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1
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Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations

100%
Kasjan S.
Fundamenta Mathematicae
|
1993
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tom 143
|
nr 3
259-279
EN
A class of stratified posets $I*_ϱ$ is investigated and their incidence algebras $KI*_ϱ$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on $I*_ϱ$ we associate with $I*_ϱ$ a bound quiver (Q, Ω) in such a way that $KI*_ϱ ≃ K(Q, Ω)$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if $I*_ϱ$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition $I_ϱ$ is three-partite a fundamental domain $I^{*+×}$ of this covering is constructed and a functorial connection between $mod_{sp} (KI^{*+×}_ϱ)$ and $mod_{sp}(KI*_ϱ)$ is given.
2
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On the problem of axiomatization of tame representation type

100%
Kasjan S.
Fundamenta Mathematicae
|
2002
|
tom 171
|
nr 1
53-67
EN
Associative algebras of fixed dimension over algebraically closed fields of fixed characteristic are considered. It is proved that the class of algebras of tame representation type is axiomatizable. Moreover, finite axiomatizability of this class is equivalent to the conjecture that the algebras of tame representation type form a Zariski-open subset in the variety of algebras.
3
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Simply connected right multipeak algebras and the separation property

100%
Kasjan S.
Colloquium Mathematicum
|
1999
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tom 82
|
nr 1
137-153
EN
Let R=k(Q,I) be a finite-dimensional algebra over a field k determined by a bound quiver (Q,I). We show that if R is a simply connected right multipeak algebra which is chord-free and $\widetilde{𝔸}$-free in the sense defined below then R has the separation property and there exists a preprojective component of the Auslander-Reiten quiver of the category prin(R) of prinjective R-modules. As a consequence we get in 4.6 a criterion for finite representation type of prin(R) in terms of the prinjective Tits quadratic form of R.
4
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Representation-finite triangular algebras form an open scheme

100%
Kasjan S.
Open Mathematics
|
2003
|
tom 1
|
nr 1
97-107
EN
Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.
5
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Tameness criterion for posets with zero-relations and three-partite subamalgams of tiled orders

100%
Kasjan S.
Colloquium Mathematicum
|
2002
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tom 91
|
nr 1
39-68
EN
A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if $Λ^{•}$ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form $q_{Λ^{•}}$ associated with $Λ^{•}$ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28] and gives an affirmative answer to [28, Question 4.7].
6
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Minimal bipartite algebras of infinite prinjective type with prin-preprojective component

100%
Kasjan S.
Colloquium Mathematicum
|
1998
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tom 76
|
nr 2
295-317
7
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On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth

64%
Kasjan S., Pastuszak G.
Colloquium Mathematicum
|
2014
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tom 136
|
nr 2
179-220
EN
Assume that k is a field of characteristic different from 2. We show that if Γ is a strongly simply connected k-algebra of non-polynomial growth, then there exists a special family of pointed Γ-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of Ziegler that Γ admits a super-decomposable pure-injective module if k is a countable field.
8
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An isomorphism problem for algebras defined by some quivers and nonadmissible ideals

64%
Kasjan S., Sędłak M.
Colloquium Mathematicum
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2008
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tom 112
|
nr 1
1-21
EN
Given a quiver Q, a field K and two (not necessarily admissible) ideals I, I' in the path algebra KQ, we study the problem when the factor algebras KQ/I and KQ/I' of KQ are isomorphic. Sufficient conditions are given in case Q is a tree extension of a cycle.
9
Content available remote

Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture

64%
Downarowicz T., Kasjan S.
Studia Mathematica
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2015
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tom 229
|
nr 1
45-72
EN
Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do all generalized Sturmian subshifts (not only the classical Sturmian subshift). We also give an example of an irregular Toeplitz subshift which meets our criterion. We give an example of a model of an odometer which is not even Toeplitz (it is weakly mixing), hence does not meet our criterion. However, for this example, we manage to produce a separate proof of Sarnak's conjecture. Next, we provide a class of Toeplitz sequences which fail Sarnak's conjecture (in a weak sense); all these examples have positive entropy. Finally, we examine the example of a Toeplitz sequence from [AKL] (which fails Sarnak's conjecture in the strong sense) and prove that it also has positive entropy (this proof has been announced in [AKL]). This paper can be considered a sequel to [AKL], it also fills some gaps of [D].
10
Content available remote

On two tame algebras with super-decomposable pure-injective modules

64%
Kasjan S., Pastuszak G.
Colloquium Mathematicum
|
2011
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tom 123
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nr 2
249-276
EN
Let k be a field of characteristic different from 2. We consider two important tame non-polynomial growth algebras: the incidence k-algebra of the garland 𝒢₃ of length 3 and the incidence k-algebra of the enlargement of the Nazarova-Zavadskij poset 𝒩 𝓩 by a greatest element. We show that if Λ is one of these algebras, then there exists a special family of pointed Λ-modules, called an independent pair of dense chains of pointed modules. Hence, by a result of Ziegler, Λ admits a super-decomposable pure-injective module if k is a countable field.
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