Computer-aided study of double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms in characteristic 2

TL;DR

This paper investigates double extensions of restricted Lie superalgebras in characteristic 2 that preserve non-degenerate closed 2-forms, using computational methods to classify and identify patterns in these algebraic structures.

Contribution

The paper provides a computational classification of double extensions of restricted Lie superalgebras in characteristic 2, revealing patterns and accelerating calculations through multigradings.

Findings

List of double extensions for 4 to 7 indeterminates

Identification of patterns in double extensions

Significant speed-up using multigradings

Abstract

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman's Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.