Simple Lie algebras arising from Leavitt path algebras

TL;DR

This paper investigates the structure of Lie algebras derived from Leavitt path algebras associated with directed graphs, providing criteria for their simplicity in specific cases.

Contribution

It offers new, easily computable conditions to determine when these Lie algebras are simple, focusing on row-finite graphs with simple Leavitt path algebras.

Findings

Criteria for simplicity of Lie algebras [L_K(E), L_K(E)]

Conditions for elements in the commutator subspace

Analysis specific to row-finite graphs with simple Leavitt path algebras

Abstract

For a field K and directed graph E, we analyze those elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E), L_K(E)]. This analysis allows us to give easily computable necessary and sufficient conditions to determine which Lie algebras of the form [L_K(E), L_K(E)] are simple, when E is row-finite (i.e., has finite out-degree) and L_K(E) is simple.