Crossed morphisms, (integration of) post-Lie algebras and the post-Lie Magnus expansion

TL;DR

This paper explores the post-Lie Magnus expansion as a crossed morphism between Lie groups and introduces combinatorial methods using planar trees to compute its coefficients.

Contribution

It provides a novel interpretation of the post-Lie Magnus expansion and introduces new combinatorial techniques for coefficient computation.

Findings

Post-Lie Magnus expansion can be viewed as a crossed morphism.

Two combinatorial methods for coefficient calculation are proposed.

The methods are based on tubings on planar trees.

Abstract

In the first part of this letter it will be shown that the post-Lie Magnus expansion can be interpreted as a crossed morphism between two (local) Lie group. The second part will be devoted to present two combinatorial methods, both based on special tubings on planar trees, to compute the coefficients of this remarkable formal series.