# RTAA

## PAST WEBINARS

16 September, 2022

## On generalized (bound) path algebras

The generalized path algebras were introduced in (Coelho, Liu, 2000), in order to generalize the well-known concept of path algebras over a quiver. In order to construct a generalized path algebra, we associate an algebra to each vertex of a quiver (instead of only the base field as it happens with ordinary path algebras), and we consider paths intercalated by elements from the algebras to form a vector space basis of the generalized path algebra. Multiplication is then naturally defined by concatenation of paths and using the multiplications of the algebras in each vertex. We have recently defined the generalized bound path algebras, which are a quotient of a generalized path algebra by an ideal generated by relations. The aim of this talk is to introduce these concepts and to discuss some ideas which appear in recent works by the authors, which relate representation-theoretical properties of a given generalized (bound) path algebra with those of the algebras used in its construction. This work was produced under supervision by Dr. Flávio Ulhoa Coelho (IME-USP) and the authors acknowledge financial support by São Paulo Research Foundation (grant FAPESP #2018/18123-5 and #2020/13925-6).

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24 June, 2022

## O estudo da estrutura das (co)homologias de Hochschild da classe de álgebras de dimensão finita $A$ com: $Hom_{A-A} (DA,A) =0$

O interesse sobre essa classe de álgebras nasceu na tentativa da generalização das sequências exatas curtas de grupos de cohomologia de Hochschild dos trabalhos [1] e [2]: \begin{equation*} 0 \longrightarrow H^0(TA,DA) \longrightarrow HH^0(TA) \longrightarrow HH^0(A) \longrightarrow 0 \end{equation*} \begin{equation*} 0 \longrightarrow H^1(TA,DA)\oplus\mathcal{E}(DA) \longrightarrow HH^1(TA) \longrightarrow HH^1(A) \longrightarrow 0 \;, \end{equation*} em que $\mathcal{E}(DA)$ é o $k$-subespaço de $Hom_{A\text{-}A}(DA, A)$ formado por todos os morfismos de $A$-bimódulos $f \colon DA \to A$ tais que, para quaisquer $x, y \in DA$, temos que $f(x)y+xf(y) =0$.

Nosso foco é estudar os grupos de cohomologia de Hochschild da classe de álgebras com a seguinte condição: o grupo dos homomorfismos de $A$-bimódulos $DA \to A$ é trivial, isto é, $Hom_{A\text{-}A}(DA,A)=0$, em que $DA$ é o dual da álgebra $A$. Uma classe de álgebras bem conhecida que satisfaz essa condição são as álgebras Schurian.

[1] ASSEM, I., GATICA, M.A., SCHIFFLER, R., & TAILLEFER, R. (2016). HOCHSCHILD COHOMOLOGY OF RELATION EXTENSION ALGEBRAS. Journal of Pure and Applied Algebra, (220):2471--2499.
[2] CIBILS, C., MARCOS, E., REDONDO, M., & SOLOTAR, A. (2003). COHOMOLOGY OF SPLIT ALGEBRAS AND OF TRIVIAL EXTENSIONS. Glasgow Mathematical Journal, 45(1), 21-40.

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17 June, 2022

## O método de Butler aplicado a $\mathbb Z_p[C_p\times C_p]$-módulos de permutação.

Seja $p$ um número primo e $G$ um $p$-grupo finito com subgrupo normal $N$ de ordem $p$. Vamos denotar por $\mathbb Z_p$ o anel dos inteiros $p$-ádicos. Em um trabalho recente, Zalesskii e MacQuarrie forneceram uma caracterização dos $\mathbb Z_pG$-módulos de permutação em termos de módulos para $\mathbb Z_p[G/N]$. A caracterização depende de duas condições, mas a necessidade dessas condições não era conhecida. Em um trabalho junto com MacQuarrie, aplicamos uma correspondência devida a Butler, que permite associar reticulados a $n$-uplas de $\mathbb Z_pG$-módulos, para mostrar a necessidade dessas condições por exibir um contraexemplo não trivial para a afirmação: se ambos os $N$-invariantes e $N$-coinvariantes de um certo reticulado $U$ são de permutação, então $U$ também o é.

