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Differential operators on the supercircle: conformally equivariant quantization and symbol
"... We consider the supercircle S 11 equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S 11 as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra osp(12). We study the space of linear differential operators on weighted densities ..."
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Cited by 13 (6 self)
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We consider the supercircle S 11 equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S 11 as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra osp(12). We study the space of linear differential operators on weighted densities as a module over osp(12). 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. 1
Shchepochkina I., How to quantize antibracket, preprint ESI875 (www.esi.ac.at
 Theor. and Math. Physics
"... Abstract. The uniqueness of (the class of) deformation of Poisson Lie algebra po(2n) has long been a completely accepted folklore. Actually this is wrong as stated, because its validity depends on the class of functions that generate po(2n) (e.g., it is true for polynomials but false for Laurent pol ..."
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Cited by 10 (8 self)
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Abstract. The uniqueness of (the class of) deformation of Poisson Lie algebra po(2n) has long been a completely accepted folklore. Actually this is wrong as stated, because its validity depends on the class of functions that generate po(2n) (e.g., it is true for polynomials but false for Laurent polynomials). We show that, unlike po(2nm), its quotient modulo center, the Lie superalgebra h(2nm) of Hamiltonian vector fields with polynomial coefficients, has exceptional extra deformations for (2nm) = (22) and only in this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). We show that, whereas the representation of the deform (the result of deformation aka quantization) of the Poisson algebra in the Fock space coincides with the simplest space on which the Lie algebra of commutation relations acts, this coincidence is not necessary for Lie superalgebras.
Natural and projectively equivariant quantizations by means of Cartan connections
 Lett. Math. Phys
, 2005
"... Abstract. The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of ThomasWhitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain a ..."
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Cited by 10 (6 self)
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Abstract. The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of ThomasWhitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an sl(m + 1, R)equivariant quantization exists in the flat situation in the sense of [18], thus solving one of the problems left open by M. Bordemann.
Conformally equivariant quantization
"... Let (M,g) be a pseudoRiemannian manifold and Fλ(M) the space of densities of degree λ on M. We study the space D2 λ,µ (M) of secondorder differential operators from Fλ(M) to Fµ(M). If (M,g) is conformally flat with signature p −q, then D2 λ,µ (M) is viewed as a module over the group of conformal t ..."
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Cited by 9 (1 self)
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Let (M,g) be a pseudoRiemannian manifold and Fλ(M) the space of densities of degree λ on M. We study the space D2 λ,µ (M) of secondorder differential operators from Fλ(M) to Fµ(M). If (M,g) is conformally flat with signature p −q, then D2 λ,µ (M) is viewed as a module over the group of conformal transformations of M. We prove that, for almost all values of µ − λ, the O(p+1,q+1)modules D2 λ,µ (M) and the space of symbols (i.e., of secondorder polynomials on T ∗M) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudoRiemannian manifolds and depends only on the conformal class [g] of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on halfdensities, as well as the wellknown Yamabe Laplacian. We also recover in this framework the multidimensional Schwarzian derivative of conformal transformations.
Mathonet P, Maximal subalgebras of vector fields for equivariant quantizations
 J. Math. Phys
"... Abstract. The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bij ..."
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Cited by 8 (2 self)
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Abstract. The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables that are polynomial in the momenta and differential operators. Here, we determine which finite dimensional graded Lie subalgebras are maximal. In order to characterize these, we make use of results of Guillemin, Singer and Sternberg and
Equivariant symbol calculus for differential operators acting on forms
 Lett. Math. Phys
"... Abstract. We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko [6]) for the spaces Dp of differential operators transforming pforms into functions. These results hold over a smooth manifold endowed with a flat projective structure. As ..."
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Cited by 8 (6 self)
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Abstract. We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko [6]) for the spaces Dp of differential operators transforming pforms into functions. These results hold over a smooth manifold endowed with a flat projective structure. As an application, we classify the Vect(M)equivariant maps fromDp toDq over any manifold M, recovering and improving earlier results by N. Poncin [9]. This provides the complete answer to a question raised by P. Lecomte about the extension of a certain intrinsic homotopy operator. 1.
