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1
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On the representation type of tensor product algebras

100%
Leszczyński Z.
Fundamenta Mathematicae
|
1994
|
tom 144
|
nr 2
143-161
EN
The representation type of tensor product algebras of finite-dimensional algebras is considered. The characterization of algebras A, B such that A ⊗ B is of tame representation type is given in terms of the Gabriel quivers of the algebras A, B.
2
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Representation-tame incidence algebras of finite posets

100%
Leszczyński Z.
Colloquium Mathematicum
|
2003
|
tom 96
|
nr 2
293-305
EN
Continuing the paper [Le], we give criteria for the incidence algebra of an arbitrary finite partially ordered set to be of tame representation type. This completes our result in [Le], concerning completely separating incidence algebras of posets.
3
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Representation-tame locally hereditary algebras

100%
Leszczyński Z.
Colloquium Mathematicum
|
2004
|
tom 99
|
nr 2
175-187
EN
Let A be a finite-dimensional algebra over an algebraically closed field. The algebra A is called locally hereditary if any local left ideal of A is projective. We give criteria, in terms of the Tits quadratic form, for a locally hereditary algebra to be of tame representation type. Moreover, the description of all representation-tame locally hereditary algebras is completed.
4
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The completely separating incidence algebras of tame representation type

100%
Leszczyński Z.
Colloquium Mathematicum
|
2002
|
tom 94
|
nr 2
243-262
EN
We prove that a completely separating incidence algebra of a partially ordered set is of tame representation type if and only if the associated Tits integral quadratic form is weakly non-negative.
5
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Incidence coalgebras of interval finite posets of tame comodule type

64%
Leszczyński Z., Simson D.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 2
261-295
EN
The incidence coalgebras $K^{□} I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•}: ℤ^{(I)} → ℤ$, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{□} I-comod$ of finite-dimensional left $K^{□} I$-modules is equivalent to the tameness of the category $K^{□} I-Comod_{fc}$ of finitely copresented left $K^{□} I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{□} I$ is deduced. Moreover, we prove that for an interval finite 𝔸̃ *ₘ-free poset I the incidence coalgebra $K^{□} I$ is of tame comodule type if and only if the quadratic form $q^{•}$ is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite 𝔸̃ *ₘ-free posets I such that $K^{□} I$ is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left $K^{□} I$-comodules M and N, $b̅_{K^{□} I} (dim M,dim N) = ∑_{j=0}^{∞} (-1)^{j} dim_{K} Ext_{K^{□} I}^{j}(M,N)$, where $b̅_{K^{□} I}: ℤ^{(I)} × ℤ^{(I)} → ℤ$ is the Euler ℤ-bilinear form of I and dim M, dim N are the dimension vectors of M and N.
6
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Tame tensor products of algebras

64%
Leszczyński Z., Skowroński A.
Colloquium Mathematicum
|
2003
|
tom 98
|
nr 1
125-145
EN
With the help of Galois coverings, we describe the tame tensor products $A ⊗_{K} B$ of basic, connected, nonsimple, finite-dimensional algebras A and B over an algebraically closed field K. In particular, the description of all tame group algebras AG of finite groups G over finite-dimensional algebras A is completed.
7
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Tame triangular matrix algebras

64%
Leszczyński Z., Skowroński A.
Colloquium Mathematicum
|
2000
|
tom 86
|
nr 2
259-303
EN
We describe all finite-dimensional algebras A over an algebraically closed field for which the algebra $T_2(A)$ of 2×2 upper triangular matrices over A is of tame representation type. Moreover, the algebras A for which $T_2(A)$ is of polynomial growth (respectively, domestic, of finite representation type) are also characterized.
8
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On the representation type of triangular matrix algebras over special algebras

50%
Leszczyński Z.
Fundamenta Mathematicae
|
1991
|
tom 137
|
nr 2
65-80
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