Poisson geometry from a Dirac perspective

Meinrenken, Eckhard

doi:10.1007/s11005-017-0977-4

Cited by 20 publications

(19 citation statements)

References 53 publications

(104 reference statements)

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“…However, the converse construction seems of more significance: one can endow the self-dual pair constructed in the Main Theorem with the structure of a local presymplectic groupoid, yielding a rather explicit construction of a local integration of a Dirac manifold; a simple version of this construction was shown to us by Eckhard Meinrenken [31], which uses the comorphism Ψ from Remark 4 to construct an action of L × −L of Σ, which corresponds to the left and right invariant vector fields on the local Lie groupoid Σ ⇒ M . This is very much in the spirit of [20], where a local symplectic groupoid integrating a Poisson manifold is constructed by using a symplectic realization.…”

Section: Further Remarksmentioning

confidence: 99%

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On dual pairs in Dirac geometry

Frejlich

Mărcuţ

2017

Math. Z.

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Abstract. In this note we discuss (weak) dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Diractheoretic version of Libermann's theorem from Poisson geometry. Our main result is an explicit construction of self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from [19], but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from [13].

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Section: Further Remarksmentioning

confidence: 99%

“…For background material on Dirac structures particularly close in spirit to the present note, we refer the reader to e.g. [2,11,23,31]. For the specific conventions and notations used in this paper, we refer the reader to Section 2.…”

Section: Introductionmentioning

confidence: 99%

On dual pairs in Dirac geometry

Frejlich

Mărcuţ

2017

Math. Z.

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“…We remark that the Moser method for Poisson manifolds (see, for example, [34]) is a special case, where A = TM, E = T M and F s = Gr(π s ) for a family of gauge equivalent Poisson structures.…”

Section: Appendix B the Moser Methods For Manin Triplesmentioning

confidence: 99%

Deformation spaces and normal forms around transversals

Bischoff¹,

Bursztyn²,

Lima³

et al. 2020

Compositio Math.

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Given a manifold M with a submanifold N , the deformation space D(M, N ) is a manifold with a submersion to R whose zero fiber is the normal bundle ν(M, N ), and all other fibers are equal to M . This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with N a submanifold transverse to the foliation. New examples include L∞-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around N , in terms of a model structure over ν(M, N ).IMPA, Estrada Dona Castorina 110, Rio de Janeiro,

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“…The proof of this result, using (weighted) deformation spaces, is a straightforward extension of the argument in [12] and [3]. Details, for more general weightings, will be given in [14]. Alternatively, one may a proof in coordinates x i , y j , z k adapted to N ⊆ M and to F ⊆ ν(M, N ), similar to the proof of Lemma 2.5.…”

Section: 3mentioning

confidence: 96%

“…In this section we indicate a generalization of Euler-like vector fields for pairs (M, N ) to include weightings. Further details will be given in a forthcoming work [14]. For simplicity, we will restrict ourselves to the degree 2 case (the filtration on functions by 'weighted degree of vanishing' is determined by its components up to filtration degree 2); the general case is slightly more involved and will be discussed in a forthcoming paper [14].…”

Section: The Weighted Settingmentioning

confidence: 99%

Euler-like vector fields, normal forms, and isotropic embeddings

Meinrenken¹

2020

Preprint

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As shown in [4], germs of tubular neighborhood embeddings for submanifolds N ⊆ M are in one-one correspondence with germs of Euler-like vector fields near N . In many contexts, this reduces the proof of 'normal forms results' for geometric structures to the construction of an Euler-like vector field compatible with the given structure. We illustrate this principle in a variety of examples, including the Morse-Bott lemma, Weinstein's Lagrangian embedding theorem, and Zung's linearization theorem for proper Lie groupoids. In the second part of this article, we extend the theory to a weighted context, with an application to isotropic embeddings.

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Poisson geometry from a Dirac perspective

Abstract: Abstract. We present proofs of classical results in Poisson geometry using techniques from Dirac geometry. This article is based on mini-courses at the Poisson summer school in Geneva, June 2016, and at the workshop Quantum Groups and Gravity at the University of Waterloo, April 2016.

Cited by 20 publications

References 53 publications

On dual pairs in Dirac geometry

On dual pairs in Dirac geometry

Deformation spaces and normal forms around transversals

Euler-like vector fields, normal forms, and isotropic embeddings

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