Partial Coactions of Weak Hopf Algebras on Coalgebras

TL;DR

This paper introduces the concept of partial coactions of weak Hopf algebras on coalgebras, explores their properties, provides examples, and investigates how to derive partial coactions from global ones, along with related algebraic structures.

Contribution

It defines partial coactions of weak Hopf algebras on coalgebras, provides examples, and studies their relationship with global coactions and related algebraic constructions.

Findings

Partial coactions generalize global coactions of weak Hopf algebras.

Conditions for constructing weak Hopf algebra structures from coproducts are identified.

A duality between partial actions and coactions is established.

Abstract

It will be seen that if $H$ is a weak Hopf algebra in the definition of coaction of weak bialgebras on coalgebras \cite{Wang}, then a definition property is suppressed giving rise to the (global) coactions of weak Hopf algebras on coalgebras. The next step will be introduce the more general notion of partial coactions of weak Hopf algebras on coalgebras as well as a family of examples via a fixed element on the weak Hopf algebra, illustrating both definitions: global and partial. Moreover, it will also be presented how to obtain a partial comodule coalgebra from a global one via projections, giving another way to find examples of partial coactions of weak Hopf algebras on coalgebras. In addition, the weak smash coproduct \cite{Wang} will be studied and it will be seen under what conditions it is possible to generate a weak Hopf algebra structure from the coproduct and the counit defined on it. Finally, a dual relationship between the structures of partial action and partial coaction of a weak Hopf algebra on a coalgebra will be established.