Quiver Algebras, Path Coalgebras and co-reflexivity

TL;DR

This paper explores the relationship between quiver algebras and path coalgebras, identifying conditions under which they are duals and how they relate to finite dimensional representations and coreflexivity.

Contribution

It provides new criteria for when the path coalgebra is the finite dual of the quiver algebra and analyzes the coreflexivity of related coalgebras.

Findings

Quiver coalgebra can be recovered as a finite dual of the quiver algebra.

Conditions are established for the path coalgebra to be the classical finite dual.

Results connect coreflexivity with tensor products of coalgebras.

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.