Nesta conversa, vamos ver como essa correspondência é feita no caso em que $G=C_p\times C_p$ e como a utilizamos para construir $\mathbb Z_pG$-reticulados que são contraexemplos para cada primo $p\geq 3$.

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10 June, 2022

## Uma trisecção das componentes do quiver de Auslander-Reiten.

O conceito de trisecções na categoria de módulos sobre uma álgebra tem sido bastante utilizado na teoria de representações. Uma dessas trisecções envolve as subcategorias ${\mathcal L}_{\Lambda}$ e ${\mathcal R}_{\Lambda}$ definidas como segue. Para uma álgebra $\Lambda$, $${\mathcal L}_{\Lambda} \ = \ \{ X \in \mbox{ ind}\Lambda \colon \mbox{ pd}_\Lambda Y \leq 1 \mbox{ quando existir um caminho } Y \leadsto X \}$$ $${\mathcal R}_{\Lambda} \ = \ \{ X \in \mbox{ ind}\Lambda \colon \mbox{ id}_\Lambda Y \leq 1 \mbox{ quando existir um caminho} X \leadsto Y \}$$ onde $M \leadsto N$ significa que existe um caminho de morfismos n\~ao nulos de $M$ a $N$. Relembre também que pd$_\Lambda N$ e id$_\Lambda N$ indicam, respectivamente, as dimensões projetiva e injetiva de um módulo $N$. Desta forma, $({\mathcal L}_{\Lambda}\setminus {\mathcal R}_{\Lambda}, {\mathcal L}_{\Lambda} \cap {\mathcal R}_{\Lambda}, {\mathcal R}_{\Lambda} \setminus {\mathcal L}_{\Lambda})$ induz uma trisecção na categoria ind$\Lambda$ para classes de álgebras como quasitilted e shod. Vamos discutir quando é que $({\mathcal L}_{\Lambda}\setminus {\mathcal R}_{\Lambda}, {\mathcal L}_{\Lambda} \cap {\mathcal R}_{\Lambda}, {\mathcal R}_{\Lambda} \setminus {\mathcal L}_{\Lambda})$ induz uma trisecção no quiver de Auslander-Reiten de uma álgebra.

[1] Coelho, F. U., A trisection in the Auslander-Reiten quiver, Colloquium Mathematicum 2022.

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03 June, 2022

## $\mathcal{P}^{<\infty}$-contravariant finiteness and strong tilting iteration via corner algebras

A sufficient condition for the verification of the finitistic dimension conjecture for an Artin algebra $\Lambda$ is that the subcategory $\mathcal{P}^{<\infty}(mod-\Lambda)$ of (finitely generated) modules of finite projective dimension be contravariantly finite in the category $mod-\Lambda$. This last condition is in turn equivalent to the existence of a strong tilting $\Lambda$-module. Two natural problems arise: 1) Give methods to construct Artin algebras $\Lambda$ with the mentioned contravariant finiteness property; 2) Assuming that property, and hence also the existence of a strong tilting $\Lambda$-module $T$, when is it true that the (strongly tilted) endomorphism algebra $\tilde{\Lambda}=End(T_\Lambda)$ has the mentioned contravariant property? and, if so, can this process of strong tilting be iterated indefinitely?. We will show that, by an adequate choice of an idempotent $e\in\Lambda$, the answers to these questions can already be checked by looking at the corner algebra $e\Lambda e$.

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27 May, 2022

## Homological approximations in persistence theory

Multiparameter persistence modules are defined over a wild algebra and therefore they do not admit a complete discrete invariant. One thus tries in persistence theory to “approximate” such a module by a more manageable class of modules. Using that approach we define a class of invariants for persistence modules based on ideas from homological algebra.

This is a report on joint work with Benjamin Blanchette and Eric Hanson. No prior knowledge of persistence theory is required.

