Irreducible representation-types of Leavitt path algebras

Abstract

Irreducible representations of both Leavitt and Cohn path algebras of an arbitrary digraph with coefficients in a commutative field is classified. They are constructed in several ways using both infinite paths on the right as well as direct limits or factors of one-sided projective ideals of the ordinary quiver algebra, respectively. Furthermore, their defining relations are described, too, whence criterions are easily given when they are finitely presented or finite dimensional. Moreover, their endomorphism rings, annihilator primitive ideals are also computed directly. In particular, the cardinality of the set of sinks or infinite emitters, respectively, is an invariant of Leavitt path algebras.