New Constructions of Exceptional Simple Lie Superalgebras with Integer Cartan Matrix in Characteristics 3 and 5 via Tensor Categories

TL;DR

This paper introduces new methods using tensor categories to construct exceptional simple Lie superalgebras with integer Cartan matrices in characteristics 3 and 5, expanding the classification of modular Lie superalgebras.

Contribution

It presents novel constructions of exceptional Lie superalgebras in characteristics 3 and 5 via tensor categories, based on the classification by Bouarroudj, Grozman, and Leites.

Findings

Constructed new exceptional Lie superalgebras in characteristic 3 and 5.

Realized superalgebras as images of Lie algebras with nilpotent derivations.

Connected the constructions to the Verlinde category and semisimplification functors.

Abstract

Using tensor categories, we present new constructions of several of the exceptional simple Lie superalgebras with integer Cartan matrix in characteristic $p = 3$ and $p = 5$ from the complete classification of modular Lie superalgebras with indecomposable Cartan matrix and their simple subquotients over algebraically closed fields by Bouarroudj, Grozman, and Leites in 2009. Specifically, let $\mathbf{\alpha}_p$ denote the kernel of the Frobenius endomorphism on the additive group scheme $\mathbb{G}_a$ over an algebraically closed field of characteristic $p$. The Verlinde category $\mathrm{Ver}_p$ is the semisimplification of the representation category $\mathrm{Rep} \ \mathbf{\alpha}_p$, and $\mathrm{Ver}_p$ contains the category of super vector spaces as a full subcategory. Each exceptional Lie superalgebra we construct is realized as the image of an exceptional Lie algebra equipped with a nilpotent derivation of order at most $p$ under the semisimplification functor from $\mathrm{Rep} \ \mathbf{\alpha}_p$ to $\mathrm{Ver}_p$.