Actions of monoidal categories and generalized Hopf smash products

Abstract

Let R be a k-algebra, and C a monoidal category. Assume given the structure of a C-category on the category RM of left R-modules; that is, the monoidal category C is assumed to act on the category RM by a coherently associative bifunctor ♢:C×RM→RM. 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 C gives rise to an R-ring A♢R (respectively an R-coring C♢R) whose modules (respectively comodules) are the A-modules (respectively C-comodules) within the category RM. 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.