Representation-finite Self-injective Algebras Research Articles

Rigidity degrees of indecomposable modules over representation-finite self-injective algebras

The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension — rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types A and E are given.

The Liftability Question for Stable Equivalences Between Representation-Finite Self-Injective Algebras

Let k be an algebraically closed field. It is known that any stable equivalence between standard representation-finite self-injective k-algebras (without block of Loewy length 2) lifts to a standard derived equivalence, in particular, it is of Morita type. We show that the same holds for any stable equivalence between nonstandard representation-finite self-injective k-algebras. We also fill a gap in the original proof in standard case. This gives a complete solution of the liftability question raised by H. Asashiba about twenty years ago.

On Simple-Minded Systems Over Representation-Finite Self-Injective Algebras

Let A be a representation-finite self-injective algebra over an algebraically closed field k. We give a new characterization for an orthogonal system in the stable module category A-\(\underline {\text {mod}}\) to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in A-\(\underline {\text {mod}}\) extends to a simple-minded system.

D-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander–Reiten theory of self-injective algebras. We give a systematic construction of (weakly) d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most d. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for k-linear categories to the setting of higher-dimensional Auslander–Reiten theory.Applications include n-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be d-representation-finite in many cases. We also get a complete classification of all d-representation-finite self-injective Nakayama algebras for arbitrary d.

Simple-minded systems, configurations and mutations for representation-finite self-injective algebras

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.

Derived equivalences for Φ-Auslander-Yoneda algebras

In this paper, we first define a new family of Yoneda algebras, called Φ \Phi -Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets Φ \Phi in N \mathbb N , which includes higher cohomologies indexed by Φ \Phi , and then present a general method to construct a family of new derived equivalences for these Φ \Phi -Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters Φ \Phi are rather abundant. Among applications of our method are the following results: (1) if A A is a self-injective Artin algebra, then, for any A A -module X X and for any admissible set Φ \Phi in N \mathbb N , the Φ \Phi -Auslander-Yoneda algebras of A ⊕ X A\oplus X and A ⊕ Ω A ( X ) A\oplus \Omega _A(X) are derived equivalent, where Ω \Omega is the Heller loop operator. (2) Suppose that A A and B B are representation-finite self-injective algebras with additive generators A X _AX and B Y _BY , respectively. If A A and B B are derived equivalent, then so are the Φ \Phi -Auslander-Yoneda algebras of X X and Y Y for any admissible set Φ \Phi . In particular, the Auslander algebras of A A and B B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between Φ \Phi -Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles.

Resolutions of mesh algebras: periodicity and Calabi–Yau dimensions

A triangulated category $${(\mathcal{T},\Sigma)}$$ is said to be Calabi–Yau of dimension d if Σ d is a Serre functor. We determine which stable module categories of self-injective algebras Λ of finite type are Calabi–Yau and compute their Calabi–Yau dimensions, correcting errors in previous work. We first show that the Calabi–Yau property of mod-Λ can be detected in the minimal projective resolution of the stable Auslander algebra Γ of Λ, over its enveloping algebra. We then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the (generalized) preprojective algebras of Dynkin graphs. For representation-finite self-injective algebras of torsion order t = 1, we obtain a complete description of their stable Calabi–Yau properties, but only partial results for those algebras of torsion order t = 2. We also obtain some new information about the periods of the representation-finite self-injective algebras of torsion order t > 1. Finally, we describe how these questions can also be approached by realizing the stable categories of representation-finite self-injective algebras as orbit categories of the bounded derived categories of hereditary algebras, and illustrate this technique with several explicit computations that our previous methods left unsettled.

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

In terms of generators and relations, the Hochschild cohomology algebra is described for a family of representation-finite self-injective algebras of tree class D 4. In calculations, the minimal projective bimodule resolution that was constructed in another authors’ paper is used. Bibliography: 10 titles.

Stable equivalence classes of self-injective algebras of tree class D n

A classification up to stable equivalence of representation-finite self-injective algebras with associated Dynkin diagram Dn is suggested. In each stable equivalence class, a representative is chosen, and the representatives are partitioned into five families of algebras, which are described in terms of quivers with relations.

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 m≥2, 3∤s lifts to a standard derived equivalence. This implies that all stable equivalences between these algebras are of Morita type.

The complexity of minimal degenerations for representation-finite selfinjective algebras

Minimal degenerations of modules over all but one exceptional class of representation-finite selfinjective algebras are given by extensions [G. Zwara, Colloq. Math. 75 (1998) 91]. This means in terms of the complexity of degenerations, that these minimal degenerations have complexity one. The minimal degenerations of modules for the exceptional class of representation-finite selfinjective algebras are studied by R. Aehle [J. London Math. Soc. 66 (2002) 73]. In this paper it is shown that these degenerations have complexity less or equal to two.

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class An, II

Up to derived equivalence, the representation-finite self-injective algebras of class A n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Mobius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Mobius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Mobius algebras.

The Derived Equivalence Classification of Representation-Finite Selfinjective Algebras

The Derived Equivalence Classification of Representation-Finite Selfinjective Algebras

Degenerations for modules over representation-finite selfinjective algebras

Degenerations for modules over representation-finite selfinjective algebras