Poisson Brackets in Kontsevich's "Lie World"

Naef, Florian

doi:10.48550/arxiv.1608.08886

Cited by 5 publications

(15 citation statements)

References 7 publications

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“…In [29], one gives an algebraic proof of this result and shows that its inverse also holds true. That is, if F ∈ TAut(L) verifies (46) then 7 The Kashiwara-Vergne problem and homomorphic expansions…”

Section: Expansions and Transfer Of Structuresmentioning

confidence: 91%

The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

Alekseev

Kawazumi

Kuno

et al. 2018

Advances in Mathematics

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In this paper, we describe a surprising link between the theory of the GoldmanTuraev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman-Turaev Lie bialgebra is defined by the Goldman bracket {−, −} and Turaev cobracket δ on the K-span of homotopy classes of free loops on Σ.Applying an expansion θ : Kπ → K x 1 , . . . , x n yields an algebraic description of the operations {−, −} and δ in terms of non-commutative variables x 1 , . . . , x n . If Σ is a surface of genus g = 0 the lowest degree parts {−, −} −1 and δ −1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that {−, −} = {−, −} −1 and δ = δ −1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral.In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).

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Section: Expansions and Transfer Of Structuresmentioning

confidence: 91%

The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

Alekseev

Kawazumi

Kuno

et al. 2018

Advances in Mathematics

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“…where g 1,2 , g 12,3 , etc. are images of g under various co-face maps, and Φ ∈ TAut 3 is related to ω 0 via ω 0 = (g 2,3 g 1,23 ).h P (C(Φ)).…”

Section: Motivation: Descent Equationsmentioning

confidence: 99%

“…We will often use a notation u.α = ρ(u)α for actions of tder n on various spaces. Equipped with the Lie bracket (12), tder n is a pro-nilpotent Lie algebra which readily integrates to a group denoted TAut n together with the group homomorphism (again denoted by ρ):…”

Section: Proofmentioning

confidence: 99%

“…Since g ∈ S is a solution of the first Kashiwara-Vergne equation, its associator (13) satisfies Φ g ∈ SAut 3 . Set ω = C(g) and apply the cocycle C to the equation g 2,3 g 1,23 = g 1,2 g 12,3 Φ g to get 0 = C(g 2,3 g 1,23 ) − C(g 1,2 g 12,3 ) = (ω 2,3 + g 2,3 .ω 1,23 ) − (ω 1,2 + g 1,2 .ω 12,3 + (g 1,2 g 12,3 ).C(Φ g )) = (ω(x 2 , x 3 ) + g 2,3 .ω(x 1 , x 2 + x 3 )) − (ω(x 1 , x 2 ) + g 1,2 .ω(x 1 + x 2 , x 3 ) + (g 1,2 g 12,3 ).C(Φ g )) = (ω(x 2 , x 3 ) + ω(x 1 , log(e x2 e x3 ))) − (ω(x 1 , x 2 ) + ω(log(e x1 e x2 ), x 3 ) + (g 1,2 g 12,3 ).C(Φ g )) = ∆(ω) − (g 1,2 g 12,3 ).C(Φ g ).…”

Section: Define a Vector Space Isomorphism γ : Tdermentioning

confidence: 99%

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Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Alekseev

Naef

et al. 2017

Lett Math Phys

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Abstract. Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g in the Chern-Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as p = F, F where F is the curvature 2-form and ·, · is an invariant scalar product on the corresponding Lie algebra g. The descent for p gives rise to an element ω = ω 3 + ω 2 + ω 1 + ω 0 of mixed degree. The 3-form part ω 3 is the Chern-Simons form. The 2-form part ω 2 is known as the Wess-Zumino action in physics. The 1-form component ω 1 is related to the canonical central extension of the loop group LG.In this paper, we give a new interpretation of the low degree components ω 1 and ω 0 . Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara-Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class p. In more detail, we define a 1-cocycle C which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara-Vergne equation F is mapped to ω 1 = C(F ). Furthermore, the component ω 0 is related to the associator corresponding to F . It is surprising that while F and Φ satisfy the highly non-linear twist and pentagon equations, the elements ω 1 and ω 0 solve the linear descent equation.

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“…They are isomorphic as filtered K-vector spaces, but not canonically. The formality question for the Goldman-Turaev Lie bialgebras is whether there exists a filtered Lie bialgebra isomorphism from g to gr g such that the associated graded map is the identity on gr g. This question has been studied during the last several years by various approaches: the study first began with formality for Goldman brackets [16,17,22,23,24,12] and then has been deepened to formality for Turaev cobrackets [21,2,4,3,13]. One motivation for considering this question comes from the study of the Johnson homomorphisms of mapping class groups [18].…”

Section: Introductionmentioning

confidence: 99%

Goldman-Turaev formality implies Kashiwara-Vergne

Alekseev¹,

Kawazumi²,

Kuno³

et al. 2018

Preprint

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Let Σ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work [3], we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism F ∈ Aut(L) of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra g(Σ) and its associated graded gr g(Σ). In this paper, we prove the converse: if F induces an isomorphism g(Σ) ∼ = gr g(Σ), then it satisfies the Kashiwara-Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.

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Poisson Brackets in Kontsevich's "Lie World"

Cited by 5 publications

References 7 publications

The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Goldman-Turaev formality implies Kashiwara-Vergne

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