Geometric post-Lie deformations of post-Lie algebras and regularity structures

TL;DR

This paper develops geometric deformations of post-Lie algebras using torsion and curvature concepts, applying to regularity structures and connecting various algebraic frameworks in differential geometry and algebra.

Contribution

It introduces a class of geometric deformations of post-Lie algebras that preserve structure and links different algebraic approaches in regularity structures and differential geometry.

Findings

Derived compatibility conditions for deformations preserving post-Lie structures

Connected deformed post-Lie algebras to pre-Lie structures in regularity theories

Showed the deformed algebraic structures encompass known geometrical post-Lie algebras

Abstract

In order to derive a class of geometric-type deformations of post-Lie algebras, we first extend the geometrical notions of torsion and curvature for a general bilinear operation on a Lie algebra, then we derive compatibility conditions which will ensure that the post-Lie structure remains preserved. This type of deformation applies in particular to the post-Lie algebra introduced in arXiv:2306.02484v3 in the context of regularity structures theory. We use this deformation to derive a pre-Lie structure for the regularity structures approach given in arXiv:2103.04187v4, which is isomorphic to the post-Lie algebra studied in arXiv:2306.02484v3 at the level of their associated Hopf algebras. In the case of sections of smooth vector bundles of a finite-dimensional manifold, this deformed structure contains also, as a subalgebra, the post-Lie algebra structure introduced in arXiv:1203.4738v3 in the geometrical context of moving frames.