Computing subalgebras and $\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields

TL;DR

This paper presents two new algorithms for computing subalgebras and $Z_2$-gradings of simple Lie algebras over finite fields, specifically applied to classify structures of small-dimensional algebras over $F_2$.

Contribution

It introduces novel algorithms for constructing $Z_2$-gradings and computing subalgebras of Lie algebras over finite fields, advancing classification methods.

Findings

All known simple Lie algebras of dimension ≤ 20 over $F_2$ have $Z_2$-gradings.

The algorithms successfully determine subalgebras and simple subquotients for these algebras.

The methods facilitate structural analysis of Lie algebras over finite fields.

Abstract

This paper introduces two new algorithms for Lie algebras over finite fields and applies them to the investigate the known simple Lie algebras of dimension at most $20$ over the field $\mathbb{F}_2$ with two elements. The first algorithm is a new approach towards the construction of $\mathbb{Z}_2$-gradings of a Lie algebra over a finite field of characteristic $2$. Using this, we observe that each of the known simple Lie algebras of dimension at most $20$ over $\mathbb{F}_2$ has a $\mathbb{Z}_2$-grading and we determine the associated simple Lie superalgebras. The second algorithm allows us to compute all subalgebras of a Lie algebra over a finite field. We apply this to compute the subalgebras, the maximal subalgebras and the simple subquotients of the known simple Lie algebras of dimension at most $16$ over $\mathbb{F}_2$ (with the exception of the $15$-dimensional Zassenhaus algebra).