Finite Dimensional Selfinjective Algebras Research Articles

The Auslander-Reiten quiver of perfect complexes for a self-injective algebra

We consider the homotopy category of perfect complexes for a finite dimensional self-injective algebra over a field, identifying many aspects of perfect complexes according to their position in the Auslander-Reiten quiver. Short complexes lie close to the rim. We characterize the position in the quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and truncated projective resolutions. We describe completely the quiver components that contain projective modules (complexes of length 1). We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We show that homology stabilizes, in a certain sense, away from the rim of the quiver.

On abelian subcategories of triangulated categories

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial t-structures, and in particular, full abelian subcategories need not arise as hearts of a t-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.

On simple-minded systems and τ-periodic modules of self-injective algebras

Let A be a finite-dimensional self-injective algebra over an algebraically closed field, C a stably quasi-serial component (i.e. its stable part is a tube) of rank n of the Auslander-Reiten quiver of A, and S be a simple-minded system of the stable module category mod_A. We show that the intersection S∩C is of size strictly less than n, and consists only of modules with quasi-length strictly less than n. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of A cannot be in any simple-minded system.

Selfinjective algebras with hereditary stable slice

We determine the structure of all finite-dimensional selfinjective algebras over a field whose Auslander-Reiten quiver admits a hereditary stable slice.

Socle deformed preprojective algebras of generalized Dynkin type

We provide a complete classification of finite-dimensional self-injective algebras which are socle equivalent to preprojective algebras of generalized Dynkin type. In particular, we conclude that these algebras are deformed preprojective algebras of generalized Dynkin type (in the sense of [5, 12]), and hence are periodic algebras.

Selfinjective algebras having a generalized standard family of quasi-tubes maximally saturated by simple and projective modules

We give a complete description of finite dimensional selfinjective algebras over an algebraically closed field whose Auslander–Reiten quiver admits a generalized standard family of quasi-tubes maximally saturated by simple and projective modules. In particular, we show that these algebras are selfinjective algebras of strictly canonical type.

Selfinjective algebras without short cycles of indecomposable modules

We describe the structure of finite dimensional selfinjective algebras over an arbitrary field without short cycles of indecomposable modules.

Invariance of the restricted p-power map on integrable derivations under stable equivalences

We show that the p-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic p commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the p-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.

Stable categories and reconstruction

We study sets of objects in a triangulated category that satisfy properties similar to simple modules when the triangulated category is the derived category of a ring or the stable category of a finite-dimensional self-injective algebra. In the first case, we construct t-structures and, in the second case, we construct a graded algebra.

Pseudo-Frobenius Graded Algebras with Enough Idempotents

We introduce and study the notion of pseudo-Frobenius graded algebra with enough idempotents, showing that it follows the pattern of the classical concept of pseudo-Frobenius (PF) and quasi-Frobenius (QF) ring, in particular finite dimensional self-injective algebras, as studied by Nakayama, Morita, Faith, Tachikawa, etc. We show that such an algebra is characterized by the existence of a graded Nakayama form. Moreover, we prove that the pseudo-Frobenius property is preserved and reflected by covering functors, a fact which makes the concept useful in representation theory.

ν-Stable τ-Tilting Modules

Inspired by τ-tilting theory [3], we introduce the notion of ν-stable support τ-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in K b (proj Λ), ν-stable support τ-tilting Λ-modules, and ν-stable functorially finite torsion classes in modΛ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in 𝒞 if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category 𝒞. We also study some properties of support τ-tilting modules over 2-CY tilted algebras, and we give a necessary condition such that algebras are 2-CY tilted in terms of support τ-tilting modules.

On selfinjective algebras of finite representation type with maximal almost split sequences

We investigate the structure of finite dimensional selfinjective algebras of finite representation type over an arbitrary field having almost split sequences of modules whose middle term admits the maximal possible number of indecomposable direct summands.

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.

Stably Calabi-Yau algebras and skew group algebras

Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some sufficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.

Support varieties and representation type of self-injective algebras

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg [7]: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in [23]. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras. 2010 Mathematics Subject Classification: 16D50, 16L60, 16G60, 16G10, 16E40, 16T05, 17B35, 17B37, 20C08

On The Number of Simple and Projective Modules in the Quasi-Tubes of Self-Injective Algebras

We establish a bound on the number of simple and projective modules in the quasi-tubes of the Auslander–Reiten quivers of finite dimensional self-injective algebras over a field.

Representations of finite dimensional self-injective algebras and related topics

Representations of finite dimensional self-injective algebras and related topics

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.

Maximal 𝑛-orthogonal modules for selfinjective algebras

Let A A be a finite-dimensional selfinjective algebra. We show that, for any n ≥ 1 n\ge 1 , maximal n n -orthogonal A A -modules (in the sense of Iyama) rarely exist. More precisely, we prove that if A A admits a maximal n n -orthogonal module, then all A A -modules are of complexity at most 1.

Invariance of selfinjective algebras of quasitilted type under stable equivalences

We prove that the class of finite dimensional selfinjective algebras over a field which admit Galois coverings by the repetitive algebras of the quasitilted algebras, with Galois groups generated by compositions of the Nakayama automorphisms with strictly positive automorphisms, is invariant under stable and derived equivalences.