Semisimplifying Lie algebras of $J$-ternary algebras in characteristic $3$

TL;DR

This paper introduces a new class of Lie superalgebras in characteristic 3, constructed via tensor products of composition algebras and semisimplification functors, expanding understanding of their structure and forms.

Contribution

It describes a novel approach to constructing Lie superalgebras in characteristic 3 using tensor products and semisimplification, extending previous methods and including new examples.

Findings

Constructed Lie superalgebras containing Elduque-Cunha and Elduque superalgebras.

Applied semisimplification functor to relate these superalgebras to classical Lie algebras of types E6, E7, E8.

Extended the construction to Lie algebras from J-ternary algebras in characteristic 3.

Abstract

We describe a class of Lie superalgebras in characteristic $3$, containing the Elduque-Cunha superalgebras $\mathfrak{g}(3,3), \mathfrak{g}(6,6)$ and the Elduque superalgebra $\mathfrak{el}(5,3)$, using the tensor product of composition algebras. For the Lie superalgebra $\mathfrak{el}(5,3)$, this allows us to move beyond the contragredient construction and it also allows us to construct more general forms. We also describe how one obtains these Lie superalgebras using the semisimplification functor on the representation category $\mathbf{Rep}(\alpha_3)$ to Lie algebras of type $E_6, E_7$ and $E_8$, in line with how Arun Kannan applied this functor to the split algebras. We further apply this functor more broadly to the class of Lie algebras coming from $J$-ternary algebras over fields of characteristic $3$.