On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

TL;DR

This paper classifies modulo 2 gradings of simple Lie algebras in characteristic 2, revealing new superalgebras and a non-trivial deformation, which aids in understanding Lie superalgebra structures.

Contribution

It provides the first complete classification of modulo 2 gradings for several series of simple Lie algebras in characteristic 2, including new superalgebras and deformation results.

Findings

Complete classification of gradings for special linear and orthogonal series.

Identification of a parametric family of gradings for Zassenhaus algebras.

Discovery of a non-trivial deformation of a 3|2-dimensional Lie superalgebra.

Abstract

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_\Pi(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).