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On characterization of Poisson and Jacobi structures

Janusz Grabowski, Paweŀ Urbański (2003)

Open Mathematics

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.

On Liouville forms

Paulette Libermann (2000)

Banach Center Publications

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

Ugo Bruzzo, Vladimir Rubtsov (2012)

Open Mathematics

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

Janusz Grabowski, Norbert Poncin (2007)

Banach Center Publications

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 Riemann-Poisson Lie groups

Brahim Alioune, Mohamed Boucetta, Ahmed Sid’Ahmed Lessiad (2020)

Archivum Mathematicum

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

Charles-Michel Marle (2000)

Banach Center Publications

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.

Manuel de León, Juan Carlos Marrero, Edith Padrón (2001)

RACSAM

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 index theorem for symplectic orbifolds

Boris Fedosov, Bert-Wolfang Schulze, Nikolai Tarkhanov (2004)

Annales de l’institut Fourier

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 linearization theorem for proper Lie groupoids

Marius Crainic, Ivan Struchiner (2013)

Annales scientifiques de l'École Normale Supérieure

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...

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