Bernstein graph algebras
type: Item
instance of: scholarly article
title: Bernstein graph algebras
zbMATH Open document ID: 1496.17025
author: Harold N. Ward
published in: Archiv der Mathematik
publication date: 2021-05-05T00:00:00Z
review text: A simply connected graph \(G=(V,E)\), with \(|V|\geq 3\), defines a Bernstein superalgebra over a field of characteristic distinct from two. Its basis is the disjoint union of \(V\) and \(E\). In addition, if \(\{x,y\}\) is the edge connecting two vertices \(x\) and \(y\), then the product of the algebra is described by bilinearity as \(x\{x,y\}=\{x,y\}x=y\), \(y\{x,y\}=\{x,y\}y=x\), \(xy=0\) and \(\{x,y\}\{u,v\}=0\), for all \(\{x,y\}\), \(\{u,v\}\in E\). Any other vertex-edge product is zero. \textit{A. Grishkov} and \textit{R. Costa} proved in [J. Math. Sci., New York 93, No. 6, 877--882 (1999; Zbl 0979.17012)] that graphs with isomorphic algebras are isomorphic. In their proof, they used techniques from algebraic groups, which restrict implicitly the base field. In this paper, this result is reproved for any base field of characteristic distinct from two. In addition, the automorphisms of the Bernstein superalgebra are described.
reviewed by: Raúl M. Falcón
Mathematics Subject Classification ID: 17D92
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 7343929
zbMATH Keywords: Bernstein algebra
zbMATH Keywords: graph
zbMATH Keywords: automorphism
MaRDI profile type: Publication
full work available at URL: s00013-020-01572-y
OpenAlex ID: W3131380124
cites work: A class of exceptional bernstein algebras associated to graphs1
cites work: Bernstein superalgebras and supermodules
cites work: Graphs and nonassociative algebras
cites work: Q3869125
DOI: 10.1007/S00013-020-01572-Y
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description: scientific article; zbMATH DE number 7343929