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 sl(m|n) with m≠n. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on sl(2|1), (ii) sl(m|1) does not admit left-symmetric superalgebras for m≥3, and (iii) sl(m+1|m) admits a left-symmetric superalgebra for every m≥1. To prove these results we combine previous results on the existence and classification of left-symmetric algebras on the Lie algebras glm with a detailed analysis of small representations of the Lie superalgebras sl(m|1). We also conjecture that sl(m|n) admits left-symmetric superalgebras if and only if m=n+1.