Generators of simple modular Lie superalgebras

TL;DR

This paper proves that most finite-dimensional simple graded Lie superalgebras of Cartan type over fields with characteristic greater than 3 can be generated by one or two elements, with specific exceptions.

Contribution

It establishes minimal generating sets for a broad class of simple Lie superalgebras, including Cartan and classical types, highlighting cases requiring two generators.

Findings

Most Cartan type superalgebras are generated by one element.

Exceptions W, HO, KO, SKO require two generators.

Certain classical Lie superalgebras can also be generated by one or two elements.

Abstract

Let $X$ be one of the finite-dimensional simple graded Lie superalgebras of Cartan type $W, S, H, K, HO, KO, SHO$ or $SKO$ over an algebraically closed field of characteristic $p>3$. In this paper we prove that $X$ can be generated by one element except the ones of type $W,$ $HO$, $KO$ or $SKO$ in certain exceptional cases, in which $X$ can be generated by two elements. As a subsidiary result, we also prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.