Multiplier Hopf algebras: Globalization for partial actions

Abstract

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.