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On generalized partial twisted smash products. (English) Zbl 1338.16036

Partial group actions were first defined by Exel in the context of operator algebras and they turned out to be a powerful tool in the study of \(C^*\)-algebras generated by partial isometries on a Hilbert space. The definitions of partial Hopf actions and coactions were introduced by S. Caenepeel and K. Janssen [see Commun. Algebra 36, No. 8, 2923-2946 (2008; Zbl 1168.16021)], using the notion of partial entwining structures. In the same article, the authors also introduced the concept of partial smash product. In the Hopf algebra setting, the theory of partial Hopf actions was done by M. M. S. Alves and E. Batista [see Commun. Algebra 38, No. 8, 2872-2902 (2010; Zbl 1226.16022)]. They also constructed a Morita context relating the fixed point subalgebra for partial actions of finite dimensional Hopf algebras, and constructed the partial smash product.
In the paper under review, the notions of a right generalized partial smash product and a generalized partial twisted smash product are introduced. Then the author discusses some properties of such partial smash products. A condition under which generalized partial twisted smash product is a Hopf algebra is given and a Morita context for partial coactions of a co-Frobenius Hopf algebra is established.

MSC:

16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions
16T15 Coalgebras and comodules; corings
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References:

[1] M. M. S. Alves, E. Batista: Enveloping actions for partial Hopf actions. Commun. Algebra 38 (2010), 2872–2902. · Zbl 1226.16022
[2] M. M. S. Alves, E. Batista: Globalization theorems for partial Hopf (co)actions, and some of their applications. Groups, Algebras and Applications. Proceedings of XVIII Latin American algebra colloquium, São Pedro, Brazil, 2009 (C. Polcino Milies, ed.). Contemporary Mathematics 537, American Mathematical Society, Providence, 2011, pp. 13–30.
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