Socle theory for Leavitt path algebras of arbitrary graphs

2008/02/08 by Gonzalo Aranda Pino, Dolores Martı́n Barquero, Pino, Gonzalo Aranda +7 · 2 citations
Mathematics · #16D70 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16D70

paper · pdf · doi:10.48550/arxiv.0802.1198

20 pgs

arxiv created 2008/02/08 · openalex publication_date 2008/02/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• The Leavitt inverse semigroup of a separated graph
• Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras

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