Doi-Hopf data, smash data, Frobenius-types and functors.

摘要

In Chapters 1 and 2 we study Doi-Hopf data. Given a morphism ;In Chapter 2 we partially answer whether ;In Chapter 3 the main issue discussed is whether the induction functor is isomorphic to the coinduction functor for two smash data. We first consider the smash products of smash data and construct four functors from the morphism between two such data. Then we study the Frobenius data, which naturally generalize 2nd-Frobenius extensions in (NT1), (Ka) and (BF) and Frobenius morphisms in (MN). As applications, we give both necessary and sufficient conditions under which the answer to this question is affirmative (see Section 3.3), and then show some homological dimension relations between two smash data and the Imprimitivity theory for Hopf algebra actions, in particular, Theorem 8.3.3 of (Mo) is strengthened. Now it is clear that the important induction functor in (Don) or (Do3) and six in (FP) for algebraic groups, three in (Me), two in (CR) or four in Chapter 1 essentially can naturally be constructed by typical bimodule and rationalizing methods. And many properties (e.g., preserving composition of morphisms) of the functors in (Don) now are the special case of ours (see Section 3.5).