# Post-Lie algebras and factorization theorems

## TL;DR

This paper investigates the properties of universal enveloping algebras of post-Lie algebras, focusing on the Magnus expansion and factorization of group-like elements, especially in the context of solutions to modified classical Yang-Baxter equations.

## Contribution

It extends the understanding of post-Lie algebra structures and their enveloping algebras, highlighting new factorization properties related to the classical Yang-Baxter equation.

## Key findings

- Analysis of group-like elements in completed Hopf algebras
- Connections between Magnus expansion and post-Lie algebra properties
- Factorization results for elements associated with classical Yang-Baxter solutions

## Abstract

In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions) of those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang-Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.