Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the partial projective $\sigma $-representations of $G$ into algebras over a field $\kappa .$ Using the relation between partial projective representations and twisted partial actions we endow $\kappa_{par}^{\sigma}G$   with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of $\kappa_{par}^{\sigma}G.$   Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral se

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.