# Socle theory for Leavitt path algebras of arbitrary graphs

### Gonzalo Aranda Pino

Universidad de Málaga, Spain### Dolores Martín Barquero

Universidad de Málaga, Spain### Cándido Martín González

Universidad de Málaga, Spain### Mercedes Siles Molina

Universidad de Málaga, Spain

## 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. 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, by using very different means.

## Cite this article

Gonzalo Aranda Pino, Dolores Martín Barquero, Cándido Martín González, Mercedes Siles Molina, Socle theory for Leavitt path algebras of arbitrary graphs. Rev. Mat. Iberoam. 26 (2010), no. 2, pp. 611–638

DOI 10.4171/RMI/611