Semiperfect and coreflexive coalgebras

TL;DR

This paper develops a duality theory for non-unital coalgebras and introduces a new notion of left coreflexivity, linking it to semiperfectness and applying it to quantum groups and dual objects.

Contribution

It introduces the finite dual of a non-unital algebra and a new left coreflexivity concept for coalgebras, extending duality theory beyond unital cases.

Findings

Left semiperfectness is equivalent to left coreflexivity.

Hopf algebras with non-zero integrals are coreflexive.

Generalizes duality relations for quiver and incidence algebras.

Abstract

We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we introduce a new notion of left coreflexivity for counital coalgebras, namely, a coalgebra is left coreflexive if $C$ is isomorphic canonically to the finite dual of its left rational dual $Rat(_{C^*}C^*)$. We show that right semiperfectness for coalgebras is in fact essentially equivalent to this left reflexivity condition, and we give the connection to usual coreflexivity. As application, we give a generalization of some recent results connecting dual objects such as quiver or incidence algebras and coalgebras, and show that Hopf algebras with non-zero integrals (compact quantum groups) are coreflexive.