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Directed graphs and Lie superalgebras of matrices. (English) Zbl 1064.17015

For a finite directed graph \(D=(V,E)\) without multiple edges but possibly with loops, with set of vertices \(V=\{1,\ldots,n\}\), one considers the vector subspace \(L\) of the \(n\times n\) matrix algebra over a field with basis \(\{e_{ij}\mid (i,j)\in E\}\). In the paper under review the author studies the following problem (related to the more general problem of characterizing gradings of matrix algebras): Let us define a mapping \(\alpha:E\to {\mathbb Z}_2=\{0,1\}\). It induces a \({\mathbb Z}_2\)-grading of \(L(D,\alpha)=L=L_0\oplus L_1\). Describe the graphs \(D\) and the mappings \(\alpha\) such that \(L(D,\alpha)\) is a Lie superalgebra.
The main result of the paper gives a complete answer. Namely, \(L(D,\alpha)\) being a Lie superalgebra is equivalent to the condition that it is an associative superalgebra with respect to the matrix multiplication. This happens if and only if the set of edges \(E\) is transitive, i.e. \((i,j),(j,k)\in E\) implies \((i,k)\in E\), and \(\alpha\) is a homomorphism, i.e. \(\alpha(i,j)+\alpha(j,k)=\alpha(i,k)\). It turns out that if \(E\) is transitive, then there exists a subset \(B\) of \(E\) (called a superbasis) such that every mapping \(B\to {\mathbb Z}_2\) induces a superstructure on \(L\).

MSC:

17B70 Graded Lie (super)algebras
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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References:

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