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On a family of vector space categories

Grzegorz Bobiński, Andrzej Skowroński (2003)

Open Mathematics

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....

On a separation of orbits in the module variety for domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.

On additive functions for stable translation quivers

Grzegorz Bobiński (1999)

Colloquium Mathematicae

The aim of this note is to give a complete description of the positive additive functions for the stable nonperiodic translation quivers with finitely many orbits. In particular, we show that all positive additive functions on the stable translation quivers of Euclidean type (respectively, of wild type) are periodic, and hence bounded (respectively, are unbounded, and hence nonperiodic).

On cyclic vertices in valued translation quivers

Piotr Malicki (2006)

Colloquium Mathematicae

Let x and y be two vertices lying on an oriented cycle in a connected valued translation quiver (Γ,τ,δ). We prove that, under certain conditions, x and y belong to the same cyclic component of (Γ,τ,δ) if and only if there is an oriented cycle in (Γ,τ,δ) passing through x and y.

On derived equivalence classification of gentle two-cycle algebras

Grzegorz Bobiński, Piotr Malicki (2008)

Colloquium Mathematicae

We classify, up to derived (equivalently, tilting-cotilting) equivalence, all nondegenerate gentle two-cycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case.

On domestic algebras of semiregular type

Alicja Jaworska-Pastuszak, Andrzej Skowroński (2013)

Colloquium Mathematicae

We describe the structure of finite-dimensional algebras of domestic representation type over an algebraically closed field whose Auslander-Reiten quiver consists of generalized standard and semiregular components. Moreover, we prove that this class of algebras contains all special biserial algebras whose Auslander-Reiten quiver consists of semiregular components.

On Hom-spaces of tame algebras

Raymundo Bautista, Yuriy Drozd, Xiangyong Zeng, Yingbo Zhang (2007)

Open Mathematics

Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules.

On large selforthogonal modules

Gabriella D'Este (2006)

Commentationes Mathematicae Universitatis Carolinae

We construct non faithful direct summands of tilting (resp. cotilting) modules large enough to inherit a functorial tilting (resp. cotilting) behaviour.

On minimal non-tilted algebras

Flávio U. Coelho, José A. de la Peña, Sonia Trepode (2008)

Colloquium Mathematicae

A minimal non-tilted triangular algebra such that any proper semiconvex subcategory is tilted is called a tilt-semicritical algebra. We study the tilt-semicritical algebras which are quasitilted or one-point extensions of tilted algebras of tame hereditary type. We establish inductive procedures to decide whether or not a given strongly simply connected algebra is tilted.

On nonstandard tame selfinjective algebras having only periodic modules

Jerzy Białkowski, Thorsten Holm, Andrzej Skowroński (2003)

Colloquium Mathematicae

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].

On path coalgebras of quivers with relations

Pascual Jara, Luis M. Merino, Gabriel Navarro (2005)

Colloquium Mathematicae

The notion of the path coalgebra of a quiver with relations introduced in [11] and [12] is studied. In particular, developing this topic in the context of the weak* topology, we give a criterion that allows us to verify whether or not a relation subcoalgebra of a path coalgebra is the path coalgebra of a quiver with relations.

On quasitilted algebras which are one-point extensions of hereditary algebras

Dieter Happel, Inger Slungård (1999)

Colloquium Mathematicae

Quasitilted algebras have been introduced as a proper generalization of tilted algebras. In an earlier article we determined necessary conditions for one-point extensions of decomposable finite-dimensional hereditary algebras to be quasitilted and not tilted. In this article we study algebras satisfying these necessary conditions in order to investigate to what extent the conditions are sufficient.

On restrictions of indecomposables of tame algebras

R. Bautista, E. Pérez, L. Salmerón (2011)

Colloquium Mathematicae

We continue the study of ditalgebras, an acronym for "differential tensor algebras", and of their categories of modules. We examine extension/restriction interactions between module categories over a ditalgebra and a proper subditalgebra. As an application, we prove a result on representations of finite-dimensional tame algebras Λ over an algebraically closed field, which gives information on the extension/restriction interaction between module categories of some special algebras Λ₀, called convex...

On self-injective algebras of finite representation type

Marta Błaszkiewicz, Andrzej Skowroński (2012)

Colloquium Mathematicae

We describe the structure of finite-dimensional self-injective algebras of finite representation type over a field whose stable Auslander-Reiten quiver has a sectional module not lying on a short chain.

On semi-invariants of tilted algebras of type Aₙ

Witold Kraśkiewicz (2001)

Colloquium Mathematicae

We prove that for algebras obtained by tilts from the path algebras of equioriented Dynkin diagrams of type Aₙ, the rings of semi-invariants are polynomial.

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