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A characterization of representation-finite biserial algebras over a perfect field

100%
Pogorzały Z.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction.........................................................................................5 1. The structure of distributive biserial algebras.....................................7 2. Almost multiplicity-free modules........................................................15 3. One point extension..........................................................................19 4. Representation-finite biserial algebras with the (S)-condition...........31 5. Representation-finite biserial algebras ............................................41 References...........................................................................................43
2
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Left-right projective bimodules and stable equivalences of Morita type

95%
Pogorzały Z.
Colloquium Mathematicum
|
2001
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tom 88
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nr 2
243-255
EN
We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for self-injective algebras are also given.
3
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Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$

95%
Pogorzały Z.
Colloquium Mathematicum
|
1993
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tom 66
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nr 2
265-281
EN
The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting equivalent algebras then their trivial extensions T(A) and T(B) are stably equivalent. Consequently, there is a special need to describe algebras that are stably equivalent to the trivial extensions of tame hereditary algebras. In the paper, there are studied algebras which are stably equivalent to the trivial extensions of hereditary algebras of type $Ã_n$, that is, algebras given by quivers whose underlying graphs are of type $Ã_n$. These algebras are isomorphic to the trivial extensions of very nice algebras (see Theorem 1). Moreover, in view of [1, 8], Theorem 2 shows that every stable equivalence of such algebras is induced in some sense by a derived equivalence of well chosen subalgebras. In our study of stable equivalence, we shall use methods and results from [11]. We shall also use freely information on Auslander-Reiten sequences which can be found in [2].
4
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On locally bounded categories stably equivalent to the repetitive algebras of tubular algebras

95%
Pogorzały Z.
Colloquium Mathematicum
|
1997
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tom 72
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nr 1
123-146
5
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Left-sided quasi-invertible bimodules over Nakayama algebras

95%
Pogorzały Z.
Open Mathematics
|
2005
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tom 3
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nr 1
125-142
EN
Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.
6
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Orbit algebras that are invariant under stable equivalences of Morita type

95%
Pogorzały Z.
Open Mathematics
|
2014
|
tom 12
|
nr 6
813-823
EN
In this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita type between self-injective algebras. There are also indicated some links between Auslander-Reiten periodicity of bimodules and noetherianity of their orbit algebras.
7
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Starting and ending components of the Auslander-Reiten quivers of a class of special biserial algebras

61%
Pogorzały Z., Sufranek M.
Colloquium Mathematicum
|
2004
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tom 99
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nr 1
111-144
EN
The class of n-fundamental algebras is introduced. It is a subclass of string algebras. For n-fundamental algebras we study the problem of when the Auslander-Reiten quiver contains, at the beginning or at the end, a component which is not generalized standard.
8
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Slice modules over minimal 2-fundamental algebras

61%
Pogorzały Z., Szmyt K.
Open Mathematics
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2007
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tom 5
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nr 1
164-180
EN
We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module.
9
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Tilting slice modules over minimal 2-fundamental algebras

61%
Pogorzały Z., Szmyt K.
Colloquium Mathematicum
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2008
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tom 113
|
nr 2
287-306
EN
A class of finite-dimensional algebras whose Auslander-Reiten quivers have starting but not generalized standard components is investigated. For these components the slices whose slice modules are tilting are considered. Moreover, the endomorphism algebras of tilting slice modules are characterized.
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