The Baker-Campbell-Hausdorff formula in the free metabelian Lie algebra

Kontsevich's Universal Formula for Deformation Quantization and the Campbell-Baker-Hausdorff Formula, I

We relate a universal formula for the deformation quantization of arbitrary Poisson structures proposed by Maxim Kontsevich to the Campbell-Baker-Hausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such universal formulae.

On the Kontsevich and the Campbell-Baker-Hausdorff deformation quantizations of a linear Poisson structure

For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K], and the formula, arising from the classical Campbell-Baker-Hausdorff formula [Ka]. It was proved in [Ka] that the last formula is exactly the part of Kontsevich's formula consisting of all the admissible graphs without (oriented) cycles between the vertices of the first type. It follows from the CBH-theorem that this part of Kontsevich's formula defines an …

Kashiwara-Vergne-Rouviere methods for symmetric spaces

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".

Higher Lie theory

We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's original functor but allows us to prove the existence of additional, previously unknown, structures and properties. Namely, we introduce a well-behaved left adjoint functor, we establish functoriality with respect to infinity-morphisms, and we construct a coherent hierarchy of higher Baker-Campbell-Hausdorff formulas. Thanks to these tools, we are able to establish the most important results of higher Lie t…

Eulerian idempotent, pre-Lie logarithm and combinatorics of trees

The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to establish explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The p…

完全罗塔-巴克斯李代数的正式积分

Formal integration of complete Rota-Baxter Lie algebras

本文提出了一种完全罗塔-巴克斯李代数的正式积分理论，通过后李马格努斯展开给出了罗塔-巴克斯算子的显式公式，并展示了如何从过滤罗塔-巴克斯群构造分层罗塔-巴克斯李环。