A Prolog assisted search for new simple Lie algebras

TL;DR

This paper uses Prolog-based constraint logic programming to discover new simple Lie algebras over GF(2), confirming conjectures, identifying new families, and classifying sporadic examples in dimension 31.

Contribution

It introduces a novel computational approach with Prolog to find and classify simple Lie algebras over GF(2), including new families and sporadic cases.

Findings

Confirmed a conjecture by Grishkov et al.

Discovered two new infinite families of simple Lie algebras.

Identified seven new sporadic simple Lie algebras in dimension 31.

Abstract

We describe some recent computer investigations with the `Constraint Logic Programming over Finite Domains' -- CLP(FD) -- library in the Prolog programming environment to search for new simple Lie algebras over the field $\GF(2)$ of $2$ elements. Motivated by a paper of Grishkov et. al., we specifically look for those with a `thin decomposition', and we settle one of their conjectures. We extrapolate from our results the existence of two new infinite families of simple Lie algebras, in addition to finding seven new sporadic examples in dimension $31$. We also better contextualise some previously discovered simple algebras, putting them into families which do not seem to have ever appeared in the literature, and give an updated table of those currently known.