47 research outputs found

    Cohomology of groups of diffeomorphims related to the modules of differential operators on a smooth manifold

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    Let MM be a manifold and T∗MT^*M be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of MM with values in the space of linear differential operators acting on C∞(T∗M).C^{\infty} (T^*M). When MM is the nn-dimensional sphere, SnS^n, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of SnS^n, with coefficients in the space of linear differential operators acting on contravariant tensor fields.Comment: arxiv version is already officia

    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

    SO(4)-symmetry of mechanical systems with 3 degrees of freedom

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    We answered the old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra o(4) by means of the Poisson brackets? We presented a system which is not centrally symmetric, but has such 6 first integrals. We showed also that not every mechanical system with 3 degrees of freedom possesses such Lie algebra o(4).Comment: 8 pages. A mistake in the last Subsection of version 1 is corrected together with its corollarie

    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

    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
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