Post-Lie-Rinehart Algebra Overview

• Post-Lie-Rinehart algebras are algebraic structures defined on a commutative algebra with a Lie bracket, a post-Lie product, and an anchor map, extending classic Lie-Rinehart theory.
• They introduce fundamental compatibility identities and a universal enveloping post-Hopf algebroid that underpins advanced geometric numerical integration methods.
• The framework offers new insights into high-order geometric integrators by unifying Lie–Butcher and aromatic S-series through innovative algebraic operations.

A post-Lie-Rinehart algebra is a prominent generalization of both post-Lie algebras and Lie-Rinehart algebras, formalizing a rich algebraic structure that simultaneously combines a Lie bracket, a post-Lie product, and an anchor map linking derivations of a commutative algebra. Post-Lie-Rinehart algebras and their universal enveloping algebras, termed post-Hopf algebroids, yield frameworks for the algebraic and combinatorial treatment of geometric integration, notably encompassing the algebraic underpinning of Lie–Butcher and aromatic S-series in numerical analysis on manifolds (Laurent et al., 26 Dec 2025).

1. Definition

Let $A$ be a commutative unital algebra over an algebraically closed field of characteristic $0$, and let $L$ be an $A$-module. A post-Lie-Rinehart algebra is specified by the quintuple

$(A, L, [\,,\,]_L, \rho, \circ)$

where:

• $[\,,\,]_L\colon L\otimes L\to L$ is an $A$-linear Lie bracket, making $L$ a Lie $A$-algebra.
• $\circ\colon L\otimes L\to L$ is an $A$-linear post-Lie product.
• $\rho\colon (L, [\,,\,]_\circ)\to \Der(A)$ is an $A$-linear anchor map (Lie action) with

$[x,y]_\circ := x\circ y - y\circ x + [x,y]_L\ .$

The structure is required to satisfy the following compatibility axioms for all $f \in A$, $x, y, z \in L$:

• (i) $A$-linearity in the first argument:

$(f x)\circ y = f(x\circ y)\ .$

• (ii) Leibniz-type rule in the second argument:

$x\circ (f y) = f(x\circ y) + \rho(x)(f)\,y\ .$

• (iii) Post-Lie and compatibility relations:

$$x\circ [y, z]_L = [x\circ y,\, z]_L + [y,\, x\circ z]_L\ ,$$

$$([x, y]_L + x\circ y - y\circ x)\circ z = x\circ (y\circ z) - y\circ (x\circ z)\ .$$

This system generalizes the classical Lie-Rinehart algebra, which is recovered when $\circ = 0$ (Laurent et al., 26 Dec 2025).

2. Fundamental Identities and Lie-Rinehart Generalization

In the classical Lie-Rinehart setting, the only product is $[\,,\,]_L$ together with the anchor $\rho$, and they must satisfy

$[x, f y]_L = f[x, y]_L + \rho(x)(f)\,y\ .$

The addition of the post-Lie product $\circ$ gives rise to new fundamental identities:

• "Left derivation" of $\circ$ over $[\,,\,]_L$:

$$x\circ [y, z]_L = [x\circ y, z]_L + [y, x\circ z]_L\ .$$

• The "right pre-Lie" law for the sub-adjacent bracket:

$$[x, y]_\circ = x\circ y - y\circ x + [x, y]_L\ ,\qquad [x, y]_\circ\circ z = x\circ (y\circ z) - y\circ (x\circ z)\ .$$

These additional structures provide the algebraic mechanism for encoding geometric properties and higher order interactions beyond those captured by classical Lie-Rinehart algebras (Laurent et al., 26 Dec 2025).

3. Universal Enveloping Algebra and Post-Hopf Algebroids

Given a post-Lie-Rinehart algebra $(A, L, [\,,\,]_L, \rho, \circ)$, the universal enveloping algebra $\mathcal U_A(L)$ is defined as the quotient of the tensor $A$-algebra on $L$ by the relations

$$xy - yx = [x, y]_L\ ,\qquad xf - fx = \rho(x)(f)\ ,\quad x, y \in L,\ f\in A\ .$$

$\mathcal U_A(L)$ acquires the structure of a Hopf algebroid over $A$ with:

• Source and target maps as the natural inclusion $A \hookrightarrow \mathcal U_A(L)$.
• Coproduct $\Delta$ satisfies:

$$\Delta(f) = f \otimes_A 1\ ,\quad \Delta(x) = x\otimes_A 1 + 1 \otimes_A x\ ,\ x\in L\ ,$$

extended multiplicatively.

• Counit $\varepsilon$ and antipode $S$ by:

$$\varepsilon(f) = f\ ,\ \varepsilon(x) = 0\ ,\ S(f) = f\ ,\ S(x) = -x\ ,\ f\in A,\, x\in L\ ,$$

and extension to products as in the data.

