Abstract. Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev [6] to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen [3]. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construct a Morita context between the partial smash product defined by the authors of [3] and the smash product related to the enveloping action. Third, we dualize the globalization theorem to partial coactions and finally, we define partial representations of Hopf algebras and show some results relating partial actions and partial representations.
In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebras. Examples are elaborated using algebraic groups.
The effectiveness of the aplication of constructions in G-graded k-categories to the computation of the fundamental group of a finite dimensional k-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a k-linear category, motivated the investigation of H-module categories, i.e., actions of a Hopf algebra H on a k-linear category. The G-graded case corresponds then to actions of the Hopf algebra k G on a k-linear category. In this work we take a step further and introduce partial H-module categories. We extend several results of partial H-module algebras to this context, such as the globalization theorem, the construction of the partial smash product and the Morita equivalence of this category with the smash product over a globalization. We also present a detailed description of partial actions of k G .
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