Simple Lie algebras arising from Steinberg algebras of Hausdorff ample groupoids

Abstract

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