Left-symmetric Structures on Complex Simple Lie Superalgebras

TL;DR

This paper investigates the existence and classification of compatible left-symmetric superalgebra structures on complex simple Lie superalgebras, revealing that such structures are rare and classifiable in specific cases.

Contribution

It demonstrates the non-existence of compatible LSSA structures on most complex simple Lie superalgebras and classifies all such structures with a right-identity on A(0,1).

Findings

No compatible LSSA on most complex simple Lie superalgebras.

Existence of compatible LSSA only on A(m,n) with m≠n and W(n) for n≥3.

Complete classification of compatible LSSAs with right-identity on A(0,1).

Abstract

A well-known fact is that there does not exist any compatible left-symmetric structures on a finite-dimensional complex semisimple Lie algebra (see \cite{Chu1974}). This result is not valid in semisimple Lie superalgebra case. In this paper, we study the compatible Left-symmetric superalgebra (LSSA for short) structures on complex simple Lie superalgebras. We prove that there is not any compatible LSSA structure on a finite-dimensional complex simple Lie superalgebra except for the classical simple Lie superalgebra $A(m,n)(m\neq n)$ and Cartan simple Lie superalgebra $W(n)(n\geq 3)$. We also classify all compatible LSSAs with a right-identity on A(0,1).