Selfinjective algebras of tubular type

Białkowski, Jerzy; Skowroński, Andrzej

doi:10.4064/cm94-2-2

Cited by 19 publications

(13 citation statements)

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“…A group G of automorphisms of the K-algebra B is called admissible if G acts freely on the set E and has finitely many orbits. Then the orbit algebra B/G is defined [12] and is a (finite dimensional) selfinjective algebra, called a selfinjective algebra of tubular type [5]. The action of the Nakayama automorphism ν B of B on the set E is given by The following proposition gives a general description of admissible groups of K-automorphisms of repetitive algebras of tubular algebras (see [18, (3.9)]).…”

Section: Corollary 3 Let a Be A Weakly Symmetric Algebra Of Tubular mentioning

confidence: 99%

“…This is the class of all nondomestic polynomial growth algebras having simply connected Galois coverings [18]. Moreover, it has been recently shown [5] that a selfinjective algebra A is of tubular type if and only if A is tame, admits a simply connected Galois covering, and s A consists only of tubes. On the other hand, in the process of classifying tame blocks of group algebras of finite groups, K. Erdmann discovered various families of tame symmetric algebras (of quaternion type) having at most three simple modules, nonsingular Cartan matrix, and the stable Auslander-Reiten quiver consisting of tubes of ranks at most 2, but only very few of them admit simply connected Galois coverings (see [7], [8], [9]).…”

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On tame weakly symmetric algebras having only periodic modules

Białkowski

Skowroński

2003

Archiv der Mathematik

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We classify (up to Morita equivalence) all tame weakly symmetric finite dimensional algebras over an algebraically closed field having simply connected Galois coverings, nonsingular Cartan matrices and the stable Auslander-Reiten quivers consisting only of tubes. In particular, we prove that these algebras have at most four simple modules.1. Introduction and the main results. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finite dimensional K-algebra with an identity, which we shall assume (without loss of generality) to be basic and connected. For an algebra A, we denote by modA the category of finite dimensional right A-modules and by D the standard duality Hom K (−, K) on modA.From Drozd's remarkable Tame and Wild Theorem [6] the class of algebras may be divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur in each dimension d, in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all finite dimensional K-algebras. Accordingly we may realistically hope to classify the indecomposable finite dimensional modules only for tame algebras.An algebra A is called selfinjective if A ∼ = D(A) in modA, that is, the projective A-modules are injective. Further, A is called symmetric if A and D(A) are isomorphic as A-bimodules. Moreover, A is called weakly symmetric if for any indecomposable projective A-module P the socle soc P of P is isomorphic to its top P / rad P . For a selfinjective algebra A, we denote by s A the stable Auslander-Reiten quiver of A, obtained from the Auslander-Reiten quiver A of A by removing all projective modules and arrows attached to them. A component of s A of the form ZA ∞ /(τ r ), r 1, is called a stable tube of rank r. It is known [13] that a component in s A is a stable tube (of rank r) if and only if it consists of indecomposable periodic modules (of period r) with respect to the action of the Auslander-Reiten translation τ A = D Tr. We also note that τ A = 2 A • N A , where A is the Heller syzygy operator and N A : mod A → mod A is an equivalence induced by the Nakayama automorphism ν A of A. In particular, τ A = 2 A if A is symmetric. Recall also

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Section: Corollary 3 Let a Be A Weakly Symmetric Algebra Of Tubular mentioning

confidence: 99%

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confidence: 99%

On tame weakly symmetric algebras having only periodic modules

Białkowski

Skowroński

2003

Archiv der Mathematik

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“…Tame selfinjective algebras for which all modules are periodic can be found in [4], [5]. Moreover, there are as well the algebras of quaternion type in [11].…”

Section: Concluding Remarks and Open Questionsmentioning

confidence: 99%

Maximal 𝑛-orthogonal modules for selfinjective algebras

Erdmann¹,

Holm

2008

Proc. Amer. Math. Soc.

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“…The module category of a finite-dimensional preprojective algebra P (∆) is known to be quite exceptional, namely all non-projective indecomposable finite-dimensional modules have Ω-period at most six, where Ω is Heller's syzygy operator which assigns to any module M the kernel of its projective cover (see [AR2], [B], [ES1], [ES2], [S]). We also note that the preprojective algebras P (∆) of Dynkin graphs ∆ other than A n (1 ≤ n ≤ 5) and D 4 are of wild representation type (see [BS1], [BS2], [DR2]), and these first give natural examples of wild algebras whose stable Auslander-Reiten quiver consists only of tubes.…”

Section: Aā (V a Vertex Of Q)mentioning

confidence: 99%