On sl(2)-equivariant quantizations

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

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

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $\alpha$. We investigate the space of symmetric contravariant tensor fields on $M$ 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 $\alpha$ 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 $sp(2n+2)$. 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 groups of diffeomorphims related to the modules of differential operators on a smooth manifold

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

Deformations of modules of differential forms

We study non-trivial deformations of the natural action of the Lie algebra $\mathrm{Vect}({\mathbb R}^n)$ on the space of differential forms on ${\mathbb R}^n$. We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of \cite{ff}.Comment: Published by JNMP at http://www.sm.luth.se/math/JNM

Classical Poisson algebra of a vector bundle : Lie-algebraic characterization

We prove that the Lie algebra $\mathcal{S}(\mathcal{P}(E,M))$ of symbols of linear operators acting on smooth sections of a vector bundle $E\to M,$ characterizes it. To obtain this, we assume that $\mathcal{S}(\mathcal{P}(E,M))$ is seen as ${\rm C}^\infty(M)-$module and that the vector bundle is of rank $n>1.$ We improve this result for the Lie algebra $\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators. We obtain a Lie algebraic characterization of vector bundles with $\mathcal{S}^1(\mathcal{P}(E,M))$ without the hypothesis of being seen as a ${\rm C}^\infty(M)-$module.Comment: 16 page

On quasi quantum Poisson algebras : Lie-algebraic characterization

We prove a Lie-algebraic characterization of vector bundle for the Lie algebra $\mathcal{D}(E,M)$ of all linear operators acting on sections of a vector bundle $E$. We obtain similar result for its Lie subalgebra $\mathcal{D}^1(E,M)$ of all linear first-order differential operators. Thanks to a well-chosen filtration, $\mathcal{D}(E,M)$ becomes $\mathcal{P}(E,M)$ and we prove that $\mathcal{P}^1(E,M)$ characterizes the vector bundle without the hypothesis of being seen as module on the space of smooth functions of $M$.Comment: 17 page