Irreducible representation-types of Leavitt path algebras

TL;DR

This paper classifies all irreducible representations of Leavitt and Cohn path algebras associated with any digraph, describing their construction, relations, and invariants like endomorphism rings and primitive ideals.

Contribution

It provides a comprehensive classification of irreducible representations of Leavitt and Cohn path algebras, including criteria for finite presentation and dimension, and computes key invariants.

Findings

Irreducible representations are constructed via infinite paths and direct limits.

Relations and criteria for finite presentation and dimension are established.

Invariants such as the number of sinks and infinite emitters are identified.

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.