On classical -matrices with parabolic carrier.
On Liouville forms
We give different notions of Liouville forms, generalized Liouville forms and vertical Liouville forms with respect to a locally trivial fibration π:E → M. These notions are linked with those of semi-basic forms and vertical forms. We study the infinitesimal automorphisms of these forms; we also investigate the relations with momentum maps.
On localization in holomorphic equivariant cohomology
We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. This encompasses various residue formulas in complex geometry, in particular we shall show that it contains as special cases Carrell-Liebermann’s and Feng-Ma’s residue formulas, and Baum-Bott’s formula for the zeroes of a meromorphic vector field.
On quantum and classical Poisson algebras
Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular a somewhat unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, whether or not the line bundles themselves are isomorphic....
On quantum de Rham cohomology theory.
On Riemann-Poisson Lie groups
A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in [4]. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.
On submanifolds and quotients of Poisson and Jacobi manifolds
We obtain conditions under which a submanifold of a Poisson manifold has an induced Poisson structure, which encompass both the Poisson submanifolds of A. Weinstein [21] and the Poisson structures on the phase space of a mechanical system with kinematic constraints of Van der Schaft and Maschke [20]. Generalizations of these results for submanifolds of a Jacobi manifold are briefly sketched.
On the geometric prequantization of brackets.
En este artículo se considera un marco general para la precuantización geométrica de una variedad provista de un corchete que no es necesariamente de Jacobi. La existencia de una foliación generalizada permite definir una noción de fibrado de precuantización. Se estudia una aproximación alternativa suponiendo la existencia de un algebroide de Lie sobre la variedad. Se relacionan ambos enfoques y se recuperan los resultados conocidos para variedades de Poisson y Jacobi.
On the geometry of the standard -symplectic and Poisson manifolds.
On the index theorem for symplectic orbifolds
We give an explicit construction of the trace on the algebra of quantum observables on a symplectiv orbifold and propose an index formula.
On the integrability of orthogonal distributions in Poisson manifolds.
On the linearization theorem for proper Lie groupoids
We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the fixed point case (known as Zung’s theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passage to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise statements of the Linearization Theorem (there has been some confusion on this, which has propagated throughout...
On the Riemann-Lie algebras and Riemann-Poisson Lie groups.
On the variety of lagrangian subalgebras, I
On the variety of lagrangian subalgebras, II
Poisson algebras of spinor functions.
Poisson geometry and deformation quantization near a strictly pseudoconvex boundary
Let be a complex manifold with strongly pseudoconvex boundary . If is a defining function for , then is plurisubharmonic on a neighborhood of in , and the (real) 2-form is a symplectic structure on the complement of in a neighborhood of in ; it blows up along . The Poisson structure obtained by inverting extends smoothly across and determines a contact structure on which is the same as the one induced by the complex structure. When is compact, the Poisson structure near...
Poisson geometry of directed networks in an annulus
As a generalization of Postnikov’s construction [P], we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in [GSV3] induce a family of Poisson structures on rational matrix-valued functions and on the space of loops in the Grassmannian. In the former case, this family includes, for a particular kind of networks, the Poisson bracket associated...
Poisson manifolds, Lie algebroids, modular classes: a survey.
Poisson structures on having only two symplectic leaves: the origin and the rest
Examples of Poisson structures with isolated non-symplectic points are constructed from classical r-matrices.