arXiv subjects

Andrzej Skowroński

Publications and source records attributed to Andrzej Skowroński.

At least 19 recordsLinked to original sources

arXiv2025

Generalized weighted surface algebras

The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory of cluster algebras, were also proved to be (with some minor exceptions) finite-dimensional tame symmetric and periodic algebras of period 4. In this paper, we introduce a new concept of a generalized triangulation quiver, extending the notion of a triangulation quiver. Inparticular, it is also shown that the generalized triangulation quivers can be constructed from triangulations of orientable surfaces with marked self-foldedtriangles. Moreover, motivated by the recent results of [28], we define and investigate so called weighted generalized triangulation algebras associated to generalized triangulation quivers, which naturally arise from mutations of weighted triangulation algebras.

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arXiv2024

Hybrid algebras

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an Auslander-Reiten component is Dynkin or Euclidean or one of the infinite tress $A_{\infty}$, $A_{\infty}^{\infty}$, or $D_{\infty}$.

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arXiv2021

Socle equivalences of weighted surface algebras

We describe the structure and properties of the finite-dimensional symmetric algebras over an algebraically closed field $K$ which are socle equivalent to the general weighted surface algebras of triangulated surfaces, investigated in Erdmann and Skowroński (J. Algebra 544:170-227, 2020). In particular, we prove that all these algebras are tame periodic algebras of period 4. The main results of this paper form an essential step towards a classification of all symmetric tame periodic algebras of period 4.

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arXiv2020

Virtual mutations of weighted surface algebras

The finite-dimensional symmetric algebras over an algebraically closed field, based on surface triangulations, motivated by the theory of cluster algebras, have been extensively investigated and applied. In particular, the weighted surface algebras and their deformations were introduced and studied in [16]-[20], and it was shown that all these algebras, except few singular cases, are symmetric tame periodic algebras of period $4$. In this article, using the general form of a weighted surface algebra from [19], we introduce and study so called virtual mutations of weighted surface algebras, which constitute a new large class of symmetric tame periodic algebras of period $4$. We prove that all these algebras are derived equivalent but not isomorphic to weighted surface algebras. We associate such algebras to any triangulated surface, first taking blow-ups of a family of edges to $2$-triangle discs, and then virtual mutations of their weighted surface algebras. The results of this paper form an essential step towards a classification of all tame symmetric periodic algebras.

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arXiv2020

Weighted surface algebras: general version. Corrigendum

In [arXiv:1902.04063] we generalize the original definition of weighted surface algebras in [arXiv:1703.02346] by allowing the possibility that arrows might not be part of the Gabriel quiver, which gives a much larger class of algebras. This means that the zero relations need modification, to make sure that the algebras are symmetric, and of the appropriate dimension. We found recently that we had missed one necessary modification for the zero relations. Here we give the correct definition, and revise the parts of [arXiv:1902.04063] which are affected by this modification.

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arXiv2019

On tame strongly simply connected algebras

In this survey we present the criterion for tameness of strongly simply connected algebras due to Brüstle, de la Peña and Skowroński. We recall relevant concepts of representation theory and discuss some applications and connections to other problems.

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arXiv2019

Cycle-finite modules over artin algebras

We describe the representation theory of finitely generated indecomposable modules over artin algebras which do not lie on cycles of indecomposable modules involving homomorphisms from the infinite Jacobson radical of the module category.

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arXiv2019

Socle deformations of selfinjective orbit algebras of tilted type

We survey recent development of the study of finite-dimensional selfinjective algebras over a field which are socle equivalent to selfinjective orbit algebras of tilted type.

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arXiv2019

Weighted surface algebras: general version

We introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.

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arXiv2018

Algebras of generalized dihedral type

We provide a complete classification of all algebras of generalised dihedral type, which are natural generalizations of algebras which occurred in the study of blocks with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.

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arXiv2018

From Brauer graph algebras to biserial weighted surface algebras

We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented triangles, investigated in [17] and [18]. Moreover we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in [17] and [19].

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arXiv2018

Self-injective algebras with hereditary stable slice

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

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arXiv2018

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.

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arXiv2018

Weakly symmetric biserial algebras

We introduce the class of generalized biserial quiver algebras and prove that they provide a complete classification of all weakly symmetric biserial algebras over an algebraically closed field.

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arXiv2018

The structure and homological properties of generalized standard Auslander-Reiten components

We describe the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic is defined and nonnegative. Further, we provide a handy criterion for an infinite Auslander-Reiten component of an artin algebra to be generalized standard. We solve also the long standing open problem concerning the structure of artin algebras admitting a~separating family of Auslander-Reiten components.

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arXiv2017

Weighted Surface Algebras

A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period $4$. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are symmetric tame periodic algebras of period $4$. The main results of the paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type. Further, the orbit closures of the weighted surface algebras (and their socle deformations) in the affine varieties of associative $K$-algebra structures contain wide classes of tame symmetric algebras related to algebras of dihedral and semidihedral types, which occur in the study of blocks of group algebras with dihedral and semidihedral defect groups.

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arXiv2017

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.

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arXiv2015

The Representation Dimension of a Selfinjective Algebra of Euclidean Type

We prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three, and give an explicit construction of the Auslander generator of its module category.

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