Semisimplifying Frank Lie algebras

TL;DR

This paper explores the semisimplification of Frank Lie algebras in characteristic 3, revealing new modular contact Lie superalgebras and simple J-ternary algebras with non-simple Jordan algebras.

Contribution

It introduces a novel application of the semisimplification functor to Frank Lie algebras, resulting in the discovery of new algebraic structures.

Findings

Semisimplification yields modular contact Lie superalgebras.

Identifies simple J-ternary algebras with non-simple Jordan algebras.

Extends understanding of Lie algebra structures in characteristic 3.

Abstract

The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category $\textbf{Rep}(\alpha_3)$. We apply the semisimplification functor to these Frank Lie algebras and obtain modular contact Lie superalgebras. We also obtain a class of simple $J$-ternary algebras whose associated Jordan algebras are not simple.