Repetitive cluster-tilted algebras

Let H be a finite-dimensional hereditary algebra over an algebraically closed field k and C_Fm be the repetitive cluster category of H with m ≥ 1. We investigate the properties of cluster tilting objects in C_Fm and the structure of repetitive cluster-tilted algebras. Moreover, we generalize Theorem 4.2 in [12] (Buan A, Marsh R, Reiten I. Cluster-tilted algebra, Trans. Amer. Math. Soc., 359(1)(2007), 323-332.) to the situation of C_Fm, and prove that the tilting graph KC_Fm of C_Fm is connected.     

Grothendieck Groups and Tilting Objects

Let C be a connected Noetherian hereditary Abelian category with a Serre functor over an algebraically closed field k, with finite-dimensional homomorphism and extension spaces. Using the classification of such categories from our 1999 preprint, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.     

Cluster-tilted algebras

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

     

The diagram for endomorphism algebras of two-term tilting complexes over self-injective algebras

We construct a diagram between the endomorphism algebra of a two-term projective module complex over a self-injective algebra and the endomorphism algebras of its homology groups and apply the diagram to the tilting theory of self-injective algebras. We give an example of recovering the endomorphism algebra of a two-term tilting complex over a self-injective algebra from the endomorphism algebras of its homology groups with some additional structures which appear in the diagram.     

The dimension vectors of indecomposable modules of cluster-tilted algebras and the Fomin-Zelevinsky denominators conjecture

The main result of this paper is that any two non-isomorphic indecomposable modules of a cluster-tilted algebra of finite representation type have different dimension vectors. As an application to cluster algebras of Types A,D,E, we give a proof of the Fomin-Zelevinsky denominators conjecture for cluster variables, namely, different cluster variables have different denominators with respect to any given cluster.     

Cluster-Tilted Algebras of Type D n

Let H be a hereditary algebra of Dynkin type D _n over a field k and 𝒞 _H be the cluster category of H. Assume that n ≥ 5 and that T and T′ are tilting objects in 𝒞 _H . We prove that the cluster-tilted algebra Γ = End_{𝒞 H} (T)^op is isomorphic to Γ′ = End_{𝒞 H} (T′)^op if and only if T = τ ⁱ T′ or T = στ ^j T′ for some integers i and j, where τ is the Auslander–Reiten translation and σ is the automorphism of 𝒞 _H defined in Section 4.     

Abelianness of the “missing part” from a sheaf category to a module category

This paper investigates the structure of the"missing part"from the category of coherent sheaves over a weighted projective line of weight type(2,2,n)to the category of finitely generated right modules on the associated canonical algebra.By constructing a t-structure in the stable category of the vector bundle category,we show that the"missing part"is equivalent to the heart of the t-structure,hence it is abelian.Moreover,it is equivalent to the category of finitely generated modules on the path algebra of type An-1.     

广义Hom-李代数的中心不变量

设L是一个广义Hom-李代数, V 是[L,L]的一个H-Hom-李理想. 本文主要研究了L的中心不变量问题. 利用Hopf代数中的方法, 得到了V 的H-不变量包含在L的中心H-不变量中, 这推广了1994年Cohen和Westreich的主要结论.     

泛代数的二次扩张

在本文中引入了泛代数的二次扩张的概念，解决了ＴＵ－ＥＣ（Ａ）和ＵＴ－ＥＣ（Ａ）的存在、真类和基数问题，范畴ＴＵ－ＥＣ（Ａ）（及ＵＴ－ＥＣ（Ａ）），并得到了有关二次扩张的几个同构定理，还对一点二次扩张作了讨论。     

From recollement of triangulated categories to recollement of abelian categories

In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T)  T,j j (T) T.     

A NOTE ON CYCLIC DUALITY AND HOPF ALGEBRAS

ABSTRACT

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes’ duality isomorphism for the cyclic category.     

非脆弱离散重复控制的二维模型方法

针对一类线性离散系统,提出一种基于二维模型的非脆弱离散重复控制设计方法.通过独立地考虑重复控制系统的控制与学习行为,建立离散重复控制系统的二维模型. 在此基础上,针对重复控制器和反馈控制器具有不确定性的离散重复控制系统,给出了基于线性矩阵不等式的系统稳定性条件和重复控制律. 最后,数值仿真实例验证了所提方法的有效性.     

关于MH中的Lie双代数和余Poisson-Hopf代数

本文研究余三角Ｈｏｐｆ代数余模范畴中的Ｌｉｅ双代数和余ＰｏｉｓｓｏｎＨｏｐｆ代数,我们主要讨论余三角Ｈｏｐｆ代数余模范畴中的Ｌｉｅ双代数和余Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数之间的关系。     

广义Radford双积Hom-Hopf代数和相关辫子张量范畴

本文研究了Radford双积的Hom-型.通过把广义smash积Hom-代数和广义smash余积Hom-余代数相结合,得到了他们成为Hom-双代数的充分必要条件,这一结果推广了著名的Radford双积.     

Axiomatizing subcategories of Abelian categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any d-abelian category is equivalent to a d-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.     

两个相关代数上的一对同构模范畴

本文研究了Artin代数A与其子代数模范畴中反变有限子范畴之间的关系.利用范畴同构,获得了代数A上投射维数有限的子模范畴P∞(A)在有限生成的左A模范畴A-mod上反变有限的一个条件,推广了关于子范畴P∞(A)反变有限性的结果.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Weak Rickart and dual weak Rickart objects in abelian categories

We introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) weak relative Rickart properties. We relate our theory to the study of relative regular objects and (dual) relative Baer objects. We also give some applications to module and comodule categories.     

Tilting subcategories in extriangulated categories

Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.