Session on Representations of Algebras:
Thursday, August 8
14.30-15.00
Christof Geiss (UNAM, México)
Quantum cluster algebras and their specializations
This is a report on joint work with B. Leclerc and J. Schröer.
We show that if a cluster algebra coincides with its upper cluster algebra and admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be viewed as a flat deformation of the classical cluster algebra.
15:00-15:30
José Vélez Marulanda (Valdosa State University, USA)
Deformations of modules for finite dimensional Jacobian algebras
It follows from results due to Ch. Geiss et al. and S. Ladkani that every Jacobian algebra over an algebraically closed field associated to a triangulation of a closed surface $S$ with a collection of marked points $M$ is tame and (weakly) symmetric (and in particular finite dimensional). In this talk, we investigate the behavior of the versal deformation rings (in the sense of F. M. Bleher and J. A. Velez-Marulanda) of finitely generated modules over such Jacobian algebras. We also investigate versal deformation rings of Gorenstein-projective modules over monomial Jacobian algebras.
15:40-16:10
Alfredo González Chaio (Universidad Nacional de Mar del Plata, Argentina)
Description of complexes in the derived category
Joint work with Claudia Chaio and Isabel Pratti.
Let $A$ be a finite dimensional algebra over an algebraically closed field. We denote by $\rm {mod}\,A$ the finitely generated module category and by $\rm{proj}\,A$ the full subcategory of $\rm{ mod}\,A$ whose objects are the finitely generated projective $A$-modules.
Let $n$ be a positive integer, with $n \geq 2$. The categories $\mathbf{C_n}(\rm{proj}\,A)$ of complexes of fixed size were defined and studied in [BSZ]. Moreover, in [CPS], the authors showed that the knitting technique used to build the Auslander-Reiten quiver of a module category can also be used to build the Auslander-Reiten quiver of the category of complexes of fixed size.
In this work, we explain how to use the mentioned knitting technique to obtain specific Auslander-Reiten triangles in the bounded derived category. To show this process, we consider two different families of finite dimensional algebras over an algebraically closed field. We describe the complexes that belong to the mouth of non-homogeneous tubes in the Auslander-Reiten quiver of their bounded derived category, whenever this algebras are either derived equivalent to hereditary algebras of type $\widetilde{A}_n$ or $\widetilde{D}_n$. In case the algebras are discrete, we describe the complexes in the mouth of components of type $\mathbb{Z}A_{\infty}$ of the Auslander-Reiten quiver of their bounded derived category.
References:
[BSZ] R. Bautista, M.J. Souto Salorio, R. Zuazua. Almost split sequences for complexes of fixed size. J. Algebra 287, 140-168, (2005).
[CPS]. C. Chaio, I. Pratti, M. J. Souto Salorio. On sectional paths in a category of complexes of fixed size. Algebras and Representation Theory 20, (2017), 289-311.
16:10-16:40
Hernán Giraldo (Universidad de Antioquia, Colombia)
String and band complexes over string almost gentle algebras
We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of string almost gentle algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by V. Bekkert and H. Merklen in [BM]. With this description, we give a characterization for a given string complex to have infinite minimal projective resolution, and we extend this characterization for the case of string algebras.
Reference:
[BM] Bekkert, V., Merklen, H.A.: Indecomposables in derived categories of gentle algebras. Algebr. Represent Theory 6 (3), 285-302 (2003).
17:10-17:40
Ricardo Rueda Robayo (Universidad de Antioquia, Colombia)
Position of String and Band Complexes in the Auslander-Reiten Quiver of $K^b(P_{\Lambda})$
We introduce the Bondarenko-type algebras, which satisfy that the string and band complexes, as introduced by Bekkert and Merklen are indecomposable in the category of perfect complexes $K^b(P_{\Lambda})$.
We discuss the shape of the components of the Auslander Reiten quiver of $K^b(P_{\Lambda})$ containing these objects.