IFFTequivariant quantizations
 J. Geom. Phys
"... Abstract. The existence and uniqueness of quantizations that are equivariant with respect to conformal and projective Lie algebras of vector fields were recently obtained by Duval, Lecomte and Ovsienko. In order to do so, they computed spectra of some Casimir operators. We give an explicit formula f ..."
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Cited by 7 (6 self)
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Abstract. The existence and uniqueness of quantizations that are equivariant with respect to conformal and projective Lie algebras of vector fields were recently obtained by Duval, Lecomte and Ovsienko. In order to do so, they computed spectra of some Casimir operators. We give an explicit formula for those spectra in the general framework of IFFTalgebras classified by Kobayashi and Nagano. We also define treelike subsets of eigenspaces of those operators in which eigenvalues can be compared to show the existence of IFFTequivariant quantizations. We apply our results to prove existence and uniqueness of quantizations that are equivariant with respect to the infinitesimal action of the symplectic (resp. pseudoorhogonal) group on the corresponding Grassmann manifold of maximal isotropic subspaces.
Cartan connections and natural and projectively equivariant quantizations
 J. Lond. Math. Soc
"... Abstract. In this paper, we analyse the question of existence of a natural and projectively equivariant symbol calculus, using the theory of projective Cartan connections. We establish a close relationship between the existence of such a natural symbol calculus and the existence of an sl(m + 1, R) ..."
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Cited by 7 (3 self)
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Abstract. In this paper, we analyse the question of existence of a natural and projectively equivariant symbol calculus, using the theory of projective Cartan connections. We establish a close relationship between the existence of such a natural symbol calculus and the existence of an sl(m + 1, R) equivariant calculus over R m in the sense of [15, 1]. Moreover we show that the formulae that hold in the noncritical situation over R m for the sl(m+1, R) equivariant calculus can be directly generalized to an arbitrary manifold by simply replacing the partial derivatives by invariant differentiations with respect to a Cartan connection. 1.
Existence of natural and conformally invariant quantizations of arbitrary symbols, available at arXiv:08113710
"... Abstract. A quantization over a manifold can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of principal symbols to the space of differential operators is moreover required to be a linear bijection. It is known that there is in general ..."
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Cited by 6 (1 self)
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Abstract. A quantization over a manifold can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of principal symbols to the space of differential operators is moreover required to be a linear bijection. It is known that there is in general no natural quantization procedure. However, considering manifolds endowed with additional structures, such as projective or pseudoconformal 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 conjectured by P. Lecomte in [19] in the context of projective and conformal geometry. The question of existence of such a quantization was addressed in a series of papers in the context of projective geometry, using the framework of ThomasWhitehead connections (see for instance [4, 14, 13, 15]). In [23, 21], we recovered the existence of a quantization that depends on a projective structure and that is natural (provided some critical situations are avoided), using the theory of Cartan projective connections. In the present work, we show that our method can be adapted to pseudoconformal geometry to yield the socalled natural and conformally invariant quantization for arbitrary symbols, still outside some critical situations. Moreover, we give new and more general proofs of some results of [21] and eventually, we notice that the method is general enough to analyze the problem of natural and invariant quantizations in the context of manifolds endowed with irreducible parabolic geometries studied in [9]. 1.
Poncin N., Decomposition of symmetric tensor fields in the presence of a flat contact projective structure
 2008), 252–269, math.DG/0703922. Conformally Equivariant Quantization in Dimension 12 11
"... Abstract. Let M be an odddimensional Euclidean space endowed with a contact 1form α. 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 con ..."
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Cited by 5 (1 self)
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Abstract. Let M be an odddimensional Euclidean space endowed with a contact 1form α. 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. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form α 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 space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2). 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 in [18]. 1.