The Cohomology of the Full Directed Graph Complex

Dolgushev, Vasily A.; Rogers, Christopher L.

doi:10.1007/s10468-019-09875-7

Cited by 3 publications

(7 citation statements)

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“…45 See [71, Appendix K] for the first item (cf. also [13]), [72] for the second and [73] for the third. 46 The first Betti number is defined as b := k − N + 1 for a connected graph with N vertices and k edges.…”

Section: The Inclusionmentioning

confidence: 91%

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

Morand¹

2022

SIGMA

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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 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.

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Section: The Inclusionmentioning

confidence: 91%

“…directed) graphs. 48 See [13,71] for the first arrow and [47] for a general statement. 49 The cocycle L3 stands for the triangle loop 2 1…”

Section: The Inclusionmentioning

confidence: 99%

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

Morand¹

2022

SIGMA

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“…A proof of Theorem 4.28 for 𝔇𝔊 𝑚 (𝒞) instead of 𝔇𝔊 𝑚 trunc (𝒞) has been sketched by Willwacher [Wil14, Appendix K] (in the case of the commutative cyclic operad 𝒞 = 𝒞om). Other related statements have been proven in detail by Živković[Živ20] and Dolgushev-Rogers[DR19]. We quickly survey what is known about the homology of the undirected graph complex 𝔘𝔊 𝑚 (ℒie) associated to the cyclic Lie operad.…”

mentioning

confidence: 81%

“…Although less ubiquitous than undirected graph complexes, directed versions have appeared in the literature before; see, for example, [Wil14, Živ20, DR19]. Surprisingly, it turns out that the homology of the truncated directed graph complex is actually isomorphic to the one of .…”

Section: Introductionmentioning

confidence: 99%

The stable cohomology of self-equivalences of connected sums of products of spheres

Stoll

2024

Forum of Mathematics, Sigma

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“…33 Here, fcGC d stands for the usual Kontsevich graph complex of connected undirected graphs. 34 See [56,10] for the first arrow and [39] for a general statement. 35 The first Betti number is defined as b := k − N + 1 for a connected graph with N vertices and k edges.…”

Section: Table 2: Cohomology In Low Dimension and Degreementioning

confidence: 99%

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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The Cohomology of the Full Directed Graph Complex

Cited by 3 publications

References 29 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

The stable cohomology of self-equivalences of connected sums of products of spheres

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

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