Algorithmic construction of Chevalley bases

TL;DR

This paper introduces an efficient algorithm for constructing Chevalley bases in Chevalley Lie algebras over finite fields, crucial for recognizing certain groups of Lie type, with complexity close to optimal.

Contribution

It provides a novel algorithm for Chevalley basis construction applicable to all Chevalley Lie algebras over finite fields, improving recognition algorithms for Lie-type groups.

Findings

Algorithm works for characteristic at least 5

Complexity involves the 7th power of Lie rank

Likely near optimal complexity

Abstract

We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic at least 5, our algorithm has complexity involving the 7th power of the Lie rank, which is likely to be close to best possible.