Michel Cahen

• Université Libre de Bruxelles

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 gene...

On 4-symmetric symplectic spaces, invariant almost complex structures -up 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 Nijenhu...

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 ge...

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up 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...

In this paper we look at the question of integrability, or not, of the two natural almost complex structures \begin{document}$ J^{\pm}_\nabla $\end{document} defined on the twistor space \begin{document}$ J(M, g) $\end{document} of an even-dimensional manifold \begin{document}$ M $\end{document} with additional structures \begin{document}$ g $\end{...

In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ defined on the twistor space $J(M,g)$ of an even-dimensional manifold $M$ with additional structures $g$ and $\nabla$ a $g$-connection. We also look at the question of the compatibility of $J^{\pm}_\nabla$ with a natural clo...

We look at methods to select triples $(M,\omega,J)$ consisting of a symplectic manifold $(M,\omega)$ endowed with a compatible positive almost complex structure $J$, in terms of the Nijenhuis tensor $N^J$ associated to $J$. We study in particular the image distribution $\Image N^J$.

We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their...

We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their...

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in K\"ahlerian Geometry. They are characterized amongst a class of symplectic...

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in K\"ahlerian Geometry. They are characterized amongst a class of symplectic...

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 pseud...

We advertise the use of the group Mpc (a circle extension of the symplectic group) instead of the metaplectic group (a double cover of the symplectic group). The essential reason is that Mpc-structures exist on any symplectic manifold. They first appeared in the framework of geometric quantization [4, 10]. In a joint work with John Rawnsley [1], we...

Given a symplectic manifold (M, ω) admitting a metaplectic structure, and choosing a positive ω-compatible almost complex structure J and a linear connection ${\nabla}$ preserving ω and J, Katharina and Lutz Habermann have constructed two Dirac operators D and ${{\tilde{D}}}$ acting on sections of a bundle of symplectic spinors. They have shown...

This is a brief summary of our work on an explicit description of (s)pin structures of real projective quadrics and on the spectrum of the Dirac operator on these spaces. Key wordsDirac operator–pin structures–real projective spaces–real projective quadrics

Symmetric symplectic spaces of Ricci type are a class of symmetric symplectic spaces which can be entirely described by reduction of certain quadratic Hamiltonian systems in a symplectic vector space. We determine, in a large number of cases, if such a space admits a subgroup of its transvection group acting simply transitively. We observe that the...

We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural star-quantization on a class of examples.

In this article we relate the construction of Ricci-type symplectic connections by reduction to the construction of star products by reduction yielding rather explicit descriptions for the star product on the reduced space. AMS Classification (2010): 53D55, 53C07, 53D20

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching...

In this paper we present a construction of Ricci-flat connections through an induction procedure. Given a symplectic manifold (M, ω) of dimension 2n, we define induction as a way to construct a symplectic manifold (P, μ) of dimension 2n + 2. Given any symplectic manifold (M, ω) of dimension 2n and given a symplectic connection ▽ on (M, ω), we defin...

We show that any symplectic manifold (M) of dimension 2n(n 2) admitting a symplectic connection of Ricci type can locally be constructed by a reduction procedure from the Euclidean space R2n+2 endowed with a constant symplectic structure and the standard flat connection. We also prove that on the bundle of symplectic frames B(M) over M, there exist...

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Khler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic....

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplect...

Given the Euclidean space ℝ2n+2 endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen [1] have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold (M, ω) of dimension...

We consider analytic curves ∇t of symplectic connections of Ricci-type on the torus T2n with ∇0 the standard connection. We show, by a recursion argument, that if ∇t is a formal curve of such connections then there exists a formal curve of symplectomorphisms ψt such that ψt·∇t is a formal curve of flat T2n-invariant symplectic connections and so ∇t...

We consider analytic curves $\nabla^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show, by a recursion argument, that if $\nabla^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms $\psi_t$ such that $\psi_t\cdot\nabla^t$ is a formal curve of...

We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive.

We consider invariant symplectic connections ∇ on homogeneous symplectic manifolds (M,ω) with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic structure. If M is...

