Related papers: Ecalle's senary relation and dimorphic structures
We explain the current situation of the relationship between the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ and the double shuffle Lie algebra $\mathfrak{dmr}$. We also show the validity of Ecalle's senary relation for small depths.
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…
In this article we prove that there exists an injective Lie morphism from the double shuffle Lie algebra ${\frak{ds}}$ into the Kashiwara-Vergne Lie algebra ${\frak{krv}}$, forming a commutative triangle with the known Lie injections of the…
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…
In this paper, we introduce certain new features of the shuffle algebra, that will allow us to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra.
The linearized double shuffle Lie algebra $\mathfrak{ls}$ is a well-studied Lie algebra, which reflects the depth-graded structure of multiple zeta values. We introduce a generalization $\mathfrak{lq}$, which is motivated from the…
In this article we define an elliptic double shuffle Lie algebra $ds_{ell}$ that generalizes the well-known double shuffle Lie algebra $ds$ to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the…
It is proved by L.~Schneps that the double shuffle Lie algebra $\mathfrak{dmr}_0$ injects to the Kashiwara-Vergne Lie algebra $\mathfrak{krv}_2$ in \cite{Schneps2012,Schneps2025}. We show that $\mathfrak{dmr}_0$ with the infinitesimal…
This article is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo and some extensions are relating…
This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this…
The real multiple zeta values $\zeta(k_1,\ldots,k_r)$ are known to form a ${\bf Q}$-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. In order to study the algebraic properties…
The goal of this article is to define a linearized or depth-graded version $\mathfrak{lkv}$, and a closely related elliptic version $\mathfrak{krv}_{ell}$, of the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ originally constructed by…
A finite-dimensional Lie algebra is called (symmetric) self-dual, if it possesses an invariant nondegenerate (symmetric) bilinear form. Symmetric self-dual Lie algebras have been studied by Medina and Revoy, who have proven a very useful…
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…
Covering Algebras of extended affine Lie algebras(EALA's) relative to finite order automorphisms are studied. Conditions are given for when the resulting algebra is again an EALA. This paper deals with affinizations of EALA's relative to…
We prove some isomorphisms between exceptional W-algebras associated with exceptional simple Lie algebras.
We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an…
We consider the algebra of Hecke correspondences (elementary transformations at a single point) acting on the algebraic K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface S. We derive quadratic relations…
We show that solutions to the Kashiwara-Vergne problem can be extended degree by degree. This can be used to simplify the computation of a class of Drinfel'd associators, which under the Alekseev-Torossian conjecture, may comprise all…
In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology defined in a previous paper and the Fourier transform isomorphism for convolution algebras in Borel-Moore homology are related by the…