Artin Group Presentations Arising from Cluster Algebras

In 2003, Fomin and Zelevinsky proved that finite type cluster algebras can be classified by Dynkin diagrams. Then in 2013, Barot and Marsh defined the presentation of a reflection group associated to a Dynkin diagram in terms of an edge-weighted, oriented graph, and proved that this group is invariant (up to isomorphism) under diagram mutations. In this paper, we extend Barot and Marsh’s results to Artin group presentations, defining new generator relations and showing mutation-invariance for these presentations.     

On the Analytic Type of Artin Algebras

In this paper we give a complete analytic classification of Artin Gorenstein almost stretched algebras, i.e., Artin Gorenstein algebras with Hilbert function {1, n, 2,…, 2, 1,…, 1}.     

On the Morita Equivalence of Tensor Algebras

We develop a notion of Morita equivalence for general C^*-correspondencesover C^*-algebras. We show that if two correspondences are Moritaequivalent, then the tensor algebras built from them are stronglyMorita equivalent in the sense developed by Blecher, Muhly andPaulsen. Also, the Toeplitz algebras are strongly Morita equivalentin the sense of Rieffel, as are the Cuntz–Pimsner algebras.Conversely, if the tensor algebras are strongly Morita equivalent,and if the correspondences are aperiodic in a fashion that generalizesthe notion of aperiodicity for automorphisms of C^*-algebras,then the correspondences are Morita equivalent. This generalizesa venerated theorem of Arveson on algebraic conjugacy invariantsfor ergodic, measure-preserving transformations. The notionof aperiodicity, which also generalizes the concept of fullConnes spectrum for automorphisms, is explored; its role inthe ideal theory of tensor algebras and in the theory of theirautomorphisms is investigated. 1991 Mathematics Subject Classification:46H10, 46H20, 46H99, 46M99, 47D15, 47D25.     

On Modules of Finite Complexity Over Selfinjective Artin Algebras

In this paper we study Auslander-Reiten sequences of modules with finite complexity over selfinjective artin algebras. In particular, we show that for all eventually Ω-perfect modules of finite complexity, the number of indecomposable non projective summands of the middle term of such sequences is bounded by 4. We also describe situations in which all non projective modules in a connected component of the Auslander-Reiten quiver are eventually Ω-perfect.     

On Morita Equivalence for Simple Generalized Weyl Algebras

We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges’ result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We study how far this link can be generalized for n ≥ 3.     

正则QB环上的Morita Contexts

假设T是Morita Contexts(A,B,M,N,φ,φ)的环,如果A和B是正则QB环,本文证明了T中任何元素都是幂等元和伪逆元之和.     

Stable Ranges For Morita Contexts

In this paper, we show that if rings A and B are (s, 2)-rings, then so is the ring of a Morita context (A, B, M, N, , ). Also we get analogous results for unit 1-stable ranges and GM-rings. These give new classes of rings satisfying such stable range conditions.2000 Mathematics Subject Classification: 16U99 16E50     

Morita Contexts with Many Units

In this paper, we investigate stable range conditions over Morita contexts. As applications, we calculate the Whitehead groups of Morita contexts for exchange rings with primitive factors artinian.     

Morita Contexts as Lax Functors

Monads are well known to be equivalent to lax functors out of the terminal category. Morita contexts are here shown to be lax functors out of the chaotic category with two objects. This allows various aspects in the theory of Morita contexts to be seen as special cases of general results about lax functors. The account we give of this could serve as an introduction to lax functors for those familiar with the theory of monads. We also prove some very general results along these lines relative to a given 2-comonad, with the classical case of ordinary monad theory amounting to the case of the identity comonad on Cat.     

Representation Theory of Artin Algebras I

This is the first of a series of papers dealing with the representation theory of artin algebras, where by an artin algebra we mean an artin ring having the property that its center is an artin ring and λ is a finitely generated module over its center. The over all purpose of this paper is to develop terminology and background material which will be used in the rest of the papers in the series. While it is undoubtedly true that much of this material can be found in the literature or easily deduced from results already in the literature, the particular development presented here appears to be new and is especially well suited as a foundation for the papers to come.     

On Quasitriangular Structures in Hopf Algebras Arising from Exact Group Factorizations

We show that bicrossed product Hopf algebras arising from exact factorizations in almost simple finite groups, so in particular, in simple and symmetric groups, admit no quasitriangular structure.     

一类Morita Contexts的M-投射左总体维数

我们引进了模的M-投射维数和环的M-左总体维数的概念,采用比较新颖简便的方法,得到了一类Morita Contexts T=R ReeR eRe,e∈R,e2=e和环的M-左总体维数之间的相等关系.     

Hopf Extensions of CM-finite Artin Algebras

Let H be a finite-dimensional Hopf algebra over a field k, and A a left $H\mbox{-}$ module $k\mbox{-}$ algebra. We show that A#H is a CM-finite algebra if and only if A is a CM-finite algebra preserving global dimension of their relative Auslander algebras when A/A ^H is an $H^{*}\mbox{-}$ Galois extension and A#H/A is separable. As application, we describe all the finitely-generated Gorenstein-projective modules over a triangular matrix artin algebra $\Lambda=\left(\begin{smallmatrix} A^{H}& A\\ 0&A\#H \end{smallmatrix}\right)$ , and obtain a criteria for Λ being Gorenstein. We also show that Hopf extensions can induce recollements between categories $A\#H\mbox{-}{\rm Mod}$ and $A^{H}\mbox{-}{\rm Mod}$ .     

Morita Endomorphism Algebras of Generators

We provide a characterization for the endomorphism algebra of a generator to be a Morita algebra. As an application, n-Auslander algebras which are Morita algebras simultaneously are characterized.     

分次Morita Contexts与分次根

设M={A=⊕_(g∈G)A_g,V=⊕_(g∈G)V_g,W=⊕_(g∈G)W_g,B=⊕_(g∈G)B_g}与(,),[,]是一个G-分次Morita Context,且满足(V,W)=A,[W,V]=B,A,B都有单位元．本文证明τG(B)：[W,ΥG(A)V]=【WΥc(A),V],ΥG(A)=(V,ΥG(B)W)=(VΥG(B),W)其中ΥG代表P_G(分次素根),J_G(分次Jacobson根),K_G(分次Koethe根),L_G(分次Levitzki根)和s_G(分次强素根),us_G(分次一致强素根)．     

一类Morita Contexts的同调维数

本文得到了一类Mortia Contexts的总体维数与环R的总体维数的相等关系,并且将这个结论推广到n×n矩阵环。     

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

We determine the Krull-Gabriel dimension of the cycle-finite categories of finitely generated modules over artin algebras and derive some consequences.     

Dualizing Modules for Orders and Artin Algebras

We provide a criterion for a -bimodule to be a dualizing module, where is an order over a commutative Gorenstein complete local domain of dim R=1. Using this criterion, we give examples of dualizing modules which are neither isomorphic to nor a dual of . Thus we can also give such examples over an Artin algebra by modulo a nonzerodivisor.