Quiver Algebras, Path Coalgebras and co-reflexivity

Abstract: We study the connection between two combinatorial notions associated to a quiver: the quiver algebra and the path coalgebra. We show that the quiver coalgebra can be recovered from the quiver algebra as a certain type of finite dual, and we show precisely when the path coalgebra is the classical finite dual of the quiver algebra, and when all finite dimensional quiver representations arise as comodules over the path coalgebra. We discuss when the quiver algebra can be recovered as the rational part of the dual of the path coalgebra. Similar results are obtained for incidence (co)algebras. We also study connections to the notion of coreflexive (co)algebras, and give a partial answer to an open problem concerning tensor products of coreflexive coalgebras.