Derived Equivalence Classification of Nonstandard Selfinjective Algebras of Domestic Type

We give a complete derived equivalence classification of all nonstandard representation-infinite domestic selfinjective algebras over an algebraically closed field. As a consequence, a complete stable equivalence classification of these algebras is obtained.     

Stable Equivalence of Selfinjective Artin Algebras of Dynkin Type

We prove new results on the stable equivalences of selfinjective Artin algebras of tilted Dynkin type, extending the main results of Skowroński and Yamagata to arbitrary tilted type.Presented by I. Reiten     

Derived Equivalence Classification of the Gentle Two-Cycle Algebras

We complete a derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski–Malicki.     

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.     

Derived Equivalence of Surface Algebras in Genus 0 via Graded Equivalence

We determine some of the derived equivalences of a class of gentle algebras called surface algebras. These algebras are constructed from an unpunctured Riemann surface of genus 0 with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. In particular, we fix a triangulation of a surface and determine when different cuts produce derived equivalent algebras.     

Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Every cluster-tilted algebra B is the relation extension \(C\ltimes \textup {Ext}^{2}_{C}(DC,C)\) of a tilted algebra C. A B-module is called induced if it is of the form M? _C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.     

On a Lift of an Individual Stable Equivalence to a Standard Derived Equivalence for Representation-Finite Self-injective Algebras

We shall show that every stable equivalence (functor) between representation-finite self-injective algebras not of type (D _3m ,s/3,1) with m2, 3s lifts to a standard derived equivalence. This implies that all stable equivalences between these algebras are of Morita type.     

Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let V be an indecomposable non-projective left Λ-module with finite dimension over k.We prove that if τΛV is the Auslander-Reiten translation of V,then the versal deformation rings R(Λ,V)and R(Λ,τΛV)(in the sense of F.M.Bleher and the second author)are isomorphic.We use this to prove that if Λ is further a cluster-tilted k-algebra,then R(Λ,V)is universal and isomorphic to k.     

Automorphic Equivalence of Many-Sorted Algebras

For every variety Θ of universal algebras we can consider the category Θ⁰ of the finite generated free algebras of this variety. The quotient group \(\mathfrak {A/Y}\), where \(\mathfrak {A}\) is a group of all the automorphisms of the category Θ⁰ and \(\mathfrak {Y}\) is a subgroup of all the inner automorphisms of this category measures difference between the geometric equivalence and automorphic equivalence of algebras from the variety Θ. In Plotkin and Zhitomirski (J. Algebra 306(2), 344–367, 2006) the simple and strong method of the verbal operations was elaborated on for the calculation of the group \(\mathfrak {A/Y} \) in the case when the Θ is a variety of one-sorted algebras. In the first part of our paper (Sections 1, 2 and 3) we prove that this method can be used in the case of many-sorted algebras. In the second part of our paper (Section 4) we apply the results of the first part to the universal algebraic geometry of many-sorted algebras and prove again and refine the results of Plotkin (2003) and Tsurkov (Int. J. Algebra Comput. 17(5/6), 1263–1271, 2007) for these algebras. For example we prove in the Theorem 4.3 that the automorphic equivalence of algebras can be reduced to the geometric equivalence if we change the operations in one of these algebras. In the third part of this paper (Section 5) we consider some varieties of many-sorted algebras. We prove that automorphic equivalence coincides with geometric equivalence in the variety of all the actions of semigroups over sets and in the variety of all the automatons, because the group \(\mathfrak {A/Y}\) is trivial for these varieties. We also consider the variety of all the representations of groups and all the representations of Lie algebras. The group \(\mathfrak {A/Y}\) is not trivial for these varieties and for both these varieties we give an examples of the representations which are automorphically equivalent but not geometrically equivalent.     

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.     

Derived Categories of Schur Algebras

Abstract

In this paper we classify the derived tame Schur and infinitesimal Schur algebras and describe indecomposable objects in their derived categories.     

Repetitive代数的导出等价

若代数A和B导出等价,则它们的Repetitive代数A和B也是导出等价,从而是稳定等价.这样利用更直接的代数方法完全回答了H.Asashiba提出的问题,并推广了Rickard,杜先能,Tachikawa-Wakamatsu等人的相应结果.进而,把以上结果推广到上有界复形的导出范畴的对称recollement的情形.     

On the Derived Categories and Quasitilted Algebras

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X ^*, we consider the set and we define the application . We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras. Presented by Raymundo Bautista.     

Dn型非线性反馈移位寄存器

研究了一类Dynkin型非线性反馈移位寄存器,给出了n长码是Dn型反馈函数的周期码的充分必要条件,并得出了所有周期码及其周期。     

On Stable Equivalence of Tensor Products of Algebras

The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ? _A  ? ? _B S: mod(A ? _K B ^op ) → mod(E ? _K F ^op ) induces a stable equivalence between A ? _K B ^op and E ? _K F ^op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field.     

不可解Lie代数完备的等价条件

目的是给出特征零域上的有限维不可解L ie代数L完备的等价条件.主要是讨论L的所有导子都是内导子的充要条件.根据L的L ev i分解式(即L分解成它的根基R和另一个半单子代数S的空间直和),先在一定条件下将根基R上的导子扩充为L上的导子,给出了L完备的一个必要条件,然后又将L上的导子诱导到半单子代数S上,利用半单子代数S的完备性,证明了上述必要条件也是L完备的充分条件.