On the Lie enveloping algebra of a post-Lie algebra

TL;DR

This paper explores the algebraic structures arising from post-Lie algebras, extending Lie enveloping algebras and revealing new Hopf algebra isomorphisms relevant to numerical integration and Runge-Kutta methods.

Contribution

It introduces a novel associative product on the Lie enveloping algebra of a post-Lie algebra, leading to new Hopf algebra isomorphisms and insights into numerical methods.

Findings

Defined a new associative product on $U(g)$ from a post-Lie structure.

Established a Hopf algebra isomorphism between $U(ar{g})$ and a constructed algebra.

Provided algebraic insights into numerical integration and Runge-Kutta order theory.

Abstract

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra $U(g)$. This permits us to define another associative product on $U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and a new Hopf algebra assembled from $U(g)$ with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.