Internally connected graphs and the Kashiwara-Vergne Lie algebra

Felder, Matteo

doi:10.1007/s11005-018-1052-5

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…Moreover, Ševera and Willwacher [36] gives a decreasing filtration sder ⊃ Ker(div) ⊃ • • • ⊃ t n by using a certain spectral sequence related to a graph complex. (See also a related work by Felder [11].) This can be thought of as a solution to the genus 0 analogue of the Johnson image problem.…”

Section: Enomoto-satoh Trace and Framed Turaev Cobracketmentioning

confidence: 99%

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: Enomoto-satoh Trace and Framed Turaev Cobracketmentioning

confidence: 99%

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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“…These two spaces together with the induced differential on the first page yield a definition of the Kashiwara-Vergne Lie algebra in terms of graphs ( [1], [17]). Furthermore, using some additional simplicial structure on ICG(n), we constructed a nested sequence of Lie subalgebras of E 0,0 1 (2) interpolating between the Kashiwara-Vergne Lie algebra and the Grothendieck-Teichmüller Lie algebra [7]. In this text we describe an application of the two-loop part, that is, (2), a one-loop graph in ICG(3) and a two-loop graph in ICG(1).…”

Section: Internally Connected Graphsmentioning

confidence: 99%

“…In weight k = 24, we expect, from Goncharov's [8], and Ihara and Takao's [9] formula, two independent linear relations for the brackets {σ 2i+1 , σ 24−1−2i } modulo F 3 grt 1 . These can be found by calculating for instance ψ(x 6 y 10 (−x − y) 6 ) and ψ(x 7 y 8 (−x − y) 7 ). The relations in F 2 grt 1 /F 3 grt 1 obtained in this way are,…”

Section: S3mentioning

confidence: 99%

Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

Felder

2018

Sel. Math. New Ser.

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We establish an isomorphism between the Grothendieck-Teichmüller Lie algebra grt 1 in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs ICG(1). In particular, we recover all linear relations satisfied by the brackets of the conjectural generators σ 2k+1 modulo depth three by considering relations among two-loop graphs.The Grothendieck-Teichmüller Lie algebra is related to the zeroth cohomology of M. Kontsevich's graph complex GC 2 via T. Willwacher's isomorphism. We define a descending filtration on H 0 (GC 2 ) and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under T. Willwacher's map.

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Internally connected graphs and the Kashiwara-Vergne Lie algebra

Cited by 2 publications

References 10 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

Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

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