arXiv subjects

Viktor Chust

Publications and source records attributed to Viktor Chust.

8 recordsLinked to original sources

arXiv2025

Mesh-comparable components of the Auslander-Reiten quiver

The idea of using Riedtmann's well-behaved functors to study compositions of irreducible morphisms has been explored in a number of articles. Here we introduce the concept of mesh-comparable components of the Auslander-Reiten quiver, which are components for which a Riedtmann functor exists without the necessity of taking a covering, such as the universal or the generic one. We show properties of this type of component, and study the problem of compositions of irreducible morphisms in this context.

math.RT↗

arXiv2025

Gr\"obner bases for mesh relations and applications to compositions of irreducible morphisms

We give a necessary condition for the existence of a path of n irreducible morphisms between indecomposable modules whose composition lies in the (n + 1)-power of the radical. In order to do that, we consider the general criterion given by C. Chaio, P. Le Meur and S. Trepode, which relates these compositions with zero paths in the mesh category, and then study morphisms in the mesh category by providing Gr\"obner bases for the subspaces generated by the mesh relations.

math.RT↗

arXiv2025

Leaps in the depth of compositions of irreducible morphisms

In this article, we give a family of examples of algebras, showing that for every $n \geq 2$ and $m \geq 0$, there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the $(n+m+3)$-th power of the radical, but not in the $(n + m + 4)$-th power. Such an algebra may be also supposed to be string and representation-finite.

math.RT↗

arXiv2025

On Riedtmann's well-behaved functors and applications to composites of irreducible morphisms

In this survey, we summarize some results in the literature involving the mesh category, which is a combinatorial representation of the category of modules over a finite-dimensional associative algebra. We discuss Riedtmann's well-behaved functors, which compare the mesh category with the module category, and discuss how the properties of these functors can be applied to study the problem of composing irreducible morphisms, which is the problem of deciding when the composition of n irreducible morphisms is non-zero and lies on the (n+1)-th power of the radical.

math.RT↗

arXiv2025

A note on the representation type of generalized path algebras

In (Ib\'a\~nez-Cobos et al., 2008), the authors describe the ordinary quiver of a given generalized path algebra, a concept introduced by Coelho and Liu in (Coelho, Liu, 2000). In this short note, we use this result to characterize which generalized path algebras have finite or tame representation type.

math.RT↗

arXiv2022

Homological invariants of generalized bound path algebras

We study some homological invariants of a given generalized bound path algebra in terms of those of the algebras used in its construction. We discuss the particular case where the algebra is a generalized path algebra and give conditions for those algebras to be shod or quasitilted.

math.RT↗

arXiv2021

Representations of generalized bound path algebras

The concept of generalized path algebras was introduced in (Coelho, Liu, 2000). Roughly speaking, these algebras are constructed in a similar way to that of the path algebras over a quiver, the difference being that we assign an algebra to each vertex of the quiver and consider paths intercalated with elements from these algebras. Then we use concatenation of paths together with the algebra structure in each vertex to define multiplication. The representations of a generalized path algebra were described in one of the main results of (Ib\'a\~nez Cobos et al., 2008), in terms of the representations of the algebras used in its construction. In this article, we continue our investigation started in (Chust, Coelho, 2021) and extend the result mentioned above to describe the representations of the generalized bound path algebras, which are a quotient of generalized path algebras by an ideal generated by relations. In particular, the representations associated with the projective and injective modules are described.

math.RT↗

arXiv2021

On the correspondence between path algebras and generalized path algebras

The concept of generalized path algebras was introduced in (Coelho and Liu, 2000). It was shown in (Ib\'a\~nez Cobos et al., 2008) how to obtain the Gabriel quiver of a given generalized path algebra. In this article, we generalize the concept of generalized path algebra to allow them to have relations, and we extend the result in (Ib\'a\~nez Cobos et al., 2008) to this new setting. Moreover, we use the extended result mentioned above to address the inverse problem: that is, the problem of determining when a given algebra is isomorphic to a generalized path algebra in a non-trivial way.

math.RT↗