关于李群积分器的Hopf代数结构
On the Hopf Algebraic Structure of Lie Group Integrators
AI总结：
本文研究基于有序根树的交换但非余交换的分次Hopf代数Hn，它推广了Butcher在Runge-Kutta方法中发展的无序根树Hopf代数Hc，并作为非交换李群作用流形上数值积分器的基础，给出了余积和对极的递归与非递归定义，并证明了Hn的对偶是Grossman-Larson Hopf代数。
摘要：
研究了一个基于有序根树的交换但非余交换的分次Hopf代数 $\Hn$。该Hopf代数推广了Butcher在其Runge-Kutta方法研究中发展的无序根树Hopf代数结构 $\Hc$，后者后来被Connes和Moscovici在非交换几何背景下以及Kreimer在量子场论重正化描述中重新发现。本文表明，$\Hn$ 自然地从非交换导子范畴中的一个泛对象得到，并且特别地，它为基于流形上非交换李群作用的数值积分器研究奠定了基础。推导了余积和对极的递归和非递归定义。还证明了 $\Hn$ 的对偶是Grossman和Larson的Hopf代数。$\Hn$ 包含两个著名的Hopf代数作为特例：Butcher-Connes-Kreimer的Hopf代数 $\Hc$ 通过树对称化算子的像被识别为 $\Hn$ 的一个真子代数；自由结合代数的Hopf代数 $\Hf$ 通过商构造从 $\Hn$ 得到。
英文摘要：
A commutative but not cocommutative graded Hopf algebra $\Hn$, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees $\Hc$, developed by Butcher in his study of Runge--Kutta methods and later rediscovered by Connes and Moscovici in the context of non-commutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that $\Hn$ is naturally obtained from a universal object in a category of non-commutative derivations, and in particular, it forms a foundation for the study of numerical integrators based on non-commutative Lie group actions on a manifold. Recursive and non-recursive definitions of the coproduct and the antipode are derived. It is also shown that the dual of $\Hn$ is a Hopf algebra of Grossman and Larson. $\Hn$ contains two well-known Hopf algebras as special cases: The Hopf algebra $\Hc$ of Butcher--Connes--Kreimer is identified as a proper subalgebra of $\Hn$ using the image of a tree symmetrization operator. The Hopf algebra $\Hf$ of the Free Associative Algebra is obtained from $\Hn$ by a quotient construction.
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