Hitchhiker’s guide to Courant algebroid relations

Vysoký, Jan

doi:10.1016/j.geomphys.2020.103635

Cited by 4 publications

(2 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The non-vanishing brackets of the (essentially unique 68 ) exotic Lie ∞ -structure of Theorem 1.1 -denoted θ in the following -take the form 69 67 Here by split Courant algebroids we mean Courant algebroids whose underlying vector bundle is a Whitney sum E ⊕ E * . 68 Up to gauge transformations and rescalings. 69 The intrinsic degree carried by each bracket is given by |θ4p+2| = −12p − 2.…”

Section: Lie Induces a Morphism Of Dg Lie Algebrasmentioning

confidence: 99%

See 1 more Smart Citation

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

Morand¹

2022

SIGMA

View full text Add to dashboard Cite

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored 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 quasi-Lie bialgebras can be generalised to quasi-Lie 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.

show abstract

Section: Lie Induces a Morphism Of Dg Lie Algebrasmentioning

confidence: 99%

“…where L E X stands for the unique derivative operator extending the action of [X, •] E to the tensor algebra of E and similarly for L 92 More general notions of morphisms of Courant algebroids over different base manifolds can be defined, see [68] for a thorough treatment.…”

Section: A Geometry Of Lie Bialgebroidsmentioning

confidence: 99%