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Domestic iterated one-point extensions of algebras by two-ray modules

100%
Bobiński G., Skowroński A.
Open Mathematics
|
2003
|
tom 1
|
nr 4
457-476
EN
In the paper, we introduce a wide class of domestic finite dimensional algebras over an algebraically closed field which are obtained from the hereditary algebras of Euclidean type , n≥1, by iterated one-point extensions by two-ray modules. We prove that these algebras are domestic and their Auslander-Reiten quivers admit infinitely many nonperiodic connected components with infinitely many orbits with respect to the action of the Auslander-Reiten translation. Moreover, we exhibit a wide class of almost sincere domestic simply connected algebras of large global dimensions.
2
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On a family of vector space categories

84%
Bobiński G., Skowroński A.
Open Mathematics
|
2003
|
tom 1
|
nr 3
332-359
EN
In continuation of our earlier work [2] we describe the indecomposable representations and the Auslander-Reiten quivers of a family of vector space categories playing an important role in the study of domestic finite dimensional algebras over an algebraically closed field. The main results of the paper are applied in our paper [3] where we exhibit a wide class of almost sincere domestic simply connected algebras of arbitrary large finite global dimensions and describe their Auslander-Reiten quivers.
3
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Cartan matrices of selfinjective algebras of tubular type

84%
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.
4
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Classification of discrete derived categories

84%
Bobiński G., Geiß C., Skowroński A.
Open Mathematics
|
2004
|
tom 2
|
nr 1
19-49
EN
The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.
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