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Projectively equivariant symbol calculus
, 1999
"... The spaces of linear differential operators Dλ(R n) acting on λdensities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + ..."
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Cited by 35 (2 self)
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The spaces of linear differential operators Dλ(R n) acting on λdensities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + 1, R)modules where sl(n + 1, R) ⊂ Vect(R n) is the Lie algebra of infinitesimal projective transformations. In addition, such an sln+1equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the sln+1equivariant symbol map to study the Vect(M)modules D k λ (M) of kthorder linear differential operators acting on λdensities, for an arbitrary manifold M and classify the quotientmodules D k λ (M)/Dℓ λ (M). 1
Formula for the projectively invariant quantization on degree three
 C. R. Acad. Sci. Paris Sér. I Math
"... We give an explicit formula for the projectively invariant quantization map between the space of symbols of degree three and the space of thirdorder linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on a manifold. Une formul ..."
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Cited by 11 (0 self)
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We give an explicit formula for the projectively invariant quantization map between the space of symbols of degree three and the space of thirdorder linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on a manifold. Une formule pour la quantification projectivement invariante en degré trois Résumé Nous donnerons une formule explicite pour la quantification projectivement invariante entre l’espace des symboles de degrés trois et l’espace des opérateurs différentiels linéaires d’ordres trois, vus comme modules sur le groupe des difféomorphismes et l’algèbre de Lie des champs de vecteurs sur une variété différentiable. 1
Conformal Schwarzian derivatives and conformally invariant quantization
 Internat. Math. Res. Not. 2002
"... Let (M, g) be a pseudoRiemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1cocycles on the group of diffeomorphisms of M related to the modules of linear differential operators. As operators, these derivatives do not depend on the ..."
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Cited by 4 (4 self)
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Let (M, g) be a pseudoRiemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1cocycles on the group of diffeomorphisms of M related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric g. In particular, if the manifold (M, g) is conformally flat, these derivatives vanish on the conformal group O(p + 1, q + 1), where dim(M) = p + q. This work is a continuation of [2, 4] where the Schwarzian derivative was defined on a manifold endowed with a projective connection. 1
Projectively Quantization Map
, 2000
"... Let M be a manifold endowed with a symmetric affine connection Γ. The aim of this paper is to describe a quantization map between the space of secondorder polynomials on the cotangent bundle T ∗ M and the space of secondorder linear differential operators, both viewed as modules over the group of ..."
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Let M be a manifold endowed with a symmetric affine connection Γ. The aim of this paper is to describe a quantization map between the space of secondorder polynomials on the cotangent bundle T ∗ M and the space of secondorder linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on M. This map is an isomorphism, for almost all values of certain constants, and it depend only on the projective class of the affine connection Γ. 1
Remarks on the Conformally Invariant Quantization by means of a Finsler Function
, 2002
"... Let (M, F) be a Finsler manifold. We construct a 1cocycle on Diff(M) with values in the space of differential operators acting on tensor fields, by means of the Finsler function F. This is a first step toward the existence of Schwarzian derivatives for Finsler structure. We, furthermore, discuss so ..."
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Let (M, F) be a Finsler manifold. We construct a 1cocycle on Diff(M) with values in the space of differential operators acting on tensor fields, by means of the Finsler function F. This is a first step toward the existence of Schwarzian derivatives for Finsler structure. We, furthermore, discuss some properties of the conformally invariant quantization map by means of a (Sazaki type) metric on the slit bundle TM\0 induced by F. 1
unknown title
, 2001
"... Cohomology of groups of diffeomorphisms related to the modules of differential operators on a smooth manifold ..."
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Cohomology of groups of diffeomorphisms related to the modules of differential operators on a smooth manifold
Remarks on the Schwarzian Derivatives and the Invariant Quantization by means of a Finsler Function
, 2004
"... Let (M, F) be a Finsler manifold. We construct a 1cocycle on Diff(M) with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function F. As an operator, it has several expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection ..."
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Let (M, F) be a Finsler manifold. We construct a 1cocycle on Diff(M) with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function F. As an operator, it has several expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection, although its cohomology class does not depend on them. This cocycle is closely related to the conformal Schwarzian derivatives introduced in our previous work. The second main result of this paper is to discuss some properties of the conformally invariant quantization map by means of a Sazaki (type) metric on the slit bundle TM\0 induced by F. 1
Conformal Schwarzian derivatives on a pseudoRiemannian manifold
, 2001
"... Let (M, g) be a pseudoRiemannian manifold. We propose a new approach for the conformal Schwarzian derivatives. These derivatives are 1cocycles on the group of diffeomorphisms of M related to the modules of linear differential operators. As operators, these derivatives depend only on the conformal ..."
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Let (M, g) be a pseudoRiemannian manifold. We propose a new approach for the conformal Schwarzian derivatives. These derivatives are 1cocycles on the group of diffeomorphisms of M related to the modules of linear differential operators. As operators, these derivatives depend only on the conformal class of the metric g. In particular if the manifold (M, g) is conformally flat, these derivatives vanish on the conformal group O(p + 1, q + 1), where dim(M) = p + q. This work is a continuation of [1, 3] where the Schwarzian derivative was defined on a manifold endowed with a projective connection. 1
transformations of the Euclidian space
, 1999
"... A remark about the Lie algebra of infinitesimal conformal ..."
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