On N-Translation Algebras And N-Abelian Categories

Recently, in the context of representation theory of finite dimensional algebras, Iyama et al. introduced higher representation theory, such as:n-Auslander algebras [63], n-Auslander-Reiten translation functor [64], these higher analogues of classical representation theory have been increasingly stud-ied. Guo introduced the n-translation algebras as a generalization of transla-tion algebras, and revealed the intrinsic link between the n-translation algebras and the higher representation theory [56]. In this thesis, we study two aspects of higher representation theory:First, we study the (n+1)-representation fi-nite n-Auslander algebras(Moreover, n-pretranslation cubic algebras) and their Koszul dual, and consider their n-translation, Koszulity, Periodicity and n-APR tilts; Second, we give a characterization of n-Auslander algebras (cate-gories) via n-abelian categories.In chapter 3, we study n-cubic quivers(algebras), and consider the projec-tive resolutions of simple modules of them and their twisted trivial extensions. In chapter 4, we study the (stable) n-Pyramid algebras, and consider their n-translation, Koszulity, Periodicity. In chapter 5, we introduce T[n]-mutation, and show that the n-APR tilting of a Koszul algebra with global dimension < n is T[n]-mutation if the Koszul dual of this algebra is an admissible (n - 1)-translation algebra.In chapter 6, we study higher Auslander correspondences via n-abelian categories, as generalizations of Iyama-Beligiannis’s n-Auslander correspon-dences. And moreover, we show that an additive category A is an n-abelian category which has quasi-enough injectives if and only if A can be embedded to an injectively-cogenerated abelian category as an n-cluster tilting subcategory.