Leavitt path algebras with at most countably many irreducible representations

作者：Ara Pere^*; Rangaswamy Kulumani M

来源：Revista Matematica Iberoamericana, 2015, 31(4): 1263-1276.

DOI：10.4171/RMI/868

• Leavitt path algebra
• irreducible representation
• socle
• von Neumann regular

Let E be an arbitrary directed graph with no restrictions on the number of vertices and edges and let K be any field. We give necessary and sufficient conditions for the Leavitt path algebra L-K(E) to be of countable irreducible representation type, that is, we determine when L-K(E) has at most countably many distinct isomorphism classes of simple left L-K(E)-modules. It is also shown that L-K(E) has finitely many isomorphism classes of simple left modules if and only if L-K(E) is a semi-artinian von Neumann regular ring with finitely many ideals. Equivalent conditions on the graph E are also given. Examples are constructed showing that for each (finite or infinite) cardinal k there exists a Leavitt path algebra L-K(E) having exactly k distinct isomorphism classes of simple right modules.

• 出版日期2015