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1
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Computational details-appendix to the article “Cartan matrices of selfinjective algebras of tubular type”

100%
Białkowski J.
Open Mathematics
|
2004
|
tom 2
|
nr 1
143-176
2
Content available remote

Cartan matrices of selfinjective algebras of tubular type

100%
Białkowski J.
Open Mathematics
|
2004
|
tom 2
|
nr 1
123-142
EN
The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class of selfinjective algebras of tubular type, that is, the orbit algebras $$ \hat B$$ /G of the repetitive algebras $$ \hat B$$ of tubular algebras B with respect to the actions of admissible groups G of automorphisms of $$ \hat B$$ . The aim of the paper is to describe the determinants of the Cartan matrices of selfinjective algebras of tubular type and derive some consequences.
3
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On the trivial extensions of tubular algebras

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Białkowski J.
Colloquium Mathematicum
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2004
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tom 101
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nr 2
259-269
EN
The aim of this note is to give an affirmative answer to a problem raised in [9] by J. Nehring and A. Skowroński, concerning the number of nonstable ℙ₁(K)-families of quasi-tubes in the Auslander-Reiten quivers of the trivial extensions of tubular algebras over algebraically closed fields K.
4
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Calabi-Yau stable module categories of finite type

64%
Białkowski J., Skowroński A.
Colloquium Mathematicum
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2007
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tom 109
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nr 2
257-269
EN
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).
5
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Selfinjective algebras of tubular type

64%
Białkowski J., Skowroński A.
Colloquium Mathematicum
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2002
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tom 94
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nr 2
175-194
EN
We classify all tame self/injective algebras having simply connected Galois coverings and the stable Auslander-Reiten quivers consisting of stable tubes. Moreover, the classification of nondomestic polynomial growth standard self/injective algebras is completed.
6
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Cycle-finite algebras of semiregular type

51%
Białkowski J., Skowroński A., Skowyrski A., Wiśniewski P.
Colloquium Mathematicum
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2012
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tom 129
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nr 2
211-247
EN
We describe the structure of artin algebras for which all cycles of indecomposable finitely generated modules are finite and all Auslander-Reiten components are semiregular.
7
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On nonstandard tame selfinjective algebras having only periodic modules

51%
Białkowski J., Holm T., Skowroński A.
Colloquium Mathematicum
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2003
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tom 97
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nr 1
33-47
EN
We investigate degenerations and derived equivalences of tame selfinjective algebras having no simply connected Galois coverings but the stable Auslander-Reiten quiver consisting only of tubes, discovered recently in [4].
8
Content available remote

Deformed mesh algebras of Dynkin type ℂₙ

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Białkowski J., Erdmann K., Skowroński A.
Colloquium Mathematicum
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2012
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tom 126
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nr 2
217-230
EN
In our recent paper (J. Algebra 345 (2011)) we prove that the deformed preprojective algebras of generalized Dynkin type 𝕃ₙ (in the sense of our earlier work in Trans. Amer Math. Soc. 359 (2007)) are exactly (up to isomorphism) the stable Auslander algebras of simple plane singularities of Dynkin type $𝔸_{2n}$. In this article we complete the picture by showing that the deformed mesh algebras of Dynkin type ℂₙ are isomorphic to the canonical mesh algebras of type ℂₙ, and hence to the stable Auslander algebras of simple plane curve singularities of type $𝔸_{2n-1}$. Moreover, we describe the minimal (periodic) bimodule projective resolutions of the canonical mesh algebras of type ℂₙ.
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