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The Universal Post-Lie-Rinehart Algebra of Planar Aromatic Trees

Ludwig Rahm

TL;DR

This work constructs the universal free object in the category of tracial post-Lie-Rinehart algebras by extending aromatic-B-series machinery to the post-Lie-Rinehart setting. It introduces planar aromatic trees and a Guin-Oudom extension framework to model endomorphisms, obtaining a concrete free object $\mathcal{APT}=S(\mathcal{PA})\otimes Lie(\mathcal{PT})$ with a tracial post-Lie-Rinehart structure and a decomposable endomorphism algebra that is freely generated over $S(\mathcal{PA})$. The freeness is established through a detailed universality construction using maps $\zeta,\beta,\gamma$ that extend any generating-set map to a morphism, guaranteeing unique factorization. The results connect to post-Lie algebroids and homogeneous-space examples, illustrating how planar-aromatic techniques generalize aromatic B-series to curved spaces and providing a robust algebraic framework for universal objects in this category.

Abstract

This paper defines the algebraic structure of tracial post-Lie-Rinehart algebras and describes the free object in this category. Post-Lie-Rinehart algebras is a generalisation of pre-Lie-Rinehart algebras, and of post-Lie algebroids.

The Universal Post-Lie-Rinehart Algebra of Planar Aromatic Trees

TL;DR

This work constructs the universal free object in the category of tracial post-Lie-Rinehart algebras by extending aromatic-B-series machinery to the post-Lie-Rinehart setting. It introduces planar aromatic trees and a Guin-Oudom extension framework to model endomorphisms, obtaining a concrete free object with a tracial post-Lie-Rinehart structure and a decomposable endomorphism algebra that is freely generated over . The freeness is established through a detailed universality construction using maps that extend any generating-set map to a morphism, guaranteeing unique factorization. The results connect to post-Lie algebroids and homogeneous-space examples, illustrating how planar-aromatic techniques generalize aromatic B-series to curved spaces and providing a robust algebraic framework for universal objects in this category.

Abstract

This paper defines the algebraic structure of tracial post-Lie-Rinehart algebras and describes the free object in this category. Post-Lie-Rinehart algebras is a generalisation of pre-Lie-Rinehart algebras, and of post-Lie algebroids.
Paper Structure (10 sections, 25 theorems, 97 equations)

This paper contains 10 sections, 25 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tracial pre-Lie Rinehart algebras
  4. The free tracial pre-Lie-Rinehart algebra
  5. Post-Lie algebras
  6. The free post-Lie algebra
  7. The Guin-Oudom extension
  8. Post-Lie Algebroids
  9. Post-Lie-Rinehart algebras
  10. The free tracial post-Lie-Rinehart algebra

Key Result

Theorem 1.1

MuntheKaasVerdier2015 Let $L$ be the canonical pre-Lie algebra of vector fields on a finite-dimensional Euclidean space. A smooth local mapping $\Phi: L \to L$ can be expanded in an aromatic B-series if and only if $\Phi \circ \xi = \xi \circ \Phi$ for all pre-Lie isomorphisms $\xi: L \to L$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 2.9
  • ...and 33 more