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

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.

arXiv comment30 pages

References in corpus (15)

• Homology of generalized partition posets
• Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras
• On post-Lie algebras, Lie--Butcher series and moving frames
• Integration and geometrization of Rota-Baxter Lie algebras
• Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
• Aromatic Butcher Series
• Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials
• Yang-Baxter operators in symmetric categories
• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds
• Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras
• A Survey on the Munthe-Kaas-Wright Hopf Algebra
• The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators
• The universal equivariance properties of exotic aromatic B-series
• Post-Lie algebras of derivations and regularity structures