Dirac structures and Nijenhuis operators

Bursztyn, Henrique; Drummond, Thiago; Netto, Clarice

doi:10.1007/s00209-022-03078-5

Cited by 6 publications

(1 citation statement)

References 31 publications

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“…Dirac structures were first defined by T. Courant [10]. Dirac structures are important because they provide a unified view of Poisson and presymplectic structures, and generalize them, see [5,8,19,24,25,29]. Courant algebroids were introduced in [27].…”

Section: Introductionmentioning

confidence: 99%

Weak $(p,k)$-Dirac manifolds

Bi¹,

Chen²

2023

Preprint

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In this paper, we introduce the notion of a weak (p, k)-Dirac structure in T M ⊕ Λ p T * M , where 0 ≤ k ≤ p − 1. The weak (p, k)-Lagrangian condition has more informations than the (p, k)-Lagrangian condition and contains the (p, k)-Lagrangian condition. The weak (p, 0)-Dirac structures are exactly the higher Dirac structures of order p introduced by N. Martinez Alba and H. Bursztyn in [23] and [6], respectively. The regular weak (p, p − 1)-Dirac structure together with (p, p − 1)-Lagrangian subspace at each point m ∈ M have the multisymplectic foliation. Finally, we introduce the notion of weak (p, k)-Dirac morphism. We give the condition that a weak (p, k)-Dirac manifold is also a weak (p, k)-Dirac manifold by pulling back.

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Section: Introductionmentioning

confidence: 99%

Weak $(p,k)$-Dirac manifolds

Bi¹,

Chen²

2023

Preprint

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Quasi-Lie Bialgebroids, Dirac Structures, and Deformations of Poisson Quasi-Nijenhuis Manifolds

do Nascimento Luiz,

Mencattini,

Pedroni

2024

Bull Braz Math Soc, New Series

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Integrating Nijenhuis structures

Pugliese

Sparano

Vitagliano

2023

Annali di Matematica

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A Nijenhuis operator on a manifold is a (1, 1) tensor whose Nijenhuis torsion vanishes. A Nijenhuis operator $${\mathcal {N}}$$ N determines a Lie algebroid that knows everything about $${\mathcal {N}}$$ N . In this sense, a Nijenhuis operator is an infinitesimal object. In this paper, we identify its global counterpart. Namely, we characterize Lie groupoids integrating the Lie algebroid of a Nijenhuis operator. We illustrate our integration result in various examples, including that of a linear Nijenhuis operator on a vector space or, which is equivalent, a pre-Lie algebra structure.

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Dirac structures and Nijenhuis operators

Cited by 6 publications

References 31 publications

Weak $(p,k)$-Dirac manifolds

Weak $(p,k)$-Dirac manifolds

Quasi-Lie Bialgebroids, Dirac Structures, and Deformations of Poisson Quasi-Nijenhuis Manifolds

Integrating Nijenhuis structures

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