Actions of monoidal categories and generalized Hopf smash products

✍ Scribed by Peter Schauenburg

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 526 KB
• Volume: 270
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00403-4

✦ Synopsis

Let R be a k-algebra, and C a monoidal category. Assume given the structure of a C-category on the category R M of left R-modules; that is, the monoidal category C is assumed to act on the category R M by a coherently associative bifunctor ♦ : C × R M → R M. 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 R M. 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.