Simple Lie algebras arising from Steinberg algebras of Hausdorff ample groupoids

TL;DR

This paper characterizes when Steinberg algebras of Hausdorff ample groupoids produce simple Lie algebras, providing criteria for simplicity and analyzing related algebraic structures like Leavitt path and Exel-Pardo algebras.

Contribution

It identifies conditions under which Steinberg algebras yield simple Lie algebras and applies these results to various classes of algebras, offering practical criteria for simplicity.

Findings

Unital simple Steinberg algebras are central.

Nonunital simple Steinberg algebras have zero center.

Criteria for simplicity of Lie algebras from Leavitt path and Exel-Pardo algebras.

Abstract

In this paper, we show that a unital simple Steinberg algebra is central, and a nonunital simple Steinberg algebra has zero center. We identify the fields $K$ and Hausdorff ample groupoids $\mathcal{G}$ for which the simple Steinberg algebra $A_K(\mathcal{G})$ yields a simple Lie algebra $[A_K(\mathcal{G}), A_K(\mathcal{G})]$. We apply the obtained results on simple Leavitt path algebras, simple Kumjian-Pask algebras and simple Exel-Pardo algebras to determine their associated Lie algebras are simple. In particular, we give easily computable criteria to determine which Lie algebras of the form $[L_K(E), L_K(E)]$ are simple, when $E$ is an arbitrary graph and the Leavitt path algebra $L_K(E)$ is simple. Also, we obtain that unital simple Exel-Pardo algebras are central, and nonunital simple Exel-Pardo algebras have zero center.