Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula

TL;DR

This paper explores the structure of enveloping algebras of preLie algebras, revealing explicit formulas for projections related to the Poincaré-Birkhoff-Witt theorem and connecting them to the Magnus formula for differential equations.

Contribution

It provides explicit computations of canonical projections in enveloping algebras of preLie algebras and links these to the Magnus formula, offering new insights into differential equations and algebraic structures.

Findings

Explicit formulas for canonical projections in enveloping algebras

Connection between Magnus formula and algebraic projections

New insights into differential equations and combinatorics

Abstract

We study the internal structure of enveloping algebras of preLie algebras. We show in particular that the canonical projections arising from the Poincar\'e-Birkhoff-Witt theorem can be computed explicitely. They happen to be closely related to the Magnus formula for matrix differential equations. Indeed, we show that the Magnus formula provides a way to compute the canonical projection on the preLie algebra. Conversely, our results provide new insights on classical problems in the theory of differential equations and on recent advances in their combinatorial understanding.