Random generation of the special linear Lie algebra over finite fields

Abstract

We prove that the special linear Lie algebra ${\mathfrak{sl}}_n({\text{F}}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|{\mathfrak{sl}}_n({\text{F}}_q)|$ tends to infinity, provided that $(n,p)\ne (3,3),(4,2)$.