Lie–Butcher Series, Geometry, Algebra and Computation
Lie–Butcher Series, Geometry, Algebra and Computation
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谎言-屠夫系列、几何、代数和计算
DOI：
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发表时间：
2017
期刊：
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通讯作者：
Kristoffer K. Follesdal
中科院分区：
文献类型：
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作者：
H. Munthe;Kristoffer K. Follesdal
Lie-Butcher(LB)级数是用树和森林表示的形式的幂函数级数。在几何方面，LB级数将经典B级数从欧氏空间推广到李群和齐次流形。在代数方面，B-级数是基于预李代数和Butcher-Connes-Kreimer Hopf代数的。取而代之的是基于后李代数及其包络代数的LB级数。在过去的十年里，LB级数的代数理论已经成熟。这篇论文的目的有两个。首先，我们的目标是以一种简明和自包含的方式来呈现其背后的代数结构。其次，我们回顾了文献中关于LB级数的一些代数运算，并将它们改写为递归公式。这是正在进行的努力的一部分，目的是创建一个广泛的软件库，用于使用编程语言Haskell在LB系列和B系列中进行计算。
Lie–Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. The LB-series are instead based on post-Lie algebras and their enveloping algebras. Over the last decade the algebraic theory of LB-series has matured. The purpose of this paper is twofold. First, we aim at presenting the algebraic structures underlying LB series in a concise and self contained manner. Secondly, we review a number of algebraic operations on LB-series found in the literature, and reformulate these as recursive formulae. This is part of an ongoing effort to create an extensive software library for computations in LB-series and B-series in the programming language Haskell.