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

### Mahdi Jasim Hasan Al-Kaabi

Mustansiriyah University, Baghdad, Iraq### Kurusch Ebrahimi-Fard

Norwegian University of Science and Technology, Trondheim, Norway### Dominique Manchon

CNRS–Université Clermont-Auvergne, Aubière Cedex, France### Hans Z. Munthe-Kaas

University of Bergen, Norway

## 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 among 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.

## Cite this article

Mahdi Jasim Hasan Al-Kaabi, Kurusch Ebrahimi-Fard, Dominique Manchon, Hans Z. Munthe-Kaas, Algebraic aspects of connections: From torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials. J. Noncommut. Geom. (2023), published online first

DOI 10.4171/JNCG/539