Actions of monoidal categories and generalized Hopf smash products

[摘要] Let R be a k-algebra, and C a monoidal category. Assume given the structure of a C-category on the category M-R of left R-modules; that is, the monoidal category C is assumed to act on the category M-R by a coherently associative bifunctor lozenge: C x M-R --> M-R. 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 lozenge R (respectively an R-coring C lozenge 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. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-12-15