Uniqueness in the Kashiwara-Vergne conjecture

Finite Type Invariants of w-Knotted Objects: From Alexander to Kashiwara and Vergne

This preprint was split in two and became the first two parts of a four-part series (arXiv:1405.1956, arXiv:1405:1955, and two in preparation). The remaining relevance of this preprint is due to the series of videotaped lectures (wClips) that are linked within.

On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture

For a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K\supset \mathbb C$, we deduce from the compatibility between cup products Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an alternative way of re-writing Kontsevich product $\star$ on $\mathrm S(\mathfrak g)$ by means of the Alekseev--Torossian flat connection (Alekseev and Torossian, 2010). We deduce a similar formula directly from the Kashiwara--Vergne conjecture (Kashiwara and Vergne, 1978).

Eulerian idempotent and Kashiwara-Vergne conjecture

By using the interplay of the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution (F,G) of the first equation of the Kashiwara-Vergne conjecture: x+y-log(exp(y)exp(x))=(1-exp(-ad x))F(x,y)+(exp(ad y)-1)G(x,y) . Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent.

Double Shuffle and Kashiwara-Vergne Lie algebras

We prove that the double shuffle Lie algebra ds, dual to the space of new formal multiple zeta values, injects into the Kashiwara-Vergne Lie algebra krv defined and studied by Alekseev-Torossian. The proof is based on a reformulation of the definition of krv, and uses a theorem of Ecalle on a property of elements of ds.

Kashiwara-Vergne解的逐次延拓

Kashiwara-Vergne solutions degree by degree

本文证明Kashiwara-Vergne问题的解可逐次延拓，借此简化一类Drinfel'd结合子的计算，并证明Kashiwara-Vergne群的关联分次李代数同构于分次Kashiwara-Vergne李代数。

Kashiwara-Vergne and dihedral bigraded Lie algebras in mould theory

We introduce the Kashiwara-Vergne bigraded Lie algebra associated with a finite abelian group and give its mould theoretic reformulation. By using the mould theory, we show that it includes Goncharov's dihedral Lie algebra, which generalizes the result of Raphael and Schneps.

Internally connected graphs and the Kashiwara-Vergne Lie algebra

It is conjectured that the Kashiwara-Vergne Lie algebra $\widehat{\mathfrak{krv}}_2$ is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}_1$ and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of $\widehat{\mathfrak{krv}}_2$ whose intersection is $\mathfrak{grt}_1$, thus giving a way to interpolate between these two Lie algebras.

On the elliptic Kashiwara-Vergne Lie algebra

We recall the definitions of two independently defined elliptic versions of the Kashiwara-Vergne Lie algebra $\frak{krv}$, namely the Lie algebra $\frak{krv}^{(1,1)}$ constructed by A.Alekseev, N.Kawazumi, Y.Kuno and F.Naef arising from the study of graded formality isomorphisms associated to topological fundamental groups of surfaces, and the Lie algebra $\frak{krv}_{ell}$ defined using mould theoretic techniques arising from multiple zeta theory by E.Raphael and L.Schneps, and show that they coincide.

A topological characterisation of the Kashiwara-Vergne groups

In 2017 Bar-Natan and the first author showed that solutions to the Kashiwara--Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in $\mathbb{R}^4$, which can be finitely presented algebraically as a circuit algebra, or, equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups $\mathsf{KV}$ and $\mathsf{KRV}$ -- the symmetry groups of Kashiwara-Vergne solutions -- as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebra…

The Elliptic Kashiwara-Vergne Lie algebra in low weights

In this paper, we study the elliptic Kashiwara-Vergne Lie Algebra $\mathfrak{krv}$, which is a certain Lie subalgebra of the Lie algebra of derivations of the free Lie algebra in two generators. It has a natural bigrading, such that the Lie bracket is of bidegree $(-1,-1)$. After recalling the graphical interpretation of this Lie algebra, we examine low degree elements of $\mathfrak{krv}$. More precisely, we find that $\mathfrak{krv}^{(2,j)}$ is one-dimensional for even $j$ and zero $j$ odd. We also compute $\operatorname{dim}(\mathfrak{krv})^{(3,m)} = \lfloor\frac{m-1}{2}\rfloor - \lfloor\fra…

