Two-generation of traceless matrices over finite fields

TL;DR

The paper resolves the 2-generation problem for traceless matrices by showing $\mathfrak{sl}_n(\mathbf{F}_q)$ is generated by two elements for all $(n,p)$ except the exceptional pairs $(3,3)$ and $(4,2)$, where obstructions force at least three generators. The authors build two complementary strategies: in the bounded-rank regime they use consistently chosen diagonal generators paired with $\mathbf{1}$, while in the unbounded-rank regime they exploit companion or normal/sharply-traceless elements whose Galois-conjugate roots form consistent sets, enabling 2-generation after passing to an extension and pulling back. Even characteristic requires semiconsistent matrices, and together with detailed analysis of exceptional cases, they establish sharp boundaries: $2$-generation holds broadly, with explicit obstruction identities in the two exceptional cases. The results connect finite-field methods, Galois theory, and Lie-polynomial generation, contributing new examples where simple Lie algebras over infinite fields fail to be 2-generated, while offering concrete, constructive generation proofs in the general finite-field setting.

Abstract

We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we establish curious identities that obstruct $2$-generation.

Theorems & Definitions (31)

• Theorem
• Proposition 2.1: bois2009generators, chistopolskaya2018nilpotent
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Example 4.1
• Lemma 4.2
• proof