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

Frédéric Chapoton (1) , Frédéric Patras (2)

1 ICJ - Institut Camille Jordan
2 JAD - Laboratoire Jean Alexandre Dieudonné

We study the internal structure of enveloping algebras of preLie algebras. We show in particular that the canonical projections arising from the Poincaré-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.

Mots clés: enveloping algebra; Lie idempotent; preLie algebra; Magnus formula

Domaines: Algèbres quantiques [math.QA]

Frédéric Chapoton, Frédéric Patras. Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula. 2012. ⟨hal-00658228v1⟩