On the Kashiwara–Vergne conjecture

  title={On the Kashiwara–Vergne conjecture},
  author={A. Alekseev and Eckhard Meinrenken},
  journal={Inventiones mathematicae},
Let G be a connected Lie group, with Lie algebra $\mathfrak{g}$. In 1977, Duflo constructed a homomorphism of $\mathfrak{g}$-modules $\text{Duf}\colon S(\mathfrak{g})\to U(\mathfrak{g})$, 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… 

Divergence and q-divergence in depth 2

The Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ encodes symmetries of the Kashiwara-Vergne problem on the properties of the Campbell-Hausdorff series. It is conjectures that $\mathfrak{krv} \cong

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

The Kashiwara-Vergne conjecture and Drinfeld’s associators

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

Kontsevich Deformation Quantization and Flat Connections

Abstract In Torossian (J Lie Theory 12(2):597–616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection ωn on the compactified configuration

The Kashiwara-Vergne Method for Lie Groups

In this chapter we motivate and explain the “Kashiwara-Vergne conjecture” for a Lie algebra and its application to analysis on the corresponding Lie group (transfer of convolution of invariant

Applications de la bi-quantification \`a la th\'eorie de Lie

This article in French, with a large English introduction, 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

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

Ribbon 2–knots, 1 + 1 = 2 and Duflo’s theorem for arbitrary Lie algebras

We explain a direct topological proof for the multiplicativity of Duflo isomorphism for arbitrary finite dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows



The Campbell-Hausdorff formula and invariant hyperfunctions

Let G be a Lie group and g its Lie algebra. We denote by V the underlying vector space of g. There is a canonical isomorphism between the ring Z(g) of the biinvariant differential operators on G and

Convolution of Invariant Distributions: Proof of the Kashiwara–Vergne conjecture

Consider the Kontsevich star product on the symmetric algebra of a finite-dimensional Lie algebra g, regarded as the algebra of distributions with support 0 on g. In this Letter, we extend this star

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the

Foundations of Lie theory ; Lie transformation groups

I.Foundations of Lie Theory.- 1. Basic Notions.- 1. Lie Groups, Subgroups and Homomorphisms.- 1.1 Definition of a Lie Group.- 1.2 Lie Subgroups.- 1.3 Homomorphisms of Lie Groups.- 1.4 Linear

Uniqueness in the Kashiwara-Vergne conjecture

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

Deformation quantization and invariant distributions