Partial corepresentations of Hopf Algebras

TL;DR

This paper introduces the concept of partial corepresentations of Hopf algebras, establishing a universal coalgebra framework and connecting it to Hopf coalgebroids, expanding the understanding of partial symmetries in algebraic structures.

Contribution

It defines partial corepresentations and constructs a universal coalgebra H^{par} with a Hopf coalgebroid structure, linking partial comodules to a new algebraic framework.

Findings

Existence of a universal coalgebra H^{par} for partial H-comodules

Category of regular partial H-comodules is isomorphic to H^{par}-comodules

H^{par} has a Hopf coalgebroid structure over a suitable coalgebra

Abstract

We introduce the notion of a partial corepresentation of a given Hopf algebra $H$ over a coalgebra $C$ and the closely related concept of a partial $H$-comodule. We prove that there exists a universal coalgebra $H^{par}$, associated to the original Hopf algebra $H$, such that the category of regular partial $H$-comodules is isomorphic to the category of $H^{par}$-comodules. We introduce the notion of a Hopf coalgebroid and show that the universal coalgebra $H^{par}$ has the structure of a Hopf coalgebroid over a suitable coalgebra.