27 research outputs found

    Cohomology of osp(1∣2)\mathfrak {osp}(1|2) acting on linear differential operators on the supercircle $S^{1|1}

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    We compute the first cohomology spaces H1(osp(1∣2);Dλ,μ)H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{\lambda,\mu}) (λ,μ∈R\lambda, \mu\in\mathbb{R}) of the Lie superalgebra osp(1∣2)\mathfrak{osp}(1|2) with coefficients in the superspace Dλ,μ\mathfrak{D}_{\lambda,\mu} of linear differential operators acting on weighted densities on the supercircle S1∣1S^{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.

    Differential operators on supercircle: conformally equivariant quantization and symbol calculus

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    We consider the supercircle S1∣1S^{1|1} equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S1∣1S^{1|1} as the Lie superalgebra of contact vector fields; it contains the M\"obius superalgebra osp(1∣2)osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1∣2)osp(1|2). We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist

    Cohomology of osp(1∣2)\frak {osp}(1|2) acting on the space of bilinear differential operators on the superspace R1∣1\mathbb{R}^{1|1}

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    We compute the first cohomology of the ortosymplectic Lie superalgebra osp(1∣2)\mathfrak{osp}(1|2) on the (1,1)-dimensional real superspace with coefficients in the superspace Dλ,ν;μ\frak{D}_{\lambda,\nu;\mu} 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 RP1\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]

    Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

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    Let MM be an odd-dimensional Euclidean space endowed with a contact 1-form α\alpha. We investigate the space of symmetric contravariant tensor fields on MM as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities, the vertical cotangent lift of contact form α\alpha is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the symbol space is invariant with respect to the action of the projective contact algebra sp(2n+2)sp(2n+2). The preceding invariant operators lead to a decomposition of the symbol space (expect for some critical density weights), which generalizes a splitting proposed by V. Ovsienko

    Cohomology of the Lie Superalgebra of Contact Vector Fields on R1∣1\mathbb{R}^{1|1} and Deformations of the Superspace of Symbols

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    Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra K(1)\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 osp(1∣2)\mathfrak{osp}(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal osp(1∣2)\mathfrak{osp}(1|2)-trivial deformations of the K(1)\mathcal{K}(1)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal osp(1∣2)\mathfrak{osp}(1|2)-trivial deformation of this K(1)\mathcal{K}(1)-module is equivalent to a polynomial one of degree ≤4\leq4. This work is the simplest superization of a result by Bouarroudj [On sl\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 osp(N∣2)\mathfrak{osp}(N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1∣N1|N-dimensional superspace

    Projectively equivariant quantizations over the superspace Rp∣q\R^{p|q}

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    We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.Comment: 19 page

    On sl(2)-equivariant quantizations

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    By computing certain cohomology of Vect(M) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-projective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M). Contrariwise, for projective embeddings sl(2)-equivariant quantization exists.Comment: 09 pages, LaTeX2e, no figures; to appear in Journal of Nonlinear Mathematical Physic
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