17:40-18:10
Alejandro Argudín Monroy (Universidad Nacional Autónoma de México)
Yoneda Ext and arbitrary coproducts
The study of extensions is a theory that has developed since 1926 from multiplicative groups. In this talk we will focus on extensions in an abelian category $\mathcal{C}$. In this context, an extension of an object $A$ by an object $C$ is a short exact sequence $$0\to A\to M\to C\to 0$$ up to equivalence, where two exact sequences are equivalent if there is a morphism from one to another with identity morphisms at the ends. This kind of approach was first made by R. Baer in 1934. The class of extensions of an object $A$ by an object $C$ is usually denoted by $\mbox{YExt}^1(C,A)$.
Later on, H. Cartan and S. Eilenberg, showed that the first derived functor of the $\mbox{Hom}$ functor is isomorphic to $\mbox{YExt}^{1}$. This result marked the beginning of a series of research works looking for ways of constructing the derived functors of the Hom functor without using proyective or inyective objects, with the spirit that resolutions should be only a calculation tool for derived functors.
One of this attempts, was based in the ideas of N. Yoneda, defining what is known today as the theory of $n$-extensions and the functor called as the Yoneda Ext. An $n$-extension of an object $A$ by an object $C$ is an exact sequence of length $n$ $$0\to A\to M_{1}\rightarrow\cdots\rightarrow M_{n}\to C\to 0$$ up to equivalence, where the equivalence of exact sequences of length $n>1$ is defined in a similar way as was defined for length $1$. In this theory, it can be proved that the class $\mbox{YExt}^{1}_{\mathcal{C}}(A)$ of $n$-extensions of $A$ by $C$ is a functor equivalent to the $n$-th derived functor of the Hom functor.
Recently, the generalization of homological techniques such as Gorenstein or tilting objects to abstract contexts, such as abelian categories that do not necessarily have proyectives or inyectives, claim for the introduction of an Ext functor that can be used without restraints. A natural candidate is the Yoneda Ext. The only problem is that some of the properties of the homological Ext are not known to be valid for the Yoneda Ext. Namely, for an Ab4 abelian category with enough proyectives $\mathcal{A}$, given an object $X$ and a set of objects $\{A_{i}\}_{i\in I}$, the following isomorphism can be built \[ \mbox{Ext}^n_{\mathcal{A}}\left(\bigoplus_{i\in I}A_{i},X\right) \cong\prod_{i\in I}\mbox{Ext}^n_{\mathcal{A}}(A_{i},X) \] where $\mbox{Ext}^{n}$ is the $n$-th derived functor of the Hom functor. The goal of this talk is to show a similar isomorphism for the $n$-th Yoneda Ext. The desired isomorphism will be constructed explicitly by using colimits, in Ab4 abelian categories with not necessarily enough projectives nor injectives. A dual result will be also stated.
18:20-18:50
Patrick Le Meur (Université Paris Diderot, France)
Group actions and Calabi-Yau duality
In the past few years, there have been several investigations of the generalised cluster categories associated to the triangulations of certain orbifold surfaces with marked points. These investigations involve an action of a group on a quiver with potential (QP). This talk will present general results on the behaviour of cluster tilting theory - from the viewpoint of generalised cluster categories - under the actions of finite groups. This includes a complete description of the skew group algebra $A(Q,W)*G$ of any Ginzburg dg algebra $A(Q,W)$ acted on by a finite group $G$ as well as the comparison of the associated generalised cluster categories. Some applications to the interaction between actions of finite groups and higher Auslander-Reiten theory will be outlined.
Friday, August 9
14.30-15.00
María Julia Redondo (Universidad Nacional del Sur, Argentina)
Deformations of monomial algebras
It is well known that deformation problems can be described in terms of differential graded Lie algebras, and that if $F$ is a quasi-isomorphism of differential graded Lie algebras then $F$ induces an equivalence between the corresponding deformation problems. The problem of deforming the multiplication of a monomial algebra $A$ can be described by the graded Lie algebra $C(A,A)[1]$ which is the shifted Hochschild complex endowed with the Gerstenhaber bracket. However, Bardzell's complex $B(A)$ has shown to be more efficient to compute Hochschild cohomology of monomial algebras. We use explict comparison morphisms between $C(A,A)$ and $B(A)$ in order to describe deformations of $A$ by computing the corresponding Maurer-Cartan equation. In the particular case of infinitesimal deformations, we describe the quiver and relations of the deformed algebra in terms of the quiver and relations of $A$.
