形式微分同胚的Hopf代数与流形上的数值积分
Hopf algebras of formal diffeomorphisms and numerical integration on manifolds
AI总结：
本文介绍形式微分同胚的Hopf代数结构及其在流形上数值积分中的应用，重点阐述Lie-Butcher理论与Connes-Kreimer Hopf代数的联系，并引入非交换Faa di Bruno双代数。
摘要：
B-级数起源于John Butcher在20世纪60年代的工作，作为分析微分方程数值积分（特别是Runge-Kutta方法）的工具。近年来，它与微扰量子场论中的重整化理论建立了联系。经典Runge-Kutta方法的代数结构由Connes-Kreimer Hopf代数描述。Lie-Butcher理论是B-级数的推广，旨在研究流形上演化微分方程的Lie群积分器。Lie群积分器基于流形上的一般Lie群作用，经典Runge-Kutta积分器在此框架下作为$\mathbb{R}^n$通过平移作用于自身的特例出现。Lie-Butcher理论将$\mathbb{R}^n$上的经典B-级数与流形上的Lie级数相结合。其底层的Hopf代数结合了Connes-Kreimer Hopf代数与自由Lie代数的shuffle Hopf代数。我们面向一般数学读者介绍Hopf代数结构及其与数值分析中结构的关系。特别地，我们探讨了Lie级数、时变Lie级数和流形上微分同胚的Lie-Butcher级数之间的紧密联系。讨论了Euler和Dynkin幂等元在数值分析中的作用。引入了非交换版本的Faa di Bruno双代数，并探讨了与非交换Bell多项式的关系。
英文摘要：
B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge-Kutta methods. Connections to renormalization theory in perturbative quantum field theory have been established in recent years. The algebraic structure of classical Runge-Kutta methods is described by the Connes-Kreimer Hopf algebra. Lie-Butcher theory is a generalization of B-series aimed at studying Lie-group integrators for differential equations evolving on manifolds. Lie-group integrators are based on general Lie group actions on a manifold, and classical Runge-Kutta integrators appear in this setting as the special case of R^n acting upon itself by translations. Lie--Butcher theory combines classical B-series on R^n with Lie-series on manifolds. The underlying Hopf algebra combines the Connes-Kreimer Hopf algebra with the shuffle Hopf algebra of free Lie algebras. We give an introduction to Hopf algebraic structures and their relationship to structures appearing in numerical analysis, aimed at a general mathematical audience. In particular we explore the close connection between Lie series, time-dependent Lie series and Lie--Butcher series for diffeomorphisms on manifolds. The role of the Euler and Dynkin idempotents in numerical analysis is discussed. A non-commutative version of a Faa di Bruno bialgebra is introduced, and the relation to non-commutative Bell polynomials is explored.
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