On converse of the Schur's theorem for nilpotent Lie superalgebras

TL;DR

This paper explores a converse to Schur's theorem within the context of finite-dimensional nilpotent Lie superalgebras, introducing a new invariant to classify their structures based on specific superdimension conditions.

Contribution

It introduces the invariant st(L) and classifies nilpotent Lie superalgebras for particular values of this invariant, extending understanding of their structure.

Findings

Classification of nilpotent Lie superalgebras for st(L) in specified sets.

Introduction of the invariant st(L) for structural analysis.

Extension of Schur's theorem to Lie superalgebras.

Abstract

In this paper, we establish a converse to Schur's theorem for Lie superalgebras \( L \), focusing on cases where the minimal generator number pairs \((p \vert q)\) of \( L/Z(L) \) are considered, and where the superdimension \( \mathrm{sdim} L^{2} \) is finite. We introduce a new invariant \( st(L) \), which plays a key role in the classification of finite-dimensional nilpotent Lie superalgebras. Specifically, we classify the structure of all such Lie superalgebras \( L \) when \( st(L) \in \{(0,0), (1,0), (0,1), (2,0), (0,2), (1,1)\} \).