15:00-15:30
Santiago Valente (UNAM, México)
Triangular matrix categories and recollements
Joint Work with M. Ortíz Morales and Alicia Leon.
We define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two additive categories $\mathcal{U}$ and $\mathcal{T}$ and $M\in \mathrm{Mod}(\mathcal{U}\otimes \mathcal{T}^{op})$ we will construct the triangular matrix category $\mathbf{\Lambda}:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ and we prove that there is an equivalence $$\Big( \mathrm{Mod}(\mathcal{T}), \mathbb{G}\mathrm{Mod}(\mathcal{U})\Big) \simeq \mathrm{Mod}(\mathbf{\Lambda}).$$ We will show that if $\mathcal{U}$ and $\mathcal{T}$ are dualizing $K$-varieties and $M\in \mathrm{Mod}(\mathcal{U}\otimes \mathcal{T}^{op})$ satisfies certain conditions then $\mathbf{\Lambda}:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ is a dualizing variety.
Finally, we will show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in Theorem 4.4 of [CZ].
References:
[CZ] Q. Chen, M. Zheng. Recollements of abelian categories and special types of comma categories. J. Algebra. 321 (9), 2474-2485 (2009).
[LOMS] A. León-Galeana, M. Ort&iacut;ez-Morales, V. Santiago, Triangular Matrix Categories I: Dualizing Varieties and generalized one-point extension. Preprint arXiv: 1903.03914v1
[LOMS] A. León-Galeana, M. Ortíz-Morales, V. Santiago, Triangular Matrix Categories II: Recollements and functorially finite subcategories.
[S] S. O. Smalø. Functorial Finite Subcategories Over Triangular Matrix Rings. Proceedings of the American Mathematical Society Vol.111. No. 3 (1991).
15:40-16:10
Victoria Guazzelli (Universidad Nacional de Mar del Plata, Argentina)
Trivial extensions, Admissible Cuts and HW Reflections
This talk is based on a work in progress initiated in the Workshop "Mathematics in the Southern Cone" at Universidad de la República, Montevideo Uruguay, December 2018.
Let $A$ be a finite dimensional algebra over an algebraically closed field. We consider the trivial extension of $A$ by its minimal injective cogenerator $D(A).$ The trivial extension of $A$ is the orbit algebra obtained under the action of the Nakayama automorphism $\nu$.
In [HW], D. Hughes and J. Waschbüsch, characterized when two algebras have the same repetitive category in terms of a sequence of $\nu$-reflection which transforms one algebra into the other. In particular if two algebras have the same trivial extension and the same repetitive category, then it can be obtained one from the other by a sequence of $\nu$-reflections.
Later, in [FP1], E. Fernández and M. I. Platzeck gave a description of the bound quiver of the trivial extension of $A$ under the hypothesis that any oriented cycle in the ordinary quiver of $A$ is zero in $A$. Moreover, in [FP2], under the same hypothesis, the authors characterized all the algebras $B$ which have the same trivial extension as $A$. They showed that $B$ can be obtained as a quotient of the trivial extension by the ideal generated by some arrows which have been cut in an admissible way. The algebra $B$ is said to be an admissible cut of the trivial extension of $A$.
The aim of this talk is to relate these two points of view. More precisely, given an algebra $B$, admissible cut of the trivial extension of the algebra $A$, such that $A$ and $B$ have the same repetitive category, we described the sequence of $\nu-$reflections which transforms the algebra $A$ into the algebra $B$.
Reciprocally, given a sequence of $\nu-$reflections which transforms $A$ into $B$, we determined the ideal generated by arrows such that $B$ is an admissible cut of the trivial extension of the algebra $A$.
References:
[FP1] Fernández, E., Platzeck, M.I. Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner. J. Algebra 249 (2002), no. 2, 326-344.