The post-Lie product $\circ$ extends uniquely (Oudom–Guin construction) to all of $\mathcal U_A(L)$ by $A$-linearity and: $$f\circ X = f X,\quad x\circ f = \rho(x)(f),\quad x\circ (Y Z) = (x\circ Y)Z + Y(x\circ Z),\quad (X Y)\circ Z = X\circ(Y\circ Z) - (X\circ Y)\circ Z.$$ This structure makes $\mathcal U_A(L)$ into a (weak) post-Hopf algebroid over $A$, and under mild hypotheses a full post-Hopf algebroid. The Grossman–Larson product $X * Y := X_{(1)} (X_{(2)} \circ Y)$ gives a second Hopf algebroid structure with the same source, target, coproduct, and counit; the extended $\circ$ makes $\mathcal U_A(L)$ a module algebra over itself (Laurent et al., 26 Dec 2025).

4. Free Post-Lie-Rinehart Algebra via Magma Algebras

Let $V$ be a vector space equipped with a non-associative binary operation (a magma), and fix a linear map $f_V: V \to \Der(A)$. Form the free post-Lie algebra $\operatorname{PostLie}(V)$ generated by $V$, whose sub-adjacent Lie algebra $(\operatorname{PostLie}(V))_\circ$ admits a unique extension $\rho_V$ of $f_V$ to a Lie algebra map into $\Der(A)$. The free post-Lie-Rinehart algebra is then: $$\operatorname{PostLR}(V) = (A,\, A\otimes \operatorname{PostLie}(V),\, [\,,\,],\, \rho_V,\, \circ)$$ where the products are defined by

$$[f X,\, g Y] = f g [X, Y],\quad (f X)\circ (g Y) = f X(g) Y + f g (X \circ Y)$$

for $X, Y \in \operatorname{PostLie}(V)$, $f, g \in A$. Any magma-morphism $V \to L$ into a post-Lie-Rinehart algebra over $A$ factors uniquely through $\operatorname{PostLR}(V)$, ensuring a universal property analogous to the classical free Lie-Rinehart algebra construction (Laurent et al., 26 Dec 2025).

5. Applications in Geometric Numerical Integration

On a smooth manifold $M$, one can select a global flat frame and define a post-Lie-Rinehart structure on the $A = C^\infty(M)$-module of vector fields as follows: $$\llbracket X, Y \rrbracket = [X, Y]_L = -T(X, Y), \quad X \circ Y = \nabla_X Y$$ where $\nabla$ is a flat connection of constant torsion $T$. The universal enveloping algebra $\mathcal U_A(L)$ then coincides with the algebra of non-commutative differential operators generated by Lie and covariant derivatives. As a post-Hopf algebroid, it controls the combinatorics of Lie–Butcher series and aromatic S-series.

The Taylor expansion of the exact and numerical flow pull-backs can be expressed as

$$\varphi_t^* = \exp^*(t F),\qquad \psi_t^* = \exp^\cdot(t F)$$

where products $*$ and $\cdot$ correspond to the classical and post-Lie compositions, respectively. The post-Hopf algebroid introduces a third product $*\circ$ arising from $\circ$, significant in the field of exotic expansions (aromatic series), where all three products interact. Volume-preserving and divergence-free integrators correspond to special characters (algebra homomorphisms) on the post-Hopf algebroid of aromatic forests, and the universal enveloping post-Hopf algebroid is the natural algebraic framework for backward error and modified vector field analysis in such contexts (Laurent et al., 26 Dec 2025).

6. Structural Interplay and Research Impact

Post-Lie-Rinehart algebras unify and extend important classical algebraic frameworks relevant for the study of differential operators, representation theory, and geometric integrators. Their enveloping post-Hopf algebroids facilitate the algebraic and combinatorial understanding needed for high-order numerical methods on manifolds, providing an explicit connection to aromatic S-series and Lie–Butcher theory.

A significant aspect is the explicit correspondence between the algebraic axioms and analytical properties of numerical schemes, such as volume preservation and the structure of modified vector fields. The approach elucidates the deep relationship between algebraic operations (e.g., the Grossman–Larson product $*$, post-Lie product $\circ$, and the $*\circ$ operation) and the composition of flows and integrators in geometric analysis (Laurent et al., 26 Dec 2025).

7. Table: Core Structures in Post-Lie-Rinehart Theory

Name	Definition/Property	Role
Post-Lie product ($\circ$)	$A$-linear, satisfies left derivation & Leibniz-type compatibilities	Encodes additional geometric data
Sub-adjacent bracket ($[\,,\,]_\circ$)	$x\circ y - y\circ x + [x, y]_L$	Links $\circ$ to Lie theory
Grossman–Larson product ($*$)	$X*Y := X_{(1)}(X_{(2)}\circ Y)$	Second Hopf algebroid structure
Universal enveloping alg ($\mathcal U_A(L)$)	Quotient of tensor $A$-algebra by specified relations	Admits post-Hopf algebroid structure
Anchor ($\rho$)	$A$-linear Lie-action into $\Der(A)$ with compatibility	Connects to derivational symmetries

These core elements characterize the algebraic and analytical power of post-Lie-Rinehart algebras and their enveloping structures in contemporary research.