Connes-Moscovici霍普夫代数的公式
Formulas for the Connes-Moscovici Hopf algebra
摘要：
给出了Connes-Moscovici霍普夫代数$\mathcal{H}_{\tmop{CM}}$的余积和对极的显式表达式.要做到这一点，我们首先限制自己的一个子霍普夫代数$\mathcal{H}^1_{\tmop{CM}}$包含非平凡的元素，即那些的余积和对极是非平凡的。有两种方法可以获得显式公式。一方面，代数$\mathcal{H}^1_{\tmop{CM}}$同构于单位切同态群上坐标的Faà di Bruno霍普夫代数，并且使用替换自同构而不是自同构使计算变得容易。另一方面，在向量场的形式共轭和解析共轭的框架下，代数$\mathcal{H}^1_{\tmop{CM}}$同构于在重构理论中自然出现的经典shuffle霍普夫代数的子霍普夫代数.利用洗牌霍普夫代数的简单结构，我们再次导出了$\mathcal{H}^1_{\tmop{CM}}$中余积和对极的显式公式.
英文摘要：
We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\mathcal{H}_{\tmop{CM}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\mathcal{H}^1_{\tmop{CM}}$ containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to the Faà di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather than diffeomorphisms. On the other hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to a sub-Hopf algebra of the classical shuffle Hopf algebra which appears naturally in resummation theory, in the framework of formal and analytic conjugacy of vector fields. Using the very simple structure of the shuffle Hopf algebra, we derive once again explicit formulas for the coproduct and the antipode in $\mathcal{H}^1_{\tmop{CM}}$.
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