Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
Adrien Busnot Laurent, Yunnan Li, Yunhe Sheng
TL;DR
This work extends Hopf-like symmetry to post-structures by introducing post-Hopf algebroids and showing that the universal enveloping algebra of a post-Lie-Rinehart algebra is naturally a weak post-Hopf algebroid. It develops action post-Hopf algebroids and braiding operators, and constructs free post-Lie-Rinehart algebras from magma algebras, establishing a robust algebraic framework that encompasses Oudom-Guin-type constructions in the Lie-Rinehart setting. The results unify algebraic foundations with practical numerical analysis on manifolds by formulating aromatic flows through post-Hopf algebroids, enabling intrinsic, high-order, geometry-preserving numerical methods via Grossman-Larson-type algebroids and their braidings. Collectively, the paper provides new categorical and operadic tools to model and analyze geometric numerical integration, with potential impact on divergence-free schemes and volume-preserving integrators on manifolds.
Abstract
In this paper, we introduce the notion of post-Hopf algebroids, generalizing the pre-Hopf algebroids introduced in [Bronasco, Laurent, 2025] in the study of exotic aromatic S-series. We construct action post-Hopf algebroids through actions of post-Hopf algebras. We show that the universal enveloping algebra of a post-Lie-Rinehart algebra (post-Lie algebroid) is naturally a post-Hopf algebroid. As a byproduct, we construct the free post-Lie-Rinehart algebra using a magma algebra with a linear map to the derivation Lie algebra of a commutative associative algebra. Applications in geometric numerical integration on manifolds are given.