Researches On Quotient Categories And Hopf Extension

The main results of this paper are divided into three parts.Firstly, we discuss the idempotent completion of pretriangulated categories. Gen-erally, we always require additive categories discussed in the representation theory with the Krull-Schmidit property. By the result in [21], an additive category C is Krull-Schmidit if and only if any idempotent morphism in C splits, and EndC(X) is semiper-fect for any X∈C. On the other hand, pretriangulated categories provide a common generalization of triangulated categories, stable categories and abelian categories. It is well-known that abelian categories are idempotent completion. The idempotent com-pletion of triangulated categories were discussed by Balmer and Schlichting. However, there exist examples in [13] which are not idempotent completion. Thus it is interesting to discuss the idempotent completion of pretriangulated categories.Secondly, we study some invariants of associative algebras under stable equiva-lences of Morita type. Let A and B be two finite dimensional self-injective k-algebras which are stably equivalent of Morita type induced by bimodules AMB and BNA. In [66], Pogorzaly proved that under stable equivalence of Morita type orbit algebras A(ΩAe;A) and A(ΩBe:B) are isomorphic, that is, stable equivalences of Morita type preserve their stable Hochschild cohomology algebras. In chapter 4, we generalize this ideal to two classes of new orbit algebras. We show that A(vAe(X); X)(?)A(vAe(Y):Y), where v is a Nakayama functor, and A(τn,Ae;X)(?)A(τn,Be;Y), whereτn is an n-Auslander-Reiten translation, respectively.Finally, we explore what common representation properties A and A#σH have. In section 5.1, we show that A and A#σH have the same derived representation type if H and its dual H* are finite dimensional semisimple Hopf algebras. Then we show that A is a Cohen-Macaulay finite type k-algebra if and only if so is A#σH. In last subsection of this paper, we show that a derived equivalence between two left H-algebras can be extended to smash products under some conditions, and construct perfect recollements of derived categories of smash product algebra from a perfect recollement of derived categories of H-module algebras.