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Partial actions of weak Hopf algebras: smash product, globalization and Morita theory. (English) Zbl 1331.16024

In the paper under review the authors introduce the notion of partial action of a weak Hopf algebra on an algebra. This notion unifies the notions of partial group action, partial Hopf action and partial groupoid action, introduced, respectively, in the work of M. Dokuchaev and R. Exel [Trans. Am. Math. Soc. 357, No. 5, 1931-1952 (2005; Zbl 1072.16025)], S. Caenepeel and K. Janssen [Commun. Algebra 36, No. 8, 2923-2946 (2008; Zbl 1168.16021)] and D. Bagio and A. Paques [Commun. Algebra 40, No. 10, 3658-3678 (2012; Zbl 1266.16045)].
It is shown that every partial module algebra over a weak Hopf algebra has a globalization, or enveloping action, extending results on partial Hopf (co)actions by M. M. S. Alves and E. Batista [Commun. Algebra 38, No. 8, 2872-2902 (2010; Zbl 1226.16022); Contemp. Math. 537, 13-30 (2011; Zbl 1232.16020)]. Partial smash products are also studied and connected with global smash products by means of a surjective Morita context. As an application, a description of all partial actions of a weak Hopf algebra on its ground field is obtained.

MSC:

16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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References:

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