Socle theory for Leavitt path algebras of arbitrary graphs

TL;DR

This paper develops a socle theory for Leavitt path algebras of arbitrary graphs, characterizing the socle via line points and providing a concrete description as a direct sum of matrix rings, with simplified proofs of key theorems.

Contribution

It extends socle theory to all Leavitt path algebras, including non-row-finite graphs, and offers new, shorter proofs of fundamental theorems.

Findings

Leavitt path algebras with nonzero socle have line points.

The socle is generated by line points.

The socle is a direct sum of matrix rings over the base field.

Abstract

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra, extending so the results for row-finite graphs in the previous paper [12] (but with different methods). A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, shorthening significantly the original ones.