Computing Split Maximal Toral Subalgebras of Lie algebras over Fields of Small Characteristic

TL;DR

This paper introduces heuristic algorithms for computing split maximal toral subalgebras in Lie algebras over fields of characteristic 2 or 3, addressing limitations of previous methods in these cases.

Contribution

The paper presents new heuristic algorithms specifically designed for reductive Lie algebras over small characteristic fields, improving the computational tools available.

Findings

Algorithms successfully find split maximal toral subalgebras in characteristic 2 and 3

Enhances recognition of reductive Lie algebras over small characteristic fields

Addresses limitations of previous algorithms in these cases

Abstract

Important subalgebras of a Lie algebra of an algebraic group are its toral subalgebras, or equivalently (over fields of characteristic 0) its Cartan subalgebras. Of great importance among these are ones that are split: their action on the Lie algebra splits completely over the field of definition. While algorithms to compute split maximal toral subalgebras exist and have been implemented [Ryb07, CM09], these algorithms fail when the Lie algebra is defined over a field of characteristic 2 or 3. We present heuristic algorithms that, given a reductive Lie algebra L over a finite field of characteristic 2 or 3, find a split maximal toral subalgebra of L. Together with earlier work [CR09] these algorithms are very useful for the recognition of reductive Lie algebras over such fields.