The classification of Leibniz superalgebras of nilindex n+m (m\neq 0)

TL;DR

This paper completes the classification of complex Leibniz superalgebras with nilindex n+m by analyzing cases with specific characteristic sequences and showing certain configurations have lower nilindex.

Contribution

It extends previous classifications by proving superalgebras with certain characteristic sequences have nilindex less than n+m, completing the overall classification.

Findings

Superalgebras with characteristic sequence where n_1 ≤ n-2 have nilindex less than n+m.

Superalgebras with characteristic sequence where m_1 ≤ m-1 have nilindex less than n+m.

The classification of Leibniz superalgebras with nilindex n+m is now complete.

Abstract

In this paper we investigate the description of the complex Leibniz superalgebras with nilindex n+m, where n and m ($m\neq 0$) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to $(n_1, ..., n_k | m_1, ..., m_s)$ (where $n_1+... +n_k=n, m_1+ ... + m_s=m$) for $n_1\geq n-1$ and $(n_1, ..., n_k | m)$ were classified in works \cite{FilSup}--\cite{C-G-O-Kh1}. Here we prove that in the case of $(n_1, ..., n_k| m_1, ..., m_s)$, where $n_1\leq n-2$ and $m_1 \leq m-1$ the Leibniz superalgebras have nilindex less than n+m. Thus, we complete the classification of Leibniz superalgebras with nilindex n+m.