Commutator Leavitt path algebras

TL;DR

This paper characterizes the elements in the commutator subspace of Leavitt path algebras over any field, classifies those equal to their own commutator subspace, and explores their ideal structures.

Contribution

It provides a complete description of commutator elements, classifies Leavitt path algebras equal to their commutator subspace, and investigates their Lie ideal properties.

Findings

Identified elements in the commutator subspace of Leavitt path algebras.

Classified all Leavitt path algebras satisfying L_K(E)=[L_K(E)].

Showed that all Lie ideals are ring-theoretic ideals in these algebras.

Abstract

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures.