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24
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.
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.
Poncin N, Automorphisms of quantum and classical Poisson algebras
 Comp. Math
"... We prove PursellShanks type results for the Lie algebraD(M) of all linear differential operators of a smooth manifold M, for its Lie subalgebra D 1 (M) of all linear firstorder differential operators of M, and for the Poisson algebra S(M) = Pol(T ∗ M) of all polynomial functions on T ∗ M, the sym ..."
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Cited by 8 (5 self)
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We prove PursellShanks type results for the Lie algebraD(M) of all linear differential operators of a smooth manifold M, for its Lie subalgebra D 1 (M) of all linear firstorder differential operators of M, and for the Poisson algebra S(M) = Pol(T ∗ M) of all polynomial functions on T ∗ M, the symbols of the operators inD(M). Chiefly however we provide explicit formulas describing completely the automorphisms of the Lie algebrasD 1 (M), S(M), andD(M). 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.
R.: Nonlocal Equivariant Star Product on the Minimal Nilpotent Orbit
 Adv. Math
"... Abstract. We construct a unique Gequivariant graded star product on the algebra S(g)/I of polynomial functions on the minimal nilpotent coadjoint orbit Omin of G where G is a complex simple Lie group and g ̸ = sl2(C). This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to c ..."
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Cited by 7 (1 self)
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Abstract. We construct a unique Gequivariant graded star product on the algebra S(g)/I of polynomial functions on the minimal nilpotent coadjoint orbit Omin of G where G is a complex simple Lie group and g ̸ = sl2(C). This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function µx with any function f. We find µx ⋆ f = µxf + 1 2 {µx, f}t + Λx (f)t2. For g different from spn(C), Λx is not a differential operator. Instead Λx is the left quotient of an explicit order 4 algebraic differential operator Dx by an order 2 invertible diagonalizable operator. Precisely, Λx = −1 1 4 E ′(E ′+1) Dx where E ′ is a positive shift of the Euler vector field. Thus µx ⋆ f is not local in f. 1.
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 5 (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.
Equivariant Deformation Quantization for the Cotangent Bundle of a Flag
 Fourier 52:3 (2002) 881–897. 33 R. Brylinski, NonLocality of Equivariant Star Products on T ∗ (RP n
, 2001
"... Abstract. Let XR be a (generalized) flag manifold of a noncompact real semisimple Lie group GR, where XR and GR have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T ∗XR) which corresponds to a GRequivariant quantization of symbols into smooth di ..."
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Cited by 3 (1 self)
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Abstract. Let XR be a (generalized) flag manifold of a noncompact real semisimple Lie group GR, where XR and GR have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T ∗XR) which corresponds to a GRequivariant quantization of symbols into smooth differential operators acting on halfdensities on XR. We show that any solution is algebraic in that it restricts to a Gequivariant graded star product ⋆ on the algebraic part R of Pol(T ∗XR). We construct, when R is generated by the momentum functions µ x for G, a preferred choice of ⋆ where µ x ⋆ φ has the form µ xφ + 1 2 {µx, φ}t + Λx (φ)t2. Here Λx are operators on R which are not differential in the known examples and so µ x ⋆ φ is not local in φ. R acquires an invariant positive definite inner product compatible with its grading. The completion of R is a new Fock space type model of the unitary representation of G on L2 halfdensities on X. 1.