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See detailExistence of natural and conformally invariant quantizations of arbitrary symbols
Mathonet, Pierre ULg; Radoux, Fabian ULg

in Journal of Nonlinear Mathematical Physics (2010), 17

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be ... [more ▼]

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diffeomorphisms is taken into account. However, considering manifolds endowed with additional structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The existence of such a quantization was proved recently in a series of papers in the context of projective geometry. Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations. [less ▲]

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See detailDecomposition of symmetric tensor fields in the presence of a flat contact projective structure
Frégier, Yaël; Mathonet, Pierre ULg; Poncin, Norbert

in Journal of Nonlinear Mathematical Physics (2008), 15(2), 252-269

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form \alpha. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact ... [more ▼]

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

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See detailSymmetries of modules of differential operators
Gargoubi, Hichem; Mathonet, Pierre ULg; Ovsienko, Valentin

in Journal of Nonlinear Mathematical Physics (2005), 12(3), 348-380

Let F_lambda(S^1) be the space of tensor densities of degree (or weight) lambda on the circle S^1. The space D_lambda,mu(k)(S^1) of k-th order linear differential operators from F_lambda(S^1) to F_mu(S^1 ... [more ▼]

Let F_lambda(S^1) be the space of tensor densities of degree (or weight) lambda on the circle S^1. The space D_lambda,mu(k)(S^1) of k-th order linear differential operators from F_lambda(S^1) to F_mu(S^1) is a natural module over Diff(S^1), the diffeomorphism group of S^1. We determine the algebra of symmetries of the modules D_lambda,mu(k)(S^1), i.e., the linear maps on D_lambda,mu(k)(S^1) commuting with the Diff(S^1)-action. We also solve the same problem in the case of straight line R (instead of S^1) and compare the results. [less ▲]

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