关于Kashiwara-Vergne李代数与双洗牌李代数的注记

Notes on Kashiwara-Vergne and double shuffle Lie algebras

本文梳理了Kashiwara-Vergne李代数$\frak{krv}$与双洗牌李代数$\frak{dmr}$的关系研究进展，并验证了Ecalle六元关系在小深度情形下的有效性。

Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations

The Kashiwara-Vergne (KV) conjecture states the existence of solutions of a pair of equations related with the Campbell-Baker-Hausdorff series. It was solved by Meinrenken and the first author over the real numbers, and in a formal version, by the first and last authors over a field of characteristic 0. In this paper, we give a simple and explicit formula for a map from the set of Drinfeld associators to the set of solutions of the formal KV equations. Both sets are torsors under the actions of prounipotent groups, and we show that this map is a morphism of torsors. When specialized to the KZ …

On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras

We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell-Hausdorff series in the form ln(e^xe^y)=x+y+[x,a(x,y)]+[y,b(x,y)], where a(x,y) and b(x,y) are Lie series in x and y. This observation explains the existence of explicit rational solutions of the quadratic KV problem (see M. Vergne, C.R.A.S. 329 (1999), no. 9, 767--772 and A. Alekseev, E. Meinrenken, C.R.A.S. 335 (2002), no. 9, 723--728 arXiv:math/0209346), whereas constructing an explicit rational solution o…

关于Kashiwara-Vergne李代数的线性化和椭圆版本

On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra

本文定义了Kashiwara-Vergne李代数的线性化和椭圆版本，并展示了其与Grothendieck-Teichmüller李代数和双shuffle李代数椭圆版本的关系。

Goldman-Turaev formality implies Kashiwara-Vergne

Let $Σ$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{g}(Σ)$ and its associated graded ${\rm gr}\, \mathfrak{g}(Σ)$. In this paper, we prove the converse: if $F$ induces an isomorphism $\mathfrak{g}(Σ) \cong {\rm gr} \, \mathfrak{g}(Σ)$, then it satisfies the Kashiwara-Vergne equations up to conjugation. As an appli…

Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra

We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more det…

On the Jacobson element and generators of the Lie algebra $\mathfrak{grt}$ in nonzero characteristic

We state a conjecture (due to M. Duflo) analogous to the Kashiwara--Vergne conjecture in the case of a characteristic $p>2$, where the role of the Campbell--Hausdorff series is played by the Jacobson element. We prove a simpler version of this conjecture using Vergne's explicit rational solution of the Kashiwara--Vergne problem. Our result is related to the structure of the Grothendieck--Teichmüller Lie algebra $\mathfrak{grt}$ in characteristic $p$: we conjecture existence of a generator of $\mathfrak{grt}$ in degree $p-1$, and we provide this generator for $p=3$ and $p=5$.

Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these …

A knot-theoretic approach to comparing the Grothendieck-Teichmüller and Kashiwara-Vergne groups

Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between low-dimensional topology and quantum algebra. For example, homomorphic expansions of parenthesised braids are in one-to-one correspondence with Drinfel'd associators (Bar-Natan 1998), and homomorphic expansions of $w$-foams are in one-to-one correspondence with solutions to the Kashiwara-Vergne (KV) equations (Bar-Natan and the first author, 2017). The sets …

Brunnian braids和从双重shuffle李代数到Kashiwara-Vergne李代数的包含

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

本文证明了Brunnian braids的双重shuffle李代数通过infinitesimal hexagon方程注入到对称Kashiwara-Vergne李代数，并扩展了这一包含到两个方向：利用Brunnian Lie代数的下正则系列以及建立pentagon方程映射、stuffle coproduct、divergence映射和necklace cobracket之间的显式联系。

Drinfeld关联子与Kashiwara-Vergne关联子在更高亏格中的研究

Drinfeld associators and Kashiwara-Vergne associators in higher genera

本文基于Gonzalez和Drinfeld关联子构造了高亏格Kashiwara-Vergne关联子，解决了Goldman-Turaev Lie bialgebra形式化问题中的高亏格情况。