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20 May, 2022

## Maranda's theorem for pure-injective modules

Let $R$ be a discrete valuation domain with maximal ideal generated by $\pi$ and let $\Lambda$ be an $R$-order inside a separable algebra. For example, $R=\mathbb{Z}_{(p)}$ and $\Lambda=\mathbb{Z}_{(p)}G$ where $G$ is a finite group.
Maranda's theorem for lattices over orders states that, under these assumptions, there exists a natural number $k$ such that the functor from the category of $\Lambda$-lattices to the category of finitely presented modules over the Artin algebra $\Lambda/\Lambda\pi^k$,given by $M\mapsto M/M\pi^k$, reflects isomorphism types and, when $R$ is complete, preserves indecomposability. Motivated by model theory of modules and a desire to study modules beyond the finitely generated ones, we prove a variant of this theorem for the class of $R$-reduced $R$-torsionfree pure-injective $\Lambda$-modules.
In this talk I will briefly motivate the study of pure-injective modules and give applications of both the classical version of Maranda's theorem and the version for pure-injective modules.

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13 May, 2022

## Full exceptional sequences in the derived category of gentle algebras

In general it is difficult to determine whether the bounded derived category of a finite dimensional algebras has full exceptional sequences or not. In this talk I will focus on the class of gentle algebras and determine for which gentle algebras the bounded derived category admits full exceptional sequences in terms of an underlying geometric surface model. Gentle algebras are finite dimensional algebras that are connected to many different areas of mathematics such as cluster theory, N=2 gauge theory and homological mirror symmetry of surfaces.

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29 April, 2022

## Triangular LIT algebras

In this talk I am going to present some results concerning Lat-Igusa-Todorov algebras (LIT algebras for short). The notion of LIT algebra, given by D. Bravo, M. Lanzilotta, O. Mendoza and J. Vivero in [1], is a way of generalizing the concept of Igusa-Todorov algebra given by J. Wei in [2]. The results I am going to present can be found in [2103.12120] Triangular Lat-Igusa-Todorov algebras (arxiv.org).

[1] D. Bravo, M. Lanzilotta, O. Mendoza, J. Vivero. Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras. Journal of Algebra Vol. 580, 63-83, 2021. DOI: https://doi.org/10.1016/j.jalgebra.2021.02.036.
[2] J. Wei. Finitistic dimension and Igusa-Todorov algebras. Adv. Math. 222, no. 6, 2215-2226, 2009.

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08 April, 2022

## $S$-homogenous triples and aplications to algebras with 2-relations

This talk is on a joint work with Yuri Volkov. We define the notion os S-homogeneous triples and $S$-homogeneous algebras and give an equivalence of categories which we apply to study koszulity, etc... The results are used for the partial classifications of homogeneous algebras whose defining ideal is generated by two relations.

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01 April, 2022

## Quiver Radial Isotropy and Applications to the Paulsen Problem in Frame Theory

Sigma-critical representations are quiver representations that satisfy certain matrix equations. They arise naturally in the context of Kempf-Ness theorem on closed orbits in Invariant Theory. After introducing all the relevant concepts, I will first present the quiver radial isotropy theorem which gives necessary and sufficient conditions for the orbit of a quiver representation to contain a sigma-critical representation. I will then explain how this result can be used to solve the Paulsen Problem for matrix frames. This is based on joint work with Jasim Ismaeel.

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03 December, 2021

## Graph monoids and preprojective roots of Coxeter groups

A graph monoid $\mathfrak{M}$ of a finite undirected graph $\Gamma$ without loops and multiple edges was introduced in 1969 by Cartier and Foata in order to give a new proof and a noncommutative version of the celebrated MacMahon Master Theorem, which is a result in enumerative combinatorics and linear algebra. If $\Gamma$ is given an acyclic orientation $\Lambda$ and thus becomes a quiver $(\Gamma, \Lambda)$, a certain subset (not a submonoid) of $\mathfrak{M}$ is closely related to the $(+)$-admissible sequences of vertices introduced in 1973 by Bernstein, Gelfand, and Ponomarev in their seminal paper on quiver representations. If $\Gamma$ is the Coxeter graph of a Coxeter group $\mathcal{W}$, we use a canonical surjective monoid homomorphism $\mathfrak{M} \rightarrow \mathcal{W}$ to study preprojective roots of $\mathcal{W}$.

