### A class of Lie and Jordan algebras realized by means of the canonical commutation relations

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This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to...

En esta nota se presenta en primer lugar una introducción autocontenida a la cohomología de álgebras de Lie, y en segundo lugar algunas de sus aplicaciones recientes en matemáticas y física.

Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials.

Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups.

Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{k}$ of characteristic 0, and $\U0001d524=LieG$. Let $(e,h,f)$ be an $\U0001d530{\U0001d529}_{2}$-triple in $\U0001d524$ with $e$ being a long root vector in $\U0001d524$. Let $(\xb7,\xb7)$ be the $G$-invariant bilinear form on $\U0001d524$ with $(e,f)=1$ and let $\chi \in {\U0001d524}^{*}$ be such that $\chi \left(x\right)=(e,x)$ for all $x\in \U0001d524$. Let $\mathcal{S}$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $\mathcal{S}$; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains an ideal...

Constructing an even Poisson algebra from a Gerstenhaber algebra by means of an odd derivation of square 0 is shown to be possible in the category of Loday algebras (algebras with a non-skew-symmetric bracket, generalizing the Lie algebras, heretofore called Leibniz algebras in the literature). Such “derived brackets” give rise to Lie brackets on certain quotient spaces, and also on certain Abelian subalgebras. The construction of these derived brackets explains the origin of the Lie bracket on...

It is already known that any filiform Lie algebra which possesses a codimension 2 solvable extension is naturally graded. Here we present an alternative derivation of this result.