Special irreducible representations of Leavitt path algebras

Several descriptions of irreducible representations of both Leavitt and hence Cohn path algebras of an arbitrary digraph with coefficients in a commutative field introduced by Chen and Rangaswamy are presented, using both infinite paths on the right and vertices as well as direct limits or factors of cyclic projective ideals of the ordinary quiver algebra. Specific properties of these irreducible representations become immediate when they are viewed as modules over the commutative subalgebras generated by symmetric idempotents of paths, thereby providing a unified way to treat them. Furthermore, their defining relations are read off, whence criteria are easily given when they are finitely presented or finite dimensional. Their endomorphism rings, and annihilator primitive ideals are also computed directly.