arXiv subjects
Publications and source records attributed to Eneilson Fontes.
In this work we define partial (co)actions on multiplier Hopf algebras, we also present examples and properties. From a partial comodule coalgebra we construct a partial smash coproduct generalizing the constructions made by the L. Delvaux, E. Batista and J. Vercruysse.
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.
In partial action theory, a pertinent question is whenever given a partial (co)action of a Hopf algebra A on an algebra R, it is possible to construct an enveloping (co)action. The authors Alves and Batista, in [2],have shown that this is always possible if R has unit. We are interested in investigating the situation where both algebras A and R are nonunitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [11], which is called multiplier Hopf algebra. Therefore, we will consider partial (co)actions of multipliers Hopf algebras on algebras not necessarily unitary and we will present globalization theorems for these structures. Moreover, Dockuchaev, Del Rio and Simón, in [5], have shown when group partial actions on nonunitary algebras are globalizable. Based in [5], we will establish a bijection between group partial actions on an algebra R not necessarily unitary and partial actions of a multiplier Hopf algebra on the algebra R.