arXiv subjects
Publications and source records attributed to Mabrouk Ben Ammar.
We compute cohomology spaces of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{λ,μ}$ of $n$-ary differential operators acting on weighted densities ($λ\in\mathbb{R}^n$ and $μ\in\mathbb{R}$).
In this paper, we consider the known question of the computation of the cohomology spaces for the natural action of the Lie algebra $\mathfrak{sl}(2)$ acting on the space $\Dlm$ of multidifferential operators on the line. We first show that these cohomology spaces are entirely characterized by a linear map $f$ between two explicit finite dimensional spaces, then we prove that this map has always maximal rank, allowing to get explicit recursive expression for their dimensions.
We study formal deformations of the natural $\osp(n|2)$-action, $n\geq 3$, on the superspace $\Sc^n_d=\bigoplus_{k\geq 0}\Fc^n_{d-\frac{k}{2}}$ of symbols of linear differential operators on weighted densities over $\R^{1|n}$. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product $\Hd^1\vee \Hd^1\to \Hd^2$ which carries the quadratic obstructions. The answer is governed by the $\osp(n|2)$-invariant operators $A_k=η_1\cdotsη_n\partial_x^{k-1}$: the two cocycles $h_k$ and $\widetilde{h}_k$ spanning the off-diagonal part of $\Hd^1$ are exactly the two derivatives of the coboundary of $A_k$ with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each $k$, by a single non-trivial 2-cocycle $Ω_k$. If $2d\notin\N$ the space $\Hd^1\vee\Hd^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\in\N$ we obtain exactly $m$ quadratic integrability conditions, $τ_{2-n-k}(t_k-\widetilde{t}_k)+τ_k\widetilde{t}_k=0$, $1\leq k\leq m$, and we prove that they are also sufficient: no condition of order $\geq 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.
We consider the spaces $\mathcal{F}_μ$ of polynomial $μ$-densities on the line as $\mathfrak{sl}(2)$-modules and then we compute the cohomological spaces $\mathrm{H}^1_\mathrm{diff}(\mathfrak{sl}(2), \mathcal{D}_{\barλ,μ})$, where $μ\in \mathbb{R}$, $\barλ=(λ_1,\dots,λ_n) \in\mathbb{R}^n$ and $\mathcal{D}_{\barλ,μ}$ is the space of $n$-ary differential operators from $\mathcal{F}_{λ_1}\otimes\cdots\otimes \mathcal{F}_{λ_n}$ to $\mathcal{F}_μ$.
We consider a nonlinear representation of a Lie algebra which is regular on an abelian ideal, we define a normal form which generalizes that defined in [D. Arnal, M. Ben Ammar, M. Selmi, {\rm Normalisation d'une représentation non linéaire d'une algèbre de Lie}. {\it Annales de la Faculté des Sciences de Toulouse}, 5e série, tome 9, No 3, 1988, p 355--379.]
We consider the $μ$-densities spaces $\mathcal{F}_μ$ with $μ\in\mathbb{R}$, we compute the space $\mathrm{H}^1_\mathrm{diff}(\mathfrak{aff}(1),\mathrm{D}_{λ,μ})$ where $λ=(λ_1,\dots,λ_n)\in\mathbb{R}^n$ and $\mathrm{D}_{λ,μ}$ is the space of $n$-ary differential operators from $\mathcal{F}_{λ_1}\otimes\cdots\otimes\mathcal{F}_{λ_n}$ to $\mathcal{F}_μ$. We also compute the super analog space $\mathrm{H}^1_\mathrm{diff}(\mathfrak{aff}(1|1),\mathfrak{D}_{λ,μ})$.
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 consider the sl(2)-module structure on the spaces of symbols of differential opera- tors acting on the spaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this structure and we prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem getting the same results.
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.
We compute the first cohomology of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{λ,ν;μ}$ of bilinear differential operators acting on weighted densities. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on $\mathbb{R}\mathbb{P}^1$ acting on bilinear differential operators, International Journal of Geometric Methods in Modern Physics (2005), {\bf 2}; N 1, 23-40].
We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_δ^n=\bigoplus_{j=0}^n \mathcal{F}_{δ-j}$. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space $\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{ν,μ})$ where ${\cal D}_{ν,μ}$ is the space of differential operators from $\mathcal{F}_ν$ to $\mathcal{F}_μ$. Necessary second-order integrability conditions of any infinitesimal deformations of $\mathcal{S}_δ^n$ are given. We describe completely the formal deformations for some spaces $\mathcal{S}_δ^n$ and we give concrete examples of non trivial deformations.
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 entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{λ,μ}$ of differential operators on the super circle, acting on densities.
We consider the action of Vect_{Pol}(R) by Lie derivative on the spaces of symbols of differential operators. We study the deformations of this action that become trivial once restricted to sl(2). Necessary and sufficient conditions for integrability of infinitesimal deformations are given.
We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{λ,μ})$ ($λ, μ\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{λ,μ}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)