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26 November, 2021

## Wide intervals and mutation

Torsion pairs in the category $\textrm{mod} A$ of finite-dimensional modules over a finite-dimensional algebra have played an important role in many aspects of modern representation theory, including the study of cluster theory, $t$-structures and stability conditions. If we consider the collection of all torsion pairs ordered by inclusion of the torsion class, then we obtain a complete lattice. In recent work by Asai and Pfeiffer, the so-called `wide' intervals in this lattice are studied from the perspective of lattice theory. These intervals arise naturally both in relation to stability conditions and also in $\tau$-tilting theory. In this talk we will discuss how the torsion pairs in $\textrm{mod} A$ are parametrised by 2-term cosilting complexes in the unbounded derived category $D(\textrm{Mod} A)$ and how wide intervals correspond to their mutations. This talk is based on ongoing joint work with Lidia Angeleri Hügel, Jan Stovicek and Jorge Vitória.

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19 November, 2021

## On Brauer configurations induced by finite groups

Aspects of the representation theory of a Brauer configuration algebra, such as the Cartan matrix and the module length of the respective indecomposable modules, have shown the existence of combinatorial relations that are satisfied by any finite group. The aim of this talk is to present both the Cartan matrix of a Brauer configuration algebra and the module length of any indecomposable projective module associated to a Brauer configuration algebra. Then using these aspects and the concept of subgroup-occurrence of an element in a group, we demonstrate a couple of combinatorial relations satisfied by any finite group, when considering the Brauer configuration induced by a finite group of an order different from a prime number.

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12 November, 2021

## Relative strongly Gorenstein objects in abelian categories

The purpose of this talk is to present a relativization of the concept of strongly Gorenstein projective module, introduced by Bennis and Mahdou in 2007. Given a pair $(\mathcal{A,B})$ of full subcategories $\mathcal{A}$ and $\mathcal{B}$ of an abelian category $\mathcal{C}$, we say that an object $C \in \mathcal{C}$ is periodic (or strongly) $(\mathcal{A,B})$-Gorenstein projective if there exists a short exact sequence $C \rightarrowtail A \twoheadrightarrow C$, with $A \in \mathcal{A}$, which remains exact after applying the contravariant functor $\text{Hom}(-,B)$ for every $B \in \mathcal{B}$. We shall show several properties, characterizations and examples of these objects in the case where $(\mathcal{A,B})$ is a hereditary pair (meaning that $\textrm{Ext}^i(A,B) = 0$ for every $A \in \mathcal{A}$, $B \in \mathcal{B}$ and $i \geq 1$). For instance, we shall see a relation with relative Gorenstein objects (2021 - Becerril, Mendoza and Santiago) and periodic objects (2020 - Bazzoni, Cortés-Izurdiaga and Estrada). Concerning specific examples, we shall exhibit a characterization of noetherian rings in terms of periodic Gorenstein injective modules relative to the pair (absolutely pure modules, injective modules). If time allows, we shall comment on a more general version of periodic $(\mathcal{A,B})$-Gorenstein projective objects, namely, $(\mathcal{A,B})$-Gorenstein projective objects with period $m \geq 1$. For these families of objects, an object is $(\mathcal{A,B})$-Gorenstein projective with periods $m$ and $n$ if, and only if, it is $(\mathcal{A,B})$-Gorenstein projective with period $\mathrm{gcd}(m,n)$.
This is a joint work in progress with Mindy Huerta (IMERL-UdelaR) and Octavio Mendoza (IMATE-UNAM).

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05 November, 2021

## $\tau$-tilting theory and stratifying systems

On the one hand, tau-tilting theory was introduced by Adachi, Iyama and Reiten in 2012 and quickly became central in representation theory of finite dimensional algebras. In this talk, after giving a brief overview of some central results in this theory, we will explain how every tau-rigid object in a module category induces at least one stratifying system, as defined by Erdmann and Sáenz. Time permitting, we will use these results to give a new proof of the fact that every complete stratifying system in the module category of a hereditary algebra is an exceptional sequence, first proven by Cadavid and Marcos.

