特殊移位 Banach–Malcev 代数中的显式 Baker–Campbell–Hausdorff 半径

User manual for bch, a program for the fast computation of the Baker-Campbell-Hausdorff and similar series

This manual describes bch, an efficient program written in the C programming language for the fast computation of the Baker-Campbell-Hausdorff (BCH) and similar Lie series. The Lie series can be represented in the Lyndon basis, in the classical Hall basis, or in the right-normed basis of E.S. Chibrikov. In the Lyndon basis, which proves to be particularly efficient for this purpose, the computation of 111013 coefficients for the BCH series up to terms of degree 20 takes less than half a second on an ordinary personal computer and requires negligible 11MB of memory. Up to terms of degree 30, wh…

Baker-Campbell-Hausdorff 乘积的积分

Integration of the Baker-Campbell-Hausdorff product

本文在完备微分分次李代数中构造了使微分与 Baker-Campbell-Hausdorff 乘积相容的群运算，并利用该结构构造 4-单形的李模型，解决了 Lawrence 和 Sullivan 在 4 维情形下提出的问题。

On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can…

Magnus 展开的收敛性估计 I. Banach 代数

Convergence estimates for the Magnus expansion I. Banach algebras

本文在一般 Banach 代数框架下改进了 Magnus 展开的收敛性估计，证明其累积收敛半径为 2，并给出 Baker–Campbell–Hausdorff 级数收敛半径 C₂=2.89847930…，同时发展了解析预解式方法。

Lie's correspondence for commutative automorphic formal loops

We develop Lie's correspondence and an explicit Baker-Campbell-Hausdorff formula for commutative automorphic formal loops.

Post-Lie-Magnus expansion and BCH-recursion

We identify the Baker-Campbell-Hausdorff recursion driven by a weight$λ=1$ Rota-Baxter operator with the Magnus expansion relativeto the post-Lie structure naturally associated to the correspondingRota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursionare reviewed before the proof of the main result.

Exponential series without denominators

For a commutative algebra which comes from a Zinbiel algebra the exponential series can be written without denominators. When lifted to dendriform algebras this new series satisfies a functional equation analogous to the Baker-Campbell-Hausdorff formula. We make it explicit by showing that the obstruction series is the sum of the brace products. In the multilinear case we show that the role the Eulerian idempotent is played by the iterated pre-Lie product.

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…

A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-…

联络的代数方面：从挠率、曲率和post-Lie代数到Gavrilov的双指数与特殊多项式

Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials

本文证明具有曲率和挠率的仿射联络产生post-Lie代数和D-代数，并在该框架下重新诠释挠率、曲率及Gavrilov的双指数，部分解答其猜想，为数值积分和粗糙路径研究开辟新途径。

Factorization of formal exponentials and uniformization

Let $\mathfrak{g}$ be a Lie algebra in characteristic zero equipped with a vector space decomposition $\mathfrak{g}=\mathfrak{g}^-\oplus \mathfrak{g}^+$, and let $s$ and $t$ be commuting formal variables. We prove that the Campbell-Baker-Hausdorff map $C:s\mathfrak{g}^- [[s,t]]\times t\mathfrak{g}^+[[s,t]]\to s\mathfrak{g}^-[[s,t]]\oplus t\mathfrak{g}^+[[s,t]]$ given by $e^{sg^-}e^{tg^+}=e^{C(sg^-,tg^+)}$ for $g^\pm\in\mathfrak{g}^\pm[[s,t]]$ is a bijection, as is well known when $\mathfrak{g}$ is finite-dimensional over $\mathbb{R}$ or $\mathbb{C}$, by geometry. It follows that there exist un…