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Sonia Trepode

Publications and source records attributed to Sonia Trepode.

At least 19 recordsLinked to original sources
arXiv2025

On the Irreducible Morphisms for Skew group algebras

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of covering that we call a Galois semi-covering functor, which becomes a Galois covering when the group action is free. We study the module category of its skew group algebra under this functor. As an application, we obtain a complete description of the irreducible morphisms and almost split sequences of skew group algebras and show that the (stable) rank is preserved under skewness. In particular, we determine the stable rank of skew-gentle algebras.

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arXiv2025

On the global dimension three endomorphism algebras of the minimal generator-cogenerator

The main goal of this paper is to study the class of algebras for which the global dimension of the endomorphism ring of the generator-cogenerator, given by the sum of the projective and injective modules, is equal to three. We will refer to these algebras as representation-hereditary algebras. We show that these algebras are torsionless-finite, as defined by Ringel. These algebras do not necessarily have finite global dimension; however, when there is no non-zero morphism from an injective to a projective module, they have global dimension less than or equal to two, with some additional homological properties. By utilizing the general framework provided by the study of the representation dimension of an algebra, we present further homological consequences. In the case where these algebras are tame quasi-tilted algebras, we prove that they belong to certain classes of tilted algebras. Although not all tilted algebras are representation-hereditary, we provide sufficient conditions for them to be representation-hereditary.

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arXiv2024

Endomorphism algebras of silting complexes

We consider endomorphism algebras of $n$-term silting complexes in derived categories of hereditary algebras, and we show that the module category of such an endomorphism algebra has a separated $n$-section. For $n=3$ we obtain a trisection in the sense of [2].

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arXiv2022

Characterisations of trivial extensions

In this paper we give a characterisation of trivial extension algebras in terms of quivers with relations. This result is based on a explicit description of the ideal of relations of the trivial extension of an algebra, given by the first author in the appendix. We also give a new proof of Wakamatsu's theorem in terms of their quiver and relations, which determines when two given algebras have isomorphic trivial extensions.

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arXiv2018

$m$-cluster tilted algebras of euclidean type

We consider $m$-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the case $\widetilde{A}$, using the geometric realization, we get a description of representation finite type in terms of $(m+2)$-angulations. We establish which $m$-cluster tilted algebras arise at the same time from quivers of type $A$ and $\widetilde{A}$. Finally, we characterize representation infinite $m$-cluster tilted algebras arising from a quiver of type $\widetilde{A}$ as $m$-relations extensions of some iterated tilted algebra of type $\widetilde{A}$.

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arXiv2017

On the Representation Dimension of Tame Cluster Tilted Algebras

The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.

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arXiv2017

Degrees of Irreducible Morphisms over Perfect Fields

The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the module category in terms of the induced natural transformations between representable functors on this graded category. When the ground ring is a perfect field, and the given morphism behaves nicely with respect to covering theory (as do irreducible morphisms with indecomposable domain or indecomposable codomain), it is shown that the degree of the morphism is finite if and only if its associated natural transformation has a representable kernel. As a corollary, generalisations of known results on the degrees of irreducible morphisms over perfect fields are given. Finally, this study is applied to the composition of paths of irreducible morphisms in relationship to the powers of the radical.

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arXiv2017

Split $t$-structures and torsion pairs in hereditary categories

We give necessary and sufficient conditions for torsion pairs in a hereditary category to be in bijection with $t$-structures in the bounded derived category of that hereditary category. We prove that the existence of a split $t$-structure with nontrivial heart in a semiconnected Krull-Schmidt category implies that this category is equivalent to the derived category of a hereditary category. We construct a bijection between split torsion pairs in the module category of a tilted algebra having a complete slice in the preinjective component with corresponding $t$-structures. Finally, we classify split $t$-structures in the derived category of a hereditary algebra.

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arXiv2016

The Representation Dimension of a Selfinjective Algebra of Wild Tilted Type

We prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective algebra admits an acyclic generalised standard Auslander-Reiten component then its representation dimension is equal to three.

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arXiv2015

On finite dimensional Jacobian Algebras

We show that Jacobian algebras arising from a sphere with $n$-punctures, with $n\geq5$, are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated Jacobian algebra is finite dimensional.

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arXiv2014

Covering techniques in Auslander-Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.

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arXiv2014

On the quiver with relations of a quasitilted algebra and applications

In this paper we discuss, in terms of quivers with relations, sufficient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension two and we give a sufficient condition for it to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and, combining both type of conditions, we are able to analyze if the given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin type $E_p$.

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arXiv2012

On the existence of a derived equivalence between a Koszul algebra and its Yoneda algebra

In this paper we focus on the relations between the derived categories of a Koszul algebra and its Yoneda algebra, in particular we want to consider the cases where these categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We consider derived discrete Koszul algebras, and we give necessary and sufficient conditions for these Koszul algebras to be derived equivalent to their Yoneda algebras. Finally, we look at the Koszul algebras such that they are derived equivalent to a hereditary algebra. In the case that the hereditary algebra is tame, we characterize when these algebras are derived equivalent to their Yoneda algebras.

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arXiv2012

On m-cluster tilted algebras and trivial extensions

In this work we study the connection between iterated tilted algebras and m-cluster tilted algebras. We show that an iterated tilted algebra induces an m-cluster tilted algebra. This m-cluster tilted algebra can be seen as a trivial extension of another iterated tilted algebra which is derived equivalent to the original one. We give a procedure to find this new iterated tilted algebra. These m-cluster tilted algebras are quotients of higher relation extensions.

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