Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials

Abstract

Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a D-algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations between the post-Lie Magnus expansion, the Grossman-Larson product and the K-map, α-map and β-map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker-Campbell-Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds.

Post-Lie algebras in Regularity Structures

“…( 𝐴 (1) ⊲ 𝐵)( 𝐴 (2) ⊲ 𝐶), where 𝐴, 𝐵, 𝐶 ∈ 𝑈 (𝔤) and 𝑥, 𝑦 ∈ 𝔤. Here, we have used Sweedler's notation for the coproduct Δ: Δ 𝐴 = ( 𝐴) 𝐴 (1) ⊗ 𝐴 (2) . This coproduct is defined for 𝑥 ∈ 𝔤 by…”

“…They were first mentioned in [42,38] on the partition of posets and in the context of Lie-Butcher series. They have also been used in many works in numerical analysis (see [39,26,17,1,2]).…”