if \(\mathrm{dim}_FV>\dim_F\mathrm{End}_A(V)\) or \(\dim_F\mathrm{End}_A(V)=1\) for each irreducible \(A\)-module \(V\) and \(\mathrm{Ext}_A^1(S,S)=0\) for all \(1\)-dimensional \(A\)-modules \(S\). As consequences of the theorem, the authors also point out that many well-known algebras are generated by idempotents. For examples, let \(A\) be a finite dimensional splitting \(F\)-algebra. If \(A\) has a finite global dimension, then \(A\) is generated by idempotents. Moreover, if \(A\) is a quasi-hereditary algebra over a splitting field \(F\), then \(A\) is generated by idempotents. Particularly, \(q\)-Schur algebras and symplectic \(q\)-Schur algebras are generated by idempotents.
reviewed by: Huynh Viet Khanh
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 7357212
zbMATH Keywords: idempotent
zbMATH Keywords: zero action determined module
zbMATH Keywords: zero product determined algebra
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106693
OpenAlex ID: W3120876908
cites work: Generalized derivations on unital algebras determined by action on zero products.
cites work: Finite dimensional zero product determined algebras are generated by idempotents
cites work: Zero product determined matrix algebras
cites work: On bilinear maps on matrices with applications to commutativity preservers.
cites work: Maps characterized by action on zero products.
cites work: Derivations modulo elementary operators.
cites work: Q5340151
cites work: Jordan algebras and 3-transposition groups
cites work: Quasi-hereditary algebras
cites work: Universal axial algebras and a theorem of Sakuma
cites work: Miyamoto involutions in axial algebras of Jordan type half
cites work: Notes on the no loops conjecture
cites work: Algebras generated by two idempotents
cites work: Griess algebras and conformal vectors in vertex operator algebras
cites work: Associative and Jordan algebras generated by two idempotents
cites work: An introduction to homological algebra
cites work: Half-axes in power associative algebras
DOI: 10.1016/J.JPAA.2021.106693
description: scientific article; zbMATH DE number 7357212