The simple positive energy representations for an affine vertex algebras associated to higher rank Lie algebras of type $A$ and admissible levels whose top spaces are induced modules of cuspidal representation of type $A$ are extensively studied. Note that all simple weight modules with finite-dimensional weight spaces are classified. However, the general classification of simple admissible weight modules of type $A$ is not know. In this talk, we explain the recent developments in the general open problem.

Let $A$ be an associative algebra over a field $F$ of characteristic zero, $F\langle X \rangle$ be the free algebra generated by the countable set $X=\{x_1,x_2,\ldots \}$ and $W$ be a unitary algebra over $F$. Then $A$ is called $W$-algebra if it has a structure of $W$-bimodule with some additional conditions. 

In 1972, Kantor introduced the class of conservative algebras, which contains many other important classes of algebras, for example, associative, Lie, Jordan, and Leibniz algebras. Initially, we will discuss some known results about conservative algebras, and especially the algebra $U(n)$ (space of bilinear multiplications on the n-dimensional space $V_n$). Then, we will present results obtained on the study of the Kantor product (product defined in $U(n)$). In particular, we will study the Kantor product of some finite-dimensional algebras.

The Monster group $M$ is the largest sporadic simple group. It is also the group of automorphisms of $196, 884$-dimensional Fischer-Norton-Griess algebra $V_M$. In 2009, A. A. Ivanov offered an axiomatic approach to studying the structure of $V_M$ by introducing the notions of Majorana algebra and Majorana representation. Later, the theory developed, and Majorana representations of several groups were constructed. Our talk is dedicated to the existence of standard Majorana representations of 3-transposition groups for the Fischer list.

A Kirkman Triple System (KTS) is called $m$-pyramidal if there exists a subgroup $G$ of its automorphism group that fixes $m$ points of the KTS and acts regularly on the other points. Such a group $G$ admits a unique conjugacy class $C$ of involutions (elements of order 2) and $|C|=m$. We call groups with this property $m$-pyramidal. We prove that, if $m$ is an odd prime power $p^k$, then every $m$-pyramidal group is solvable if and only if either $m=9$ or $k$ is odd. We also determine the sizes of the vertex sets of the $m$-pyramidal KTS when $m$ is a prime number.

Seguindo a ideia no artigo de Hu e Vercruysse [1], introduzimos morfismos parciais em uma categoria arbitrária $\textbf{C}$, de modo que ações parciais de um monoide $M$ em um conjunto $X$ correspondem a certas funções de $M$ para o conjunto de classes de isomorfismo de morfismos parciais de $X$ para $X$ na categoria de conjuntos.

In this talk we present some properties of generalized torsion elements in groups. Moreover, we try connect this "new'' concept with the usual concept of torsion in some standard class of groups (e.g., nilpotent, FC-groups). This presentation is mainly based in the following papers [1,2,3,4]. This is joint work with C. Schneider and D. Silveira.   
      

References

[1] R. Bastos, C. Schneider and D. Silveira. Generalized torsion elements in groups.  To appear in Arch. Math. Basel (2023), arXiv:2302.09589.  

Let us consider the action of the general linear group $\mathrm{GL}_n(\mathbb{C})$ on the direct product $\mathcal{M}_n^d$
of $d$ copies of $\mathcal{M}_n$ by simultaneous conjugation sending $(X_1,\ldots, X_d)$ to $(gX_1g^{-1},\ldots,gX_dg^{-1})$
for any $g\in \mathrm{GL}_n(\mathbb{C})$ . This induces an action of $\mathrm{GL}_n(\mathbb{C})$ on the algebra $\mathbb{C}[\mathcal{M}_n^d]$ of polynomial
functions on $\mathcal{M}_n^d$. The algebra of invariants under this action, $\mathbb{C}[\mathcal{M}_n^d]^{\mathrm{GL}_n}$, is an important