For any i ntegers n and m (m 4) such that n+m is odd we exhibit triangular solutions of the classical Yang-Baxter equation on sl((n +1)(m+2), C) parametrized by points of a quotient of complex projective space P # # (C)by the action of the symmetric group Sym(# and we prove that no two of these solutions are isomorphic. 1

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown.

We consider invariant symplectic connections $\nabla$ on homogeneous symplectic manifolds $(M,\omega)$ with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic stru...

We determine the isomorphism classes of symmetric symplectic manifolds of dimension at least 4 which are connected, simply-connected and have a curvature tensor which has only one non-vanishing irreducible component -- the Ricci tensor.

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equati...

We define a variational principle for symplectic connections of the Yang-Mills type. When the symplectic manifold is a compact surface we show that the moduli space of the connections which are extremals of the functional coincides with the Teichmüller space of the surface. We indicate that the noncompact situation is very different.

We prove the existence of at least one G-invariant preferred symplectic connection on any coadjoint orbit of a compact semisimple Lie group G. We look at the case of the orbits of SU(3) and show that in this case the invariant preferred connection is unique.

We give an elementary proof of the fact that equivalence classes of smooth or differentiable star products on a symplectic manifold M are parametrized by sequences of elements in the second de Rham cohomology space of the manifold. The parametrization is given explicitly in terms of Fedosov's construction which yields a star product when one choose...

We study a variational principle for symplectic connections and describe the moduli space of solutions of the field equations in the case of compact surfaces. Furthermore we show what happens in the Kähler situation and study some particular solutions in the purely symplectic context. © 1998 American Institute of Physics.

A variational principle introduced to select some symplectic connections lead-s to field equations which, in the case of the Levi Civita connection of Kähler manifolds, are equivalent to the condition that the Ricci tensor is parallel. This condition, which is stronger than the field equations, is studied in a purely sym-plectic framework. 1. A sym...

We derive necessary conditions on a Lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the Lie algebra for the coadjoint Poisson structure which is both differential and tangential to all the coadjoint orbits. In particular we show that when the Lie algebra is semisimple there are no differential and ta...

For any integers n and m (m 4) such thatn +m is odd we exhibit trian- gular solutions of the classical Yang-Baxter equation on sl((n +1 )(m +2 );C) parametrized by points of a quotient of complex projective space Pb n 2 c (C )b y the action of the symmetric group Sym(bn+1 2 c) and we prove that no two of these solutions are isomorphic.

General theorems on pin structures on products of manifolds and on homogeneous (pseudo-) Riemannian spaces are given and used to find explicitly all such structures on odd-dimensional real projective quadrics, which are known to be non-orientable (Cahen et al. 1993). It is shown that the product of two manifolds has a pin structure if and only if b...

We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.

We give a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller or equal to two.

In this paper, which we are happy to dedicate to our friend Guy Rideau, we introduce the notion of symplectic symmetric spaces.

We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the unit disk in ℂ. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.

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.

We use Berezin’s dequantization procedure to define a formal *- product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous Kähler manifold M. We prove that this formal *-product isconvergent when M isa hermitian symmetric space.

Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particular, the (projective) real quadric Qp,q = (Sp × Sq)/2 is associated, in this manner, with the flat space p+q endowed with a metric tensor of signature (p, q). For p and q positive, the quadric Qp,q is orientable iff p + q is even. The quadric has two natu...

We show that Lu and Weinstein's Iwasawa Poisson Lie structure on a simple compact Lie group K is not linearizable when K is not SU(2).

We suggest a construction of a Moyal type * product on Hermitian symmetric spaces ; this construction is motivated by what happens for coadjoint orbits of the Heisenberg group.

We give a geometric interpretation of Berezin's symbolic calculus on Kähler manifolds in the framework of geometric quantization. Berezin's covariant symbols are defined in terms of coherent states and we study a function ϴ on the manifold which is the central object of the theory. When this function is constant Berezin's quantization rule coincide...

We give examples of Lorentz manifolds modelled on an indecomposable Lorentz symmetric space which are geodesically complete and not locally homogeneous.

The aim of this Letter is twofold. On the one hand, we discuss two possible definitions of complex structures on Poisson-Lie groups and we give a complete classification of the isomorphism classes of complex Lie-Poisson structures on the group SL(2, ). On the other hand, we give an algebraic characterization of a class of solutions of the Yang-Baxt...

