Actions of monoidal categories and generalized Hopf smash products |
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Authors: | Peter Schauenburg |
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Institution: | Mathematisches Institut der Universität München, Theresienstr. 39, 80333, München, Germany |
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Abstract: | Let R be a k-algebra, and
a monoidal category. Assume given the structure of a
-category on the category
of left R-modules; that is, the monoidal category
is assumed to act on the category
by a coherently associative bifunctor
. We assume that this bifunctor is right exact in its right argument. In this setup we show that every algebra A (respectively coalgebra C) in
gives rise to an R-ring AR (respectively an R-coring CR) whose modules (respectively comodules) are the A-modules (respectively C-comodules) within the category
. We show that this very general scheme for constructing (co)associative (co)rings gives conceptual explanations for the double of a quasi-Hopf algebra as well as certain doubles of Hopf algebras in braided categories, each time avoiding ad hoc computations showing associativity. |
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Keywords: | |
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