The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group...

In this paper we use cohomology of Lie algebras to study the variety of laws associated with filiform Lie algebras of a given dimension. As the main result, we describe a constructive way to find a small set of polynomials which define this variety. It allows to improve previous results related with the cardinal of this set. We have also computed explicitly these polynomials in the case of dimensions 11 and 12.

Leibniz algebras are a non-commutative version of usual Lie algebras. We introduce a notion of (pre)crossed Leibniz algebra which is a simultaneous generalization of notions of representation and two-sided ideal of a Leibniz algebra. We construct the Leibniz algebra of biderivations on crossed Leibniz algebras and we define a non-abelian tensor product of Leibniz algebras. These two notions are adjoint to each other. A (co)homological characterization of these new algebraic objects enables us to...

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...

On présente une définition et une construction unifée des homologies et cohomologies d’algèbres et de modules sur ces algèbres et de modules sur ces algèbres dans le cas d’algèbres associatives ou commutatives ou de Lie ou de Gertsenhaber. On sépare la construction linéaire des cogèbres ou bicogèbres qui traduisent les symétries des relations de définition de la structure de la partie structure qui apparaît ici comme une codérivation de degré 1 et de carré nul de la cogèbre ou de la bicogèbre.

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.

Considering a Poisson algebra as a nonassociative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this nonassociative algebra. We give a natural interpretation of deformations which preserve the underlying associative structure and of deformations which preserve the underlying Lie algebra and we compare the associated cohomologies with the Poisson cohomology parametrizing the general deformations of Poisson algebras.

In this paper a construction of characteristic classes for a subfoliation $(F_1,F_2)$ is given by using Kamber-Tondeur’s techniques. For this purpose, the notion of $(F_1,F_2)$-foliated principal bundle, and the definition of its associated characteristic homomorphism, are introduced. The relation with the characteristic homomorphism of $F_i$-foliated bundles, $i=1,2$, the results of Kamber-Tondeur on the cohomology of $g$-$DG$-algebras. Finally, Goldman’s results on the restriction of foliated bundles to the leaves of a foliation...

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space $V$. The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on $V$ (resp. graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras and observe that the ${\text{Lod}}_{\infty }$ category contains the ${\text{L}}_{\infty }$ category as...

Dans cet article, nous définissons des modules de (co)-homologie ${ℌ}_0(𝔊,𝔄)$, ${ℌ}_1(𝔊,𝔄)$, ${ℌ}^{\circ }(𝔊$, $𝔄)$, ${ℌ}^1(𝔊,𝔄)$ où $𝔊$ et $𝔄$ sont des algèbres de Lie munies d’une structure supplémentaire (algèbres de Lie croisées), qui satisfont les propriétés usuelles des foncteurs cohomologiques. Si $A$ est une $k$-algèbre, nous utilisons ces modules d’homologie pour comparer le groupe d’homologie cyclique $H{C}_1\left(A\right)$ avec un analogue additif du groupe de $K$-théorie de Milnor $K_2^{\mathrm{Madd}}\left(A\right)$.