The Tits construction for short $\mathfrak{sl}_2$-super-structures

TL;DR

This paper extends the Tits construction to short lf2 b5-super-structures, providing a unified framework for lf2 Lie superalgebras with lf2 b5-module decompositions.

Contribution

It generalizes previous Tits constructions for lf2 Lie superalgebras using lf2 b5-ternary superalgebras, extending the Tits-Kantor-Koecher and Tits-Allison-Gao frameworks.

Findings

Unified construction for lf2 b5-Lie superalgebras

Described lf2 b5-structures via lf2 b5-ternary superalgebras

Generalized previous models including those by Elduque et al. and Shang.

Abstract

In this paper, we generalize the Tits construction for Lie superalgebras such that $\mathfrak{sl}_2$ acts by even derivations and decompose, as $\mathfrak{sl}_2$-module, into a direct sum of copies of the adjoint, the natural and the trivial representations. This construction generalizes the one provided by Elduque et al in \cite{EBCC23}, and it is possible to described the $\mathfrak{sl}_2$-Lie superstructure in terms of $\mathcal{J}$-ternary superalgebras as a super version of the defined by Allison. We extend the Tits-Kantor-Koecher construction and the Tits-Allison-Gao functor that define a short $\mathfrak{sl}_2$-Lie superalgebra from a $\mathcal{J}$-ternary superalgebra $(\mathcal{J},\mathcal{M})$. Our setting includes and generalizes both \cite{EBCC23} and Shang's \cite{S22}.