Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration

TL;DR

This paper introduces post-Hopf algebroids, generalizes existing structures, and explores their applications in geometric numerical integration on manifolds, providing new algebraic tools for advanced geometric computations.

Contribution

It defines post-Hopf algebroids, constructs action versions, and links them to post-Lie-Rinehart algebras, expanding the algebraic framework for geometric numerical methods.

Findings

Post-Hopf algebroids generalize pre-Hopf algebroids.

Universal enveloping algebra of a post-Lie-Rinehart algebra forms a post-Hopf algebroid.

Applications demonstrated in geometric numerical integration 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.