缩减共作用李代数、双洗牌李代数和非交换krv2方程

Polylogarithmic characterizations of the reduced coaction Lie algebra and the double shuffle Lie algebra, and their relation to the Kashiwara--Vergne Lie algebra

本文研究了缩减共作用李代数与双洗牌李代数之间的关系，并展示了非交换krv2方程的构造方法。

零 genus Kashiwara-Vergne 解从辫子中来

Genus Zero Kashiwara-Vergne Solutions from Braids

本文通过moperads重新解释了Alekseev-Enriquez-Torossian构造Kashiwara-Vergne解的方法，展示等价关系如何生成零 genus解，并讨论Grothendieck-Techmüller模块群的作用。

w-纽结对象的有限型不变量 II：缠结、泡沫与 Kashiwara-Vergne 问题

Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem

本文研究 w-缠结与 w-泡沫的有限型不变量，借箭头图将其联系于非有度 Lie 代数，并证明泛有限型不变量等价于 Kashiwara-Vergne 猜想的解。

一个 functorial 方法用于 Kashiwara-Vergne

A functorial approach to Kashiwara-Vergne

本文通过 functorial 方法从 operads 推导 Lie bialgebras，并建立了 GRT 和 KRV 高阶组之间的关系。

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

In this paper, we describe a surprising link between the theory of the Goldman-Turaev 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 $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie bialgebra is defined by the Goldman bracket $\{ -,- \}$ and Turaev cobracket $δ$ on the $\mathbb{K}$-span of homotopy classes of free loops on $Σ$. Applying an expansion $θ: \mathbb{K}π\to \mathbb{K}\langle x_1, \dots, x_n \rangle$ yields an algebraic description of the operatio…

Drinfeld结合子方程的涌现版本

Emergent version of Drinfeld's associator equations

本文提出Drinfeld结合子方程的弱版本（即涌现版本），证明其线性化解仍能导出线性化Kashiwara-Vergne方程的解，并探讨其在涌现辫拓扑背景下的意义。

高亏格下的Goldman-Turaev Lie双代数与Kashiwara-Vergne问题

The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

本文研究高亏格曲面上的Goldman-Turaev形式化问题，通过定义并求解新的高亏格Kashiwara-Vergne问题，建立了GT形式化映射，并揭示了其与Grothendieck-Teichmüller群的联系。

线性化 KV 的数值计算以及 Deligne-Drinfeld 和 Broadhurst-Kreimer 猜想

Numerical computation of linearized KV and the Deligne-Drinfeld and Broadhurst-Kreimer conjectures

本文在低权重下数值计算了线性化Kashiwara-Vergne李代数的分级商维数，证实了相关猜想，并借此在更广范围内检验了Deligne-Drinfeld和Broadhurst-Kreimer等关于Grothendieck-Teichmüller群及多重zeta值的猜想。

Deformation quantization and invariant distributions

In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and g=Lie(G), respectively. They also proved their conjecture for G solvable. Using Kontsevich's deformation quantization we establish directly this result for distributions on any real Lie group G. In turn this gives a new proof of Duflo's result on the local solvability of bi-invariant differential operators on G.

Uniqueness of quantization of complex contact manifolds

Using the language of algebroid stacks, we will show that Kashiwara's quantization of a complex contact manifold is unique.

Solution non universelle pour le problème KV-78

In 78' M. Kashiwara and Vergne conjectured some property on the Campbell-Hausdorff series in such way a trace formula is satisfied. They proposed an explicit solution in the case of solvable Lie algebras. In this note we prove that this "solvable solution" is not universal. Our method is based on computer calculation. Furthermore our programs prove up to degree 16, Drinfeld's Lie algebra $\mathfrak{grt}_1$ coincides with the Lie algebra $\hat{kv_2}$ defined in \cite{AT}.

闭曲面上Goldman-Turaev李双代数的形式性

The formality of the Goldman-Turaev Lie bialgebra on a closed surface

该研究利用非交换联络工具重述高亏格Kashiwara-Vergne群与配极子，解决了闭曲面情形下Goldman-Turaev李双代数相伴分次的幂幺自同构群的确定问题。