Extensions of Lie brackets

Brahic, Olivier

doi:10.1016/j.geomphys.2009.10.006

Cited by 26 publications

(70 citation statements)

References 21 publications

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“…The proof follows immediately from the results in [1,Sec. 4] One may illustrate the homotopy condition appearing in Theorem 5.4 in the following way:…”

Section: 1mentioning

confidence: 80%

“…Since Hor ⊂ Im ♯, it follows that p * is surjective and covers the surjective submersion p : E → B. By its very definition (see [1]), we obtain a Lie algebroid extension with kernel ker(dp • ♯). The fiber non-degeneracy condition, shows that:…”

Section: Coupling Dirac Structures As Extensionsmentioning

confidence: 98%

“…The decomposition (16) gives a complementary vector subundle to this kernel, i.e., an Ehresmann connection in the sense of [1].…”

Section: Coupling Dirac Structures As Extensionsmentioning

confidence: 99%

“…In this section, we will take advantage of the fact that L fits into a Lie algebroid extension, to reformulate the construction given in Section 4.3, without any mentioning to to these infinite dimensional group quotients. We follow the ideas of [1] in order to describe L-paths and L-homotopies.…”

Section: Integration Of Coupling Dirac Structures IImentioning

confidence: 99%

“…We now use the constructions of [1,13] in order to obtain the obstructions to integrability of a coupling Dirac structure L.…”

Section: The Monodromy Groupoidmentioning

confidence: 99%

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Integration of coupling Dirac structures

Brahic

Fernandes

2015

Pacific J. Math.

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Abstract. Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic groupoid integrating coupling Dirac structures. We find the obstructions to integrability and we give explicit geometric descriptions of the integration.

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“…The proof follows immediately from the results in [1,Sec. 4] One may illustrate the homotopy condition appearing in Theorem 5.4 in the following way:…”

Section: 1mentioning

confidence: 80%

Section: Coupling Dirac Structures As Extensionsmentioning

confidence: 98%

“…The decomposition (16) gives a complementary vector subundle to this kernel, i.e., an Ehresmann connection in the sense of [1].…”

Section: Coupling Dirac Structures As Extensionsmentioning

confidence: 99%

Section: Integration Of Coupling Dirac Structures IImentioning

confidence: 99%

“…We now use the constructions of [1,13] in order to obtain the obstructions to integrability of a coupling Dirac structure L.…”

Section: The Monodromy Groupoidmentioning

confidence: 99%

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Integration of coupling Dirac structures

Brahic

Fernandes

2015

Pacific J. Math.

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Almost regular Poisson manifolds and their holonomy groupoids

Androulidakis

Zambon

2017

Sel. Math. New Ser.

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We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them almost regular Poisson structures and determine them completely. They include regular Poisson and log symplectic manifolds, as well as several other Poisson structures whose symplectic foliation presents singularities.We show that the holonomy groupoid associated with an almost regular Poisson structure is a Poisson groupoid, integrating a naturally associated Lie bialgebroid. The Poisson structure on the holonomy groupoid is regular, and as such it provides a desingularization. The holonomy groupoid is "minimal" among Lie groupoids which give rise to the submodule generated by Hamiltonian vector fields. This implies that, in the case of log-symplectic manifolds, the holonomy groupoid coincides with the symplectic groupoid constructed by Gualtieri and Li. Last, we focus on the integrability of almost regular Poisson manifolds and exhibit the role of the second homotopy group of the source-fibers of the holonomy groupoid.

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Lie and Courant algebroids on foliated manifolds

Vaisman

2011

Bull Braz Math Soc, New Series

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This is an exposition of the subject, which was developed in the author's papers [19,20]. Various results from the theory of foliations (cohomology, characteristic classes, deformations, etc.) are extended to subalgebroids of Lie algebroids that generalize the tangent integrable distributions. We also suggest a definition of foliated Courant algebroids and give some corresponding results and constructions.

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Extensions of Lie brackets

Cited by 26 publications

References 21 publications

Integration of coupling Dirac structures

Integration of coupling Dirac structures

Almost regular Poisson manifolds and their holonomy groupoids

Lie and Courant algebroids on foliated manifolds

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