Split Courant algebroids as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si4.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub></mml:math>-structures

Antunes, P.; Costa, Joana M. Nunes da

doi:10.1016/j.geomphys.2020.103790

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…57 The intrinsic degree carried by each bracket is given by |θ 4p+2 | = −12p − 2. Pulling back the brackets along the suspension map s : A E Lie [2] → A E Lie of degree 2 yields a series of brackets θ4p+2 on A E Lie [2] with the usual degree −4p. 58 Recall that minimal Lie∞-algebras are characterised by a vanishing differential θ 1 ≡ 0.…”

Section: Structure Black Red Cohomologymentioning

confidence: 99%

“…for example the Lie bialgebroid on Poisson manifolds defined in Example A.2 . If furthermore, Λ is of constant rank, the triangular Lie bialgebroid is said to be regular.Quasi-Lie, Lie-quasi and proto-Lie bialgebroids The above mentioned characterisation of Lie bialgebroids as dg Poisson structures on Γ (∧ • E)[1], ∧ calls for several natural generalisations, as summarised in the following table 73 see e.g [30,50,18,2]. :…”

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confidence: 99%

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A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

Morand

2021

Preprint

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Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantization of (finite dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via onecolored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their "quasi" generalisations. Using results due to T. Willwacher and M. Živković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichmüller group on Lie bialgebras and Lie-quasi bialgebras can be generalised to Lie-quasi bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie∞algebra structure non-trivially deforming the "big bracket" for Lie bialgebroids. This exotic Lie∞-structure can be interpreted as the equivalent in d = 3 of the Kontsevich-Shoikhet obstruction to the quantization of infinite dimensional Poisson manifolds (in d = 2). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.

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Section: Structure Black Red Cohomologymentioning

confidence: 99%

mentioning

confidence: 99%

A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

Morand

2021

Preprint

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“…Recently, Grützmann, Michel, and Xu [16] studied Weyl quantization of degree 2 symplectic graded manifolds. Given a pseudo-Euclidean vector bundle (E, •, • ) over M , each metric connection ∇ on E determines a degree 2 symplectic graded manifold (T * [2]M ⊕ E [1], ω ∇ ). They proved that the Weyl quantization of this degree 2 symplectic graded manifold establishes a bijection between Hamiltonian generating functions and skew-symmetric Dirac generating operators.…”

Section: Introductionmentioning

confidence: 99%

“…We hope our results will be of some use in this subject. Notably, it is shown in the papers [2,14] that each split Courant algebroid…”

Section: Introductionmentioning

confidence: 99%

Dirac generating operators of split Courant algebroids

Cai¹,

Chen²,

Lang³

et al. 2022

Preprint

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Given a vector bundle A over a smooth manifold M such that the square root L of the line bundle ∧ top A * ⊗ ∧ top T * M exists, the Clifford bundle associated to the split pseudo-Euclidean vector bundle (E = A ⊕ A * , •, • ), admits a spinor bundle ∧ • A ⊗ L, whose section space can be thought of as that of Berezinian half-densities of the graded manifold A * [1]. We give an explicit construction of Dirac generating operators of split Courant algebroid (or proto-bialgebroid) structures on A ⊕ A * introduced by Alekseev and Xu. We also prove that the square of the Dirac generating operator gives rise to an invariant of the split Courant algebroid.

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The deformation L∞ algebra of a Dirac–Jacobi structure

Tortorella

2022

Differential Geometry and its Applications

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Split Courant algebroids as L∞-structures

Cited by 5 publications

References 17 publications

A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

Dirac generating operators of split Courant algebroids

The deformation L∞ algebra of a Dirac–Jacobi structure

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