L∞-formality check for the Hochschild complex of certain universal enveloping algebras
Journal of Geometry and Physics, May 1, 2023
About the lifting of smooth functions
Functiones et Approximatio Commentarii Mathematici, 1987
info:eu-repo/semantics/publishe
Cohomology of orbits of compact Lie groups
Bulletin de la Classe des sciences. Académie royale de Belgique, 1983
We give a new, purely algebraic proof of the vanishing of the third de Rham cohomology group of a... more We give a new, purely algebraic proof of the vanishing of the third de Rham cohomology group of an orbit of a compact Lie group. This proof is based on two lemmas on root systems and on a cubic identity satisfied by the structure constants of a simple complex Lie algebra.
Some non commutative topics related to symmetric spaces
The procedure of * quantization introduces the notions of mathematical equivalence and of * spect... more The procedure of * quantization introduces the notions of mathematical equivalence and of * spectrum. We prove that mathematical equivalence, as a change of ordering for quantum operators to which it is related, does not preserve * spectrum unless it reduces to an automorphism of the * product. Suggestions about the correct.~choice of * products are made. 0. INTRODUCTION The notion of mathematical equivalence for * products~2] is the usual notion of equivalence for deformations of algebras as defined originally by M. Gerstenhaber [5]. In the particular context of the Moyal * product on~2n equivalence has been related to the Weyl correspondence. More precisely, it was observed (°)Chercheur qualifié au F.N.R.S.
On 4-symmetric symplectic spaces, invariant almost complex structuresup to sign-arise in pairs. W... more On 4-symmetric symplectic spaces, invariant almost complex structuresup to sign-arise in pairs. We exhibit some 4-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.
Moment map for the space of symplectic connections
In this paper, we consider the space E(M,ω) of symplectic connections on a simply connected sympl... more In this paper, we consider the space E(M,ω) of symplectic connections on a simply connected symplectic manifold (M,ω) and we build a moment map for the action of a group of symplectomorphisms of (M,ω) on E(M,ω). We also study the principal symbol of the equation defining Ricci-type symplectic connections in E(M,ω) and show that it has a kernel of dimension 1. ∗Universite Libre de Bruxelles, Campus Plaine CP 218, Bvd du Triomphe, B-1050 Brussels, Belgium ∗∗Universite de Metz, Departement de mathematiques, Ile du Saulcy, F-57045 Metz Cedex 01, France Email: [email protected], [email protected] 1 Moment map for the space of symplectic connections A symplectic connection on a symplectic manifold (M,ω) is a torsionless linear connection ∇ on M for which the symplectic 2–form ω is parallel. To see the existence of such a connection, take ∇ to be any torsion free linear connection (for instance, the Levi Civita connection associated to a metric g on M). Consider the tensor N on M defined by ∇Xω(Y, Z) =: ω(N(X, Y ), Z) where X, Y, Z are vector fields on M (i.e. ∈ χ(M)). Since ω is closed, one has + XY Z ω(N(X, Y ), Z) = 0. Define ∇XY := ∇ 0 XY + 1 3 N(X, Y ) + 1 3 N(Y,X). Then ∇ is a symplectic connection on (M,ω). To see how (non)-unique is a symplectic connection, take ∇ symplectic; then any other linear connection reads
We define a transverse Dolbeault cohomology associated to any almost complex structure j on a smo... more We define a transverse Dolbeault cohomology associated to any almost complex structure j on a smooth manifold M. This we do by extending the notion of transverse complex structure and by introducing a natural j-stable involutive limit distribution with such a transverse complex structure. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of (M, j) introduced by Cirici and Wilson in [3], showing that the (p, 0) cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure.
In this paper we look at the question of integrability, or not, of the two natural almost complex... more In this paper we look at the question of integrability, or not, of the two natural almost complex structures J ± ∇ defined on the twistor space J(M, g) of an evendimensional manifold M with additional structures g and ∇ a g-connection. We also look at the question of the compatibility of J ± ∇ with a natural closed 2-form ω J (M,g,∇) defined on J(M, g). For (M, g) we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection ∇. In all cases J(M, g) is a bundle of complex structures on the tangent spaces of M compatible with g and we denote by π : J(M, g) −→ M the bundle projection. In the case M is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive. The linear connection ∇ on M defines a horizontal space H ∇ j ≃ T π(j) M at any point j in the twistor space so that TjJ(M, g) is isomorphic to H ∇ j ⊕ Vj where Vj = Ker π * j is the vertical space at j. Since both Vj and T M π(j) carry complex structures defined by j, they add together to give the complex structure denoted by (J + ∇)j on TjJ(M, g). The almost complex structure denoted (J − ∇)j is defined by reversing the sign on the horizontal space. We examine the integrability, or not, of the J ± ∇ by looking at their Nijenhuis tensors N J ± ∇ and measure their non-integrability by the dimension of the span of the values of N J ± ∇. The natural closed 2-form ω J (M,g,∇) is defined on the twistor space as the trace of the curvature of a connection D E defined on the pull-back bundle bundle E = π −1 T M. This bundle E is endowed with the complex vector bundle structure defined by the natural section Φ of End(E) whose value at j is j, and the connection D E , built from the pullback connection π −1 ∇ E , satisfies D End E Φ = 0. We recall, as in Reznikov [10], when this 2-form is symplectic in the pseudo-Riemannian setting and we determine, in the pseudo-Riemannian and in the symplectic setting, when ω J (M,g,∇) is of type (1, 1) with respect to J ± ∇ .
We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault o... more We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group M p c and classify G-invariant M p c-structures on symplectic spaces with a Gaction. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space. 2 Subgroups of Sp(V, Ω) and lifts to M p c (V, Ω, j) 2.1 The symplectic Clifford algebra Let (V, Ω) be a finite-dimensional real symplectic vector space of dimension 2n. The symplectic Clifford Algebra Cl(V, Ω) is the associative unital complex algebra generated by V with the relation u • v − v • u = i Ω(u, v)1 (1) where h is a positive real number and = h 2π. A symplectic spinor space is a vector space carrying a representation of the symplectic Clifford algebra. This representation, called Clifford multiplication and denoted by cl, is derived from an irreducible unitary representation of the Heisenberg group. There are many
Transactions of the American Mathematical Society, 1993
We use Berezin's dequantization procedure to define a formal *product on a dense subalgebra of th... more We use Berezin's dequantization procedure to define a formal *product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kahler manifold M. We prove that this formal »-product is convergent when M is a hermitian symmetric space.
This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not co... more This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements-with intersection-the introduction to formal deformation quantization and group actions published in [38], corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are implemented by a moment map and we give indications to build a corresponding quantum moment map. Reduction is a construction in classical mechanics with symmetries which allows to reduce the dimension of the manifold; we describe one of the various quantum analogues which have been considered in the framework of formal deformation quantization. We end up by some considerations about convergence of star products.
We define a natural class of star products: those which are given by a series of bidifferential o... more We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance and give necessary and sufficient conditions for them to yield a quantum moment map. We show that Kravchenko's sufficient condition [18] for a moment map for a Fedosov star product is also necessary. This research was partially supported by an Action de Recherche Concertée de la Communauté française de Belgique and by the Belgian FNRS.
Connexions symplectiques à courbure de type ricci (Unpublished doctoral dissertation). Université... more Connexions symplectiques à courbure de type ricci (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences, Bruxelles.
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