Partial actions of weak Hopf algebras: smash products, globalization and Morita theory
arXiv:1409.0232 · doi:10.1016/j.jpaa.2015.05.031
Abstract
In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action ([2],[3],[9]) and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as, we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show that there is a bijective correspondence between globalizable partial groupoid actions and symmetric partial groupoid algebra actions, extending similar result for group actions [9]. Moreover, as an application we give a complete description of all partial actions of a weak Hopf algebra on its ground field, which suggests a method to construct more general examples.
27 pages
Cited by in corpus (5)
- Partial actions of C*-quantum groups I: Restriction and Globalization
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- Partial Hopf-Galois theory
- Classifying partial (co)actions of Taft and Nichols Hopf algebras on their base fields
- Partial Actions of Weak Hopf Algebras on Coalgebras