Bernstein graph algebras
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.
scientific article; zbMATH DE number 7343929