Tracial Post-Lie-Rinehart Algebras

• Tracial Post-Lie-Rinehart algebras are algebraic systems that extend Lie-Rinehart frameworks by incorporating post-Lie, pre-Lie, and trace structures to model connections and volume-preserving operations.
• They employ combinatorial constructions such as planar rooted and aromatic trees to generate free objects and establish universal properties for morphisms across algebras.
• Their formulation underpins Lie–Butcher series, enabling applications in numerical integration and invariant discretization in computational geometry.

A tracial post-Lie-Rinehart algebra is a unification and extension of several classical algebraic structures, integrating the Lie-Rinehart framework with post-Lie, pre-Lie, and trace notions. These algebras formalize connections, brackets, and volume-preserving structures on modules over commutative algebras, and provide the natural setting for universal expressions—such as Lie–Butcher series—parametrized by combinatorics of planar aromatic trees. The free tracial post-Lie-Rinehart algebra is fully characterized in terms of such trees, establishing a universal property for morphisms to other tracial post-Lie-Rinehart algebras (Rahm, 26 Jan 2026).

1. Definition and Basic Structure

Given a unital commutative $\Bbbk$-algebra $A$, a post-Lie-Rinehart algebra over $A$ is a pair $(L,A)$ where $L$ is an $A$-module equipped with:

• an $A$-bilinear Lie bracket $[\,\cdot\,,\cdot\,]: L\otimes_A L\to L$;
• an $A$-linear anchor $\rho: L\to\Der_{\Bbbk}(A)$;
• an $A$-bilinear post-Lie product $\rhd: L\otimes_A L\to L$, denoted $X\rhd Y = \nabla_X Y$.

These operations satisfy the following for all $X,Y,Z\in L$ and $f \in A$:

1. $(L,[\,,\,])$ is a $\Bbbk$-Lie algebra.
2. The anchor is a Lie algebra morphism:

$\rho\bigl([X,Y]\bigr) = [\rho(X),\rho(Y)]_{\Der(A)}.$

1. Compatibility with the $A$-module structure (Leibniz rule):

$[X, fY] = (\rho(X)f)\,Y + f\,[X,Y].$

2. Flatness (zero curvature):

$$\mathcal R(X,Y)Z := \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z = 0.$$

3. Constant torsion:

$\mathcal T(X,Y) := \nabla_X Y - \nabla_Y X - [X,Y]$

is $A$-bilinear and $\nabla$-constant.

These conditions equivalently assemble into two “post-Lie identities”:

\begin{align} &X\rhd [Y,Z] = [X\rhd Y,\, Z] + [Y, X\rhd Z], \tag{PL1} \ &[X,Y]\rhd Z = X\rhd (Y\rhd Z) - (X\rhd Y)\rhd Z - Y\rhd (X\rhd Z) + (Y\rhd X)