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29 October, 2021

## Homological theory of k-idempotent ideals in dualizing varieties

In this talk, we develop the theory of $k$-idempotent ideals in the setting of dualizing varieties. Several results given previously by M. Auslander, M. I. Platzeck, and G. Todorov are extended to this context. Given an ideal $I$ (which is the trace of a projective module), we construct a canonical recollement which is the analogous to a well-known recollement in categories of modules over artin algebras. Moreover, we study the homological properties of the categories involved in such a recollement. Consequently, we find conditions on the ideal $\mathcal{I}$ to obtain quasi-hereditary algebras in such a recollement. Applications to bounded derived categories are also given. (Joint work with V. Santiago Vargas and M. L. S. Sandoval).

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22 October, 2021

## Triangular matriz categories and recollements

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 $M \in$ $\operatorname{Mod}\left(\mathcal{U} \otimes \mathcal{T}^{o p}\right)$ we will construct the triangular matrix category $\boldsymbol{\Lambda}:=\left[\begin{array}{ll}\mathcal{T} & 0 \\ M & \mathcal{U} \end{array}\right]$ and we prove that there is an equivalence $(\operatorname{Mod}(\mathcal{T}), \mathbb{G M o d}(\mathcal{U})) \simeq \operatorname{Mod}(\boldsymbol{\Lambda})$. We will show that if $\mathcal{U}$ and $\mathcal{T}$ are dualizing $K$-varieties and $M \in \operatorname{Mod}\left(\mathcal{U} \otimes \mathcal{T}^{o p}\right)$ satisfies certain conditions then $\boldsymbol{\Lambda}:=\left[\begin{array}{ll}\mathcal{T} & 0 \\ M& \mathcal{U}\end{array}\right]$ 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 [Q. Chen, M. Zheng. Recollements of abelian categories and special types of comma categories. J. Algebra. 321 (9), 2474-2485 (2009), Theorem 4.4].

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15 October, 2021

## Representation type via quiver Grassmannians

The representation type of a quiver $Q$ can be characterized by the geometric properties of the associated quiver Grassmannians:
(a) $Q$ is representation finite iff. all quiver Grassmannians are smooth and have cell decompositions into affine spaces;
(b) $Q$ is tame iff. all quiver Grassmannians have cell decompositions into affine spaces and if there exist singular quiver Grassmannians;
(c) $Q$ is wild iff. every projective variety occurs as a quiver Grassmannian.

With the exception of (extended) Dynkin type $E$, this result is proven in joint work with Thorsten Weist. The missing parts for Type $E$ were completed by Cerulli Irelli-Esposito-Franzen-Reineke. In this talk, we will explain this result and parts of its proof.

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08 October, 2021

## Generation of the unbounded derived category of a ring

If $R$ is a ring, recall that the derived category $D(R)$ is a category whose objects are chain complexes of $R$-modules. In recent decades, $D(R)$ has been recognized to be of great importance in studying the homological algebra of R.

There are several ways in which a class of objects might be said to “generate” $D(R)$. For example, it is an exercise to show that the projective $R$-modules generate $D(R)$ as a triangulated category with coproducts (or that the injective modules do if we use products in place of coproducts). It seems more unnatural to ask whether the injective modules generate $D(R)$ as a triangulated category with coproducts, but we will discuss how this question is related to well-known open questions such as the Finitistic Dimension Conjecture.

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01 October, 2021

## A link between representations of peak posets and cluster algebras

In this talk, we show a link between the theory of cluster algebras and the theory of representations of posets. Particularly, we give a geometric interpretation of the category of finitely generated socle-projective representations over the incidence algebra of a poset of type $A$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the realization of the cluster category of type $A$ as the category of all diagonals by Caldero-Chapoton-Schiffler. Moreover, given a poset $P$ of type $A$, we define a subalgebra $B(P)$ of a cluster algebra $B$ and we establish a sufficient condition to conclude $B=B(P)$. This is a joint work with R. Schiffler.

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24 September, 2021

## Um relacionamento funtorial entre álgebras e seus quocientes pelas $n$-ésimas potências do radical de Jacobson

A famosa construção de Gabriel provou que uma álgebra de dimensão finita pode ser vista como um quociente de uma álgebra de caminhos por um ideal admissível. A construção permitiu olhar para teoria de representação de álgebras de dimensão finita de um modo combinatório. Iusenko, MacQuarrie e Quirino exibiram uma adjunção que explica de maneira muito precisa a filosofia da construção de Gabriel. A adjunção deles é obtida utilizando uma relação de equivalência nos morfismos das álgebras. Nesta palestra, mostraremos que a construção de Gabriel pode ser obtida utilizando uma relação mais fina. Além disso, através de uma adjunção, provaremos que há uma classe de álgebras que podem ser aproximadas por álgebras quadráticas, isto é, um quociente de uma álgebra de caminhos por um ideal homogêneo de grau 2.

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17 September, 2021

## Invariants of gentle algebras

Gentle and locally gentle algebras are quadratic Koszul algebras. The isomorphism classes of gentle algebras are in bijection with dissections of certain types of surfaces.
Homological methods provide important information about this family of algebras. The invariants obtained in this way are in fact derived invariants.
In this talk I will describe derived invariants -some of them already known and some of them new- of gentle algebras obtained via their Hochschild cohomology and homology, taking all the structure into account: the homology and cohomology vector spaces, the cohomology graded commutative algebra, the Gerstenhaber bracket and the cap product.
This is a joint work with Cristian Chaparro Acosta, Sibylle Schroll and Mariano Suárez-Álvarez.

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10 September, 2021

## Álgebras seriais sem ciclos truncadas são derivadamente equivalentes a álgebras de incidência de posets

Dada uma álgebra serial sem ciclos truncada $\Lambda$, denotamos sua categoria derivada por $\mathcal{D}^b(\Lambda).$ Nesta palestra apresentaremos a construção de um complexo inclinante $T_{\Lambda}$ e definiremos um conjunto parcialmente ordenado $S_{\Lambda}$ tais que a álgebra de endomorfismos $End_{\mathcal{D}^b(\Lambda)} T_{\Lambda}$ é isomorfa à álgebra de incidência de poset $kS_{\Lambda}^{op}$. Com isso, usando o Teorema de Rickard podemos concluir que $\Lambda$ é derivadamente equivalente a $kS_{\Lambda}^{op}$.

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03 September, 2021

## Ação do grupo de tranças em sequências excepcionais e aplicações para álgebras hereditárias por partes

Nesta palestra, vamos introduzir o conceito de mutação de sequências excepcionais em categorias de feixes coerentes. Estas mutações permitem definir a ação do grupo de tranças sobre sequências excepcionais. Com o uso eficiente do Teorema de Riemann-Roch e a Teoria de Auslander-Reiten, podemos tirar bons resultados sobre as órbitas desta ação. Estes resultados nos trazem um melhor entendimento sobre as álgebras que são derivadamente equivalente a categoria de feixes coerentes sobre retas projetivas com peso.

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30 July, 2021

## Topics on the Igusa-Todorov functions

One of the most important conjecture in the representation theory of Artin algebras is the finitistic dimension conjecture. It states that $sup\{pd(M) : M\ \text{is a f.g module of finite projective dimension}\}$ is finite. In an attempt to prove the conjecture this conjecture Igusa and Todorov defined in “On finitistic global dimension conjecture for artin algebras”, two functions from the objects of modA to the natural numbers, which generalize the notion of projective dimension. Nowadays they are known as the Igusa-Todorov functions. In this talk I will introduce the Igusa-Todorov functions and their main properties. We will also see how to use these functions to prove the finitistic dimension conjecture in several contexts.

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23 July, 2021

## Feixes instantons e representações de aljavas

O seminário terá como tema apresentar uma nova compactificação para o espaço de módulos (moduli spaces) de feixes intantons de carga $1$ sobre $\mathbb P^3$. Isto será feito identificando estes feixes com representações de uma aljava com três vértices e oito flechas que atendem certas propriedades e olhando essas representações no espaço de módulos de King de representações teta-estáveis sobre esta aljava.

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16 July, 2021

## On the modular automorphism of twisted Calabi-Yau algebras

A twisted Calabi-Yau algebra $A$ is, by definition, endowed with an automorphism $\sigma:A\to A$, well-defined only up to inner automorphisms, that controls the duality theory for representations of $A$. The algebra is Calabi-Yau, and not only twistedly so, precisely when $\sigma$ can be taken to be inner and even the identity map of $A$, but in general it cannot. The objective of the talk will be to present some results that give information about this automorphism in the general situation, and show how it can be used to construct interesting and non-trivial invariants of the algebra, of automorphisms and derivations of the algebra, and of related objects. Some applications of these invariants to the problem of computing Hochschild cohomology and the automorphism groups of our algebras will be discussed.

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25 June, 2021

## Variety defined by the Groebner basis of a module

This is a joint project with Ed. Green and Schroll Sibylle.
Our set up is an algebra of the form $kQ/I$, ($I$ is admissible, for simplification).
Given a right module M we define, using its right groebner basis, a variety. Each point in this variety corresponds to a module which has the same tip set. Each module corresponding to a point in the variety has the same dimension vector of the original module. We denote this variety by $V(M)$. There is a natural injection of sets $V(M) \to V_d(KQ)$, where $V_d(kQ)$ is the usual variety of representations Q representations of dimension vector $d$. We do not know if the injection is a morphism of varieties ( I conjecture it is). We also do not know which subgroup of $Gl_d$ let the subset of the variety be invariant, so we would have a grupo acting on it. Some curious results, for instance, is that if the module is an indecomposable projective then our variety consists of one point. I am open to suggestions etc...

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18 June, 2021

## Álgebra de Árvore de Brauer e Blocos de Grupos Profinitos

Sejam $G$ um grupo profinito , $k$ um corpo de característica $p> 0$. A álgebra de grupo completa de G, $k[[G]]$, possui uma decomposição como um produto direto de álgebras pseudocompactas indecomponíveis, que chamaremos de blocos. Cada bloco de $k[[G]]$ tem associado um pro-$p$ subgrupo de $G$ chamado grupo de defeito. O grupo de defeito proporciona uma medida que tão próximo estará um bloco de ser uma álgebra semisimples. Nesta palestra explicaremos como associar a cada bloco um grupo de defeito e descreveremos os blocos cujo grupo de defeito é um grupo cíclico usando a estrutura de álgebra de árvore de Brauer.

Este trabalho foi feito junto com o John Macquarrie.

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11 June, 2021

## Preprojective algebras: classical and higher

I will give an introduction to representations of quivers, aimed at non-experts, and then introduce preprojective algebras of quivers. Iyama showed how many nice aspects of the representations of quivers generalise to certain algebras of higher global dimension, and the preprojective algebra can be defined in this setting. I will explain some results on higher preprojective algebras obtained in joint work with Osamu Iyama.

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04 June, 2021

## Moduli of flags of sheaves: a quiver description

In [1], the moduli spaces of framed flags of sheaves on $\mathbb P^2$ were described by means of representations of the so-called enhanced ADHM quiver. We first review those results, with a recent refinement concerning the chamber structure in the space of stability parameters. We shall then discuss the obstruction theory for these moduli spaces, showing in general that they have a perfect obstruction theory, and providing for specific choices of invariants a class of unobstructed points. Finally, open problems and possible further developments will be presented. This is a joint work with Rodrigo von Flach and Marcos Jardim.

1. R. A. von Flach and M. Jardim, Moduli spaces of framed flags of sheaves on the projective plane. Journal of Geometry and Physics 118 (2017), 138–168.

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14 May, 2021

## Higher Auslander-Reiten theory: what is it and how to understand it

Higher Auslander-Reiten theory was introduced by Osamu Iyama in the 2000s as a generalization of the theory developed by Maurice Auslander and Idun Reiten in the 1970s. In this talk, we will discuss what is this theory and how to understand it. In doing so, we will examine both the definitions of $n$-almost split sequence and of $n$-cluster tilting subcategory. In the end, we will also look at some open problems of the theory.

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07 May, 2021

## UFMG

Uma extensão de álgebras associativas é simplesmente uma álgebra associativa $A$ com subálgebra $B$. Em uma sequência de artigos recentes, Cibils, Lanzilotta, Marcos e Solotar colocaram condições sobre a extensão (extensões "limitadas") e mostraram que, sobre essas condições, duas propriedades homológicas valem para $B$ se, e somente se, valem para $A$: nomeadamente "dimensão global finita" e "suporte da homologia de Hochschild finita". Em particular, segue disso que a "Conjectura de Han" vale para $B$ se, e somente se, vale para $A$.