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Galois Theory for Multiplier Hopf Algebras with Integrals

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Abstract

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.

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Authors and Affiliations

  1. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001, Heverlee, Belgium

    A. Van Daele & Y. H. Zhang

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  1. A. Van Daele
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  2. Y. H. Zhang
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Daele, A.V., Zhang, Y.H. Galois Theory for Multiplier Hopf Algebras with Integrals. Algebras and Representation Theory 2, 83–106 (1999). https://doi.org/10.1023/A:1009938708033

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