We give explicit formulas for the spectrum of the Dirac operator on complex projective space P 2q-1(ℂ).

We classify, up to isomorphism, those exact bialgebra structures on the (2n + 1) - dimensional Heisenberg algebra which satisfy the classical Yang-Baxter equation. In the case n = 1, we classify all bialgebra structures and describe the singular locus and the symplectic leaves of the corresponding Lie-Poisson structure on the Heisenberg group.

It is shown that the necessary and sufficient condition for the Yang–Mills equations (associated with an arbitrary group G) to be derivable from a Lagrangian (which is polynonial in the derivatives of the connection) is that the Lie algebra g of G possesses an invariant nondegenerate quadratic form Γ. It is well known that for semisimple groups suc...

We realise the *-representation program for compact semi simple Lie groups, using a Kodaira's map and Berezin's symbols. We study the Fourier transform it induces. We show that injectivity of Kodaira's map selects the data of geometric quantization. In this case an inverse of Kodaira's map is given by a moment map. This relates our construction to...

A deformation of the polynomial algebra S()on * when S() is a free I() module (I() = algebra of invariant polinomials). This deformation restricts nicely to a large class of orbits. We also give an example to show that deformations of S() restricting to orbits may not always be defined by bidifferential operators.

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 a...

Not Available Chercheur qualifié du F.N.R.S.

This volume contains the text of the lectures which were given at the Differential Geometry Meeting held at Liege in 1980 and at the Differential Geometry Meeting held at Leuven in 1981. The first of these meetings was more orientated toward mathematical physics; the second has a stronger flavour of analysis. The Editors are pleased to thank the le...

Nous donnons une classification algbrique des espaces symtriques pseudo-riemanniens localement conformment plats. Nous en dduisons la liste exhaustive ( revtement homothtique prs) des espaces symtriques pseudo-riemanniens possdant soit un champ de vecteurs conforme non homothtique soit une transformation conforme non homothtique.

We prove the existence of a * product on the cotangent bundle of a parallelizable manifold M. When M is a Lie group the properties of this * product allow us to define a linear representation of the Lie algebra of this group on L 2(G), which is, in fact, the one corresponding to the usual regular representation of G.

In this article we give a solution of Yang-Mills equations for the group G = SU(2) on the pseudo-riemannian manifold S1xS3 which is a conformai compactification of Minkowski space.

In this paper, we give an explicit basis of the invariant bidifferential operators on the Hermitian symmetric spaces of rank 1. As an application we prove that on noncompact Hermitian symmetric spaces of rank 1, an invariant*-product coincides with the usual product for holomorphic functions.

We determine all smooth solutions of Maxwell's equation in Segal's universe; furthermore we show that the group of diffeomorphisms stabilizing the space of solutions of these equations is the conformal group of Segal's model.

A multilinear version of Peetre's theorem on local operators is the key to prove the equality between the local and differentiable Hochschild cohomology on the one hand, and on the other hand the equality between the second and third local Chevalley cohomology groups and their differentiable counterpart.

This book presents the text of the lectures which were given at the NATO Advanced Study Institute on Representations of Lie groups and Harmonic Analysis which was held in Liege from September 5 to September 17, 1977. The general aim of this Summer School was to give a coordinated intro­ duction to the theory of representations of semisimple Lie gro...

Every simply connected Lorentz symmetric space of dimensionn=3 has a symmetric quotient which can be conformally imbedded in a quadric of the projective space Pn+1 (R).

Nous montrons que tout espace lorentzien symétrique simplement connexe de dimension n > 3 admet un quotient symétrique qui peut être plongé par un difféomorphisme conforme dans un certain revêtement d'une quadrique de Pn+1(R). La composante connexe du groupe conforme de cette quadrique est SO0(2,n)/Z2.

We modify a construction of graded Lie algebra suggested by S. Sternberg and J. Wolf to obtain the conformai graded algebra.

On the occasion of the sixtieth birthday of Andre Lichnerowicz a number of his friends, many of whom have been his students or coworkers, decided to celebrate this event by preparing a jubilee volume of contributed articles in the two main fields of research marked by Lichnerowicz's work, namely differential geometry and mathematical physics. Limit...