[FP2] Fernández, E., Platzeck, M.I. Isomorphic trivial extensions of finite dimensional algebras. Journal of Pure and Applied Algebra 204 (2006), no. 1, 9-20.
[HW] Hughes, D., Washbüsch, J. Trivial extensions of tilted algebras. Proc. London Math. Soc. 46, (1983), 347-364.
16:10-16:40
Yohny Calderón Henao (Universidad de Antioquia, Colombia)
Shapes of the irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras
This is a joint-work with Hernán Giraldo and José Vélez-Marulanda.
For the stable category of modules on a repetitive algebra, we have that the irreducible morphisms are divided into three canonical forms: first, all the component morphisms are split monomorphisms (smonic case), second, they are all split epimorphims (sepic case), and third, there is exactly an irreducible component (sirreducible case). Finally, we describe the shape of the Auslander-Reiten triangles using the properties of the irreducible morphisms as hown above.
17:10-17.40
Jesús Jiménez González (CIMAT, México)
Discrete geometry in the indecomposable modules of a hereditary algebra
In the talk we review a classical result by Dlab and Ringel on representations of quivers, and apply it to the analysis of triples of indecomposable modules belonging to exact sequences. It is shown that these triples constitute a Fisher space in the finite representation case, and we relate this to recent constructions of commutative nonassociative algebras.
17:40-18.10
Sinem Odabaşı (Universidad Austral de Chile, Chile)
On monoids with enough idempotents and geometrical purity
Let $\mathcal{V}$ be a closed symmetric monoidal Grothendieck category. In this talk, we present a study on several aspects of the theory of $\mathcal{V}$-enriched categories: We firstly introduce and develope the theory of monoids with enough idempotents, and accordingly, unitary modules over a monoid with enough idempotent. Besides, we prove a $\mathcal{V}$-enriched analogous of the Mitchell's result: there exists a bijection $$\{\textrm{monoids with enough idempotents in } \mathcal{V}\} \longleftrightarrow \{\textrm{small } \mathcal{V}\textrm{-categories}\},$$ and for a given small $\mathcal{V}$-category $\mathcal{A}$, there exists a monoid $A $ with enough idempotents in $\mathcal{V}$ such that $$\mathcal{V}\mbox{-Fun}(\mathcal{A}, \mathcal{V}) \cong A\mbox{-Mod},$$ where $\mathcal{V}\mbox{-Fun}(\mathcal{A}, \mathcal{V})$ is the category of $\mathcal{V}$-functors from $\mathcal{A}$ to $\mathcal{V}$, and $A\mbox{-Mod}$ is the category of unitary left $A$-modules.
In the second part, we develop further the theory of geometrical purity, which was also studied in [EGO]. We let $\mathcal{E}_{\otimes}$ denote the \textit{geometrical pure exact structure} on $\mathcal{V}$, which consists of short exact sequences in $\mathcal{V}$ that remain exact under `tensor product' by any object. We show that, under mild conditions, there exists a small category $\mathcal{A}$, which also has a $\mathcal{V}$-category structure, and an embedding $$(\mathcal{V}, \mathcal{E}_{\otimes}) \longrightarrow \mathcal{V}\mbox{-Fun}(\mathcal{A}, \mathcal{V})$$ such that $\mathcal{E}_{\otimes}\mbox{-Inj} \cong \mbox{Inj}(\mathcal{V}\mbox{-Fun}(\mathcal{A}, \mathcal{V}) )$.
The talk is formed by certain results of two joint works-in-progress with Henrik Holm and Simone Virili, which has been supported by the grant CONICYT/FONDECYT/Iniciación/11170394.
References
[EGO] Estrada, S.; Gillespie, J & Odabaşı, S. (2014). Pure exact structures and the pure derived category of a scheme. Mathematical Proceedings of the Cambridge Philosophical Society, 1-14. doi:10.1017/S0305004116000980
18:20-18.50
Ibrahim Assem (Université de Sherbrooke, Canadá)
From the potential to the first Hochschild Cohomology group of a cluster tilted algebra
This is a report on joint work with Juan Carlos Bustamante, Sonia Trepode y Yadira Valdivieso-Diaz.
We give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.