Revista Matemática Iberoamericana


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Volume 26, Issue 2, 2010, pp. 611–638
DOI: 10.4171/RMI/611

Socle theory for Leavitt path algebras of arbitrary graphs

Gonzalo Aranda Pino[1], Dolores Martín Barquero[2], Cándido Martín González[3] and Mercedes Siles Molina[4]

(1) Departamento de Algebra, Geometría y Topología, Universidad de Málaga, 29071, MÁLAGA, SPAIN
(2) Departamento de Matemática Aplicada, Universidad de Málaga, 29071, MÁLAGA, SPAIN
(3) Departamento de Algebra, Geometría y Topología, Universidad de Málaga, 29071, MÁLAGA, SPAIN
(4) Departamento de Algebra, Geometría y Topología, Universidad de Málaga, 29071, MÁLAGA, SPAIN

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.

Keywords: Leavitt path algebra, graph C*-algebra, socle, arbitrary graph, minimal left ideal

Aranda Pino G, Martín Barquero D, Martín González C, Siles Molina M. Socle theory for Leavitt path algebras of arbitrary graphs. Rev. Mat. Iberoamericana 26 (2010), 611-638. doi: 10.4171/RMI/611