Algebras Of Finite Representation Type Research Articles

Tilting Mutation of Weakly Symmetric Algebras and Stable Equivalence

We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky for quivers with potentials. By definition, weakly symmetric algebras connected by a sequence of tilting mutations are derived equivalent, and hence stably equivalent. The second aim of this article is to study these stable equivalences via a result of Okuyama describing the images of the simple modules. As an application we answer a question of Asashiba on the derived Picard groups of a class of self-injective algebras of finite representation type. We conclude by introducing a mutation procedure for maximal systems of orthogonal bricks in a triangulated category, which is motivated by the effect that a tilting mutation has on the set of simple modules in the stable category.

Examples of degenerations of Cohen-Macaulay modules

We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type(An)(A_n)are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

Stable autoequivalences of self-injective algebras of finite representation type

In this work, a subgroup of the group of autoequivalences of the stable category for all standard self-injective algebras of finite representation type (which is referred to as the group of monomial autoequivalences) in computed, as well as the quotient group of this group modulo natural isomorphisms. If some restrictions on the type of the algebra are imposed, this subgroup coincides with the whole group of autoequivalences. Furthermore, these results are generalized to the case of mesh-categories associated with the quiver of the form $ {{{\mathbb{Z}T}} \left/ {G} \right.} $ , where T is an arbitrary tree and the group G is generated by the Auslander-Reiten translate. Bibliography: 4 titles.

The number of complete exceptional sequences for a Dynkin algebra

The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type $\v

Representations of littleq-Schur algebras

In [18] and [24], little q-Schur algebras were introduced as homomorphic images of the infinitesimal quantum groups. In this paper, we will investigate representations of these algebras. We will classify simple modules for little q-Schur algebras and classify semisimple little q-Schur algebras. Moreover, through the classification of the blocks of little q-Schur algebras for n = 2, we will determine little q-Schur algebras of finite representation type in the odd roots of unity case.

Hochschild cohomology for nonstandard self-injective algebras of tree class D n

Minimal projective himodule resolutions for nonstandard self-injective algebras of finite representation type are constructed. The dimensions of the Hochschild cohomology groups are calculated, and a description of the Hochschild cohomology algebra in terms of generators with relations is obtained for the algebras under consideration through the instrumentality of these resolutions. The constructed resolutions are periodic, and, accordingly, the Hochschild cohomology for these algebras is periodic as well. Bibliography: 13 titles.

Super cocommutative Hopf algebras of finite representation type

Given an algebraically closed field k of characteristic p>5, we classify the finite algebraic k-supergroups whose algebras of measures are of finite representation type. Let G be such a supergroup and G̲ the largest ordinary algebraic k-group determined by G. We show that both G̲ and u(Lie(G)), the restricted enveloping algebra of Lie superalgebra of G, are of finite representation type. Moreover, only some special representation-finite algebraic k-groups of dimension zero are shown to appear if G≠G̲. The structure of G is almost determined by G̲ and u(Lie(G)). The Auslander–Reiten quivers are determined by showing that they are Nakayama algebras.

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.

On self-injective algebras of finite representation type

We describe the structure of finite-dimensional self-injective algebras of finite representation type over a field whose stable Auslander–Reiten quiver has a sectional module not lying on a short chain.

Formula omitted]-structures via recollements for piecewise hereditary algebras

Following the work of Beilinson, Bernstein and Deligne, we study restriction and induction of t -structures on triangulated categories with respect to recollements. For derived categories of piecewise hereditary algebras we give a necessary and sufficient condition for a bounded t -structure to be induced from recollements by derived categories of algebras. As a corollary we prove that for hereditary algebras of finite representation type all bounded t -structures can be obtained in this way.

Indecomposable Calabi-Yau objects in stable module categories of finite type

The authors give a discription of the stable categories of selfinjective algebras of finite representation type over an algebraically closed field, which admits indecomposable Calabi-Yau obdjects. For selfinjective algebras with such properties, the ones whose stable categories are not Calabi-Yau are determined. For the remaining ones, i.e., those selfinjective algebras whose stable categories are actually Calabi-Yau, the difference between the Calabi-Yau dimensions of the indecomposable Calabi-Yau objects and the Calabi-Yau dimensions of the stable categories is described.

Quivers, Quasi-Quantum Groups and Finite Tensor Categories

We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory.

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an ( A , A ) -bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B → A , B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi–Yau dimensions.

Self-injective algebras and the second Hochschild cohomology group

In this paper we study the second Hochschild cohomology group HH 2 ( Λ ) of a finite dimensional algebra Λ. In particular, we determine HH 2 ( Λ ) where Λ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field K and show that this group is zero for most such Λ; we give a basis for HH 2 ( Λ ) in the few cases where it is not zero.

An Auslander-type result for Gorenstein-projective modules

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105–179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269–310].

Hochschild cohomology for self-injective algebras of tree class D n . I

The minimal projective bimodule resolution is constructed for algebras in a family of self-injective algebras of finite representation type with tree class Dn. Using this resolution, we calculate the dimensions of the Hochschild cohomology groups for the algebras under consideration. The described resolution is periodic, and thus the Hochschild cohomology of these algebras is periodic as well. Bibliography: 11 titles.

Calabi–Yau stable module categories of finite type

We describe the stable module categories of the self-injective finite-dimensional algebras of finite representation type over an algebraically closed field which are Calabi–Yau (in the sense of Kontsevich).

Cluster-tilted algebras of finite representation type

We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

Bimodule resolutions of Möbius algebras

The minimal projective bimodule resolution of Möbius algebras that form a class of self-injective algebras of finite representation type is constructed. Bibliography: 7 titles.