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.
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).
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.
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…
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 …
This is a Bourbaki's seminar text. We introduce the combinatorial Kashiwara-Vergne conjecture on the Baker-Campbell-Hausdorff serie. After recalling previous results and consequences, we explain the Alekseev-Meinrenken's proof [math.QA/0506499]
We prove that a universal symmetric solution of the Kashiwara-Vergne conjecture is unique up to order one. in the Appendix by the second author, this result is used to show that solutions of the Kashiwara-Vergne conjecture for quadratic Lie algebras existing in the literature are not universal.
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…
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.
Consider the Kontsevich $\star$-product on the symmetric algebra of a finite dimensional Lie algebra $\mathfrak g$, regarded as the algebra of distributions with support 0 on $\mathfrak g$. In this paper, we extend this $\star$-product to distributions satisfying an appropriate support condition. As a consequence, we prove a long standing conjecture of Kashiwara-Vergne on the convolution of germs of invariant distributions on the Lie group $G$.
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.
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.
Let $G$ be a connected Lie group, with Lie algebra $g$. In 1977, Duflo constructed a homomorphism of $g$-modules $Duf: S(g) -> U(g)$, which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group $G$. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara-Vergne conjecture for any Lie group $G$. (2) We give a reformulation of the Kashiwa…
This article follows our previous work on Campbell-Hausdorff formula. We study the case of symmetric spaces. We recover, by using a Kontsevich's deformation of the Baker-Campbell-Hausdorff formula, Rouviere's results on the convolution of invariant distributions, for solvable symmetric spaces and "very symmetric spaces".
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…
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.
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…
The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite dimensional groups KV_n, and an extension \hat{KV}_2 of the group KV_2. We show that the group \hat{KV}_2 contains the Grothendieck-Teichmueller group GRT as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem Sol(KV). Furthermore, we prove that Sol(KV) is …
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…
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 …
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 …
Polylogarithmic characterizations of the reduced coaction Lie algebra and the double shuffle Lie algebra, and their relation to the Kashiwara--Vergne Lie algebra
Université de Genève（日内瓦大学）；Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心联合数学高级研究所）
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 $p=5$.
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…
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}.
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.
