arXiv subjects
Publications and source records attributed to Nizar Ben Fraj.
The purpose of this paper is to study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We introduce a suitable cohomology and discuss Infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide some examples.
Over the $(1,n)$-dimensional real supercircle, we consider the $\mathcal{K}(n)$-modules of linear differential operators, $\frak{D}^n_{λ,μ}$, acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules for $n=1$. We also prove that $\frak{D}^{n}_{λ,μ}$ and $\frak{D}_{ρ,ν}^{n}$ are isomorphic for $ρ=\frac{2-n}{2}-μ$ and $ν=\frac{2-n}{2}-λ$. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of Differential Operators on the Real Line, Functional Analysis and Its Applications, Vol. 35, No. 1, pp. 13--18, 2001.]
We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1,\,2)-dimensional real superspace. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. This work is a simplest generalization of a result by Basdouri and Ben Ammar [Cohomology of $\frak {osp}(1|2)$ with coefficients in $\frak{D}_{λ,μ}$.
Over the $(1,n)$-dimensional real superspace, $n>1$, we classify $\mathcal{K}(n)$-invariant binary differential operators acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of %the Lie superalgebra $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities--a superisation of a result by Feigin and Fuchs. We explicitly give 1-cocycles spanning these cohomology spaces.
Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that any generic formal $\mathfrak{osp}(1|2)$-trivial deformation of this $\mathcal{K}(1)$-module is equivalent to a polynomial one of degree $\leq4$. This work is the simplest superization of a result by Bouarroudj [On $\mathfrak{sl}$(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127]. Further superizations correspond to $\mathfrak{osp}(N|2)$-relative cohomology of the Lie superalgebras of contact vector fields on $1|N$-dimensional superspace.
We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie superalgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced on K(1) by the canonical central extension of $SΨDO$.
We investigate the first cohomology space associated with the embedding of the Lie superalgebra $\cK(2)$ of contact vector fields on the (1,2)-dimensional supercircle $S^{1\mid 2}$ in the Lie superalgebra $\cSΨ\cD \cO(S^{1\mid 2})$ of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we show that this space is ten-dimensional with only even cocycles and we give explicit expressions of the basis cocycles.