Left-symmetric Superalgebras on Special Linear Lie Superalgebras

Abstract

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras $\mathfrak{sl}(m|n)$ with $m\ne n$. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on $\mathfrak{sl}(2|1)$, (ii) $\mathfrak{sl}(m|1)$ does not admit left-symmetric superalgebras for $m\ge 3$, and (iii) $\mathfrak{sl}(m+1|m)$ admits a left-symmetric superalgebra for every $m\ge 1$. To prove these results we combine existing results on the existence and classification of left-symmetric algebras on the Lie algebras ${\mathfrak{gl}}_m$ with a detailed analysis of small representations of the Lie superalgebras $\mathfrak{sl}(m|1)$. We also conjecture that $\mathfrak{sl}(m|n)$ admits left-symmetric superalgebras if and only if $m=n+1$.

24 pages, 3 tables

References in corpus (1)

• Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras

Cited by in corpus (2)

• Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2
• Cohomology of left-symmetric color algebras