Post-Lie algebras and factorization theorems

Abstract

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.

Post-Lie-Magnus expansion and BCH-recursion

“…x y := [R(x), y], (19) for all x, y ∈ A. We leave it to the reader to show that the operations in ( 18), ( 19) satisfy the post-Lie identities ( 5) and ( 6) (see [4, § 5.2]).…”

“…We close this introduction by noting that the Magnus expansion, in its various forms (classical [24,27], pre-Lie [1,10,16] and post-Lie [17,18,19,26]), has been studied in applied mathematics, control theory, physics and chemistry. See reference [6] for details on the classical Magnus expansion in applied mathematics.…”

The Magnus expansion and post-Lie algebras

“…Remark 1. We remark that identity (14) encodes backward error analysis for the forward exponential Euler method [11,21]. Indeed, the Lie-Euler integration scheme is the numerical method that approximates solutions of the initial value problem (21) below by following Lie group exponentials, i.e.,…”

Post-Lie Algebra Structures on the Witt Algebra

“…Munthe-Kaas [14], post-Lie algebras also arise naturally from the differential geometry of homogeneous spaces and Klein geometries, topics that are closely related to Cartans method of moving frames. In addition, post-Lie algebras also turned up in relations with Lie groups [5,14], classical Yang-Baxter equation [1], Hopf algebra and classical r-matrices [10] and Rota-Baxter operators [11].…”