In the previous talk, we revisited some general aspects of the representation theory of the Lie algebra $\mathrm{gl}(n)$ and used them as motivation to study Gelfand-Tsetlin modules via Drinfeld categories. In this seminar, we will take a step back to build a solid foundation before diving into Gelfand-Tsetlin modules. We will start by recalling key results from the classical representation theory of $\mathrm{sl}(2)$, ensuring that we have the necessary tools and intuition. With this groundwork in place, we will then explore how Gelfand-Tsetlin modules appear in this setting.

Let $\mathcal{C}\subseteq\mathbb{N}^p$ (for a non-zero natural number $p$) be a non-negative integer cone. A monoid $S \subseteq \mathcal{C}$ is called a $\mathcal{C}$-semigroup if its complement in the cone is finite. This structure naturally extends the classical notion of numerical semigroups, which are submonoids of the natural numbers with a finite complement in the set of natural numbers.

Nos últimos anos, houve um grande interesse em detectar propriedades do grupo fundamental $\pi_1(M)$ de uma $3$-variedade por meio de seus quocientes finitos ou, mais conceitualmente, pelo seu completamento profinito. Isso motiva o estudo do completamento profinito $\widehat{\pi_1(M)}$ do grupo fundamental de uma $3$-variedade. Um trabalho recente de 2017 de H. Wilton e P. Zalesskii mostra que as decomposições típicas de grupos de $3$-variedades, como produtos livres com amalgamação, extensões HNN e grafos de grupos, são preservadas sob o completamento profinito.

In this talk, extensions of solvable Lie algebras are considered. The method of central extension is mostly used to obtain a classification of nilpotent algebras. This method was generalized for the solvable Lie algebras by T. Sund in 1979. We investigate extensions of solvable Lie algebras with naturally graded filiform nilradicals. Moreover, we generalize this method for the solvable Leibniz algebras and find extensions of solvable Lie algebras with null-filiform and filiform nilradicals.

In the 1960's. W. Leavitt studied universal algebras which do not satisfy the \emph{Invariant Basis Number Property (IBN)}. These are algebras that do not have a well-defined rank, that is, algebras for which $R^m\cong R^n$ ($m\neq n$) as $R$-modules which are later called the \emph{Leavitt algebras of module type} $(m,n)$. In 2005, the Leavitt algebra of type $(1,n)$ was found to be the so-called \emph{Leavitt path algebra} of a certain directed graph. 

In the representation theory of finite-dimensional simple Lie algebras $\mathfrak{g}$, two categories of modules stand out due to their contrasting nature. The first is the category of weight modules, consisting of $\mathfrak{g}$-representations where a fixed Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ acts semisimply. This category has been extensively studied over the past decades, with a classification of simple modules having finite-dimensional weight spaces obtained by O. Mathieu through the introduction of a special class of modules known as coherent families.

The rational numbers have been used to measure quantities since ancient times; however, their implementation in computer languages raises a significant problem: zero has no inverse. To address this issue, J. Bergstra and J. Tucker introduced an algebraic structure called a meadow, which allows for the inversion of zero.

In this seminar we will talk about results concerning faithful representations of abelian groups as automorphisms of one-rooted trees. In particular, we show the self-similarity of the free abelian group of uncountable rank and of the Baer-Specker group.

Let $G$ be a group and consider the group $\chi(G)$ obtained from the free product $G \ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. In the last 44 years, this group has been a formidable tool for obtaining finiteness conditions in Group Theory. In this talk, we want to present some important results related to the group $\chi (G)$. Moreover, we want to establish some properties of the exponent of $\chi(G)$ when $G$ has finite exponent. 

In the context of the theory of Gelfand-Tsetlin modules, Drinfeld categories were introduced in 2017 by V. Futorny et al. to prove that every irreducible 1-singular Gelfand-Tsetlin module is isomorphic to a subquotient of the universal 1-singular Gelfand-Tsetlin module. The authors also observed that these categories could be used to generalize the classification of Gelfand-Tsetlin modules for $\mathrm{sl}(n)$, which, at that time, was only known for $\mathrm{sl}(3)$.