Leavitt path algebras with at most countably many irreducible representations
Pere AraUniversitat Autònoma de Barcelona, Bellaterra, Spain
Kulumani M. RangaswamyUniversity of Colorado at Colorado Springs, USA
Let be an arbitrary directed graph with no restrictions on the number of vertices and edges and let be any field. We give necessary and sufficient conditions for the Leavitt path algebra to be of countable irreducible representation type, that is, we determine when has at most countably many distinct isomorphism classes of simple left -modules. It is also shown that has finitely many isomorphism classes of simple left modules if and only if is a semi-artinian von Neumann regular ring with finitely many ideals. Equivalent conditions on the graph are also given. Examples are constructed showing that for each (finite or infinite) cardinal there exists a Leavitt path algebra having exactly distinct isomorphism classes of simple right modules.
Cite this article
Pere Ara, Kulumani M. Rangaswamy, Leavitt path algebras with at most countably many irreducible representations. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1263–1276DOI 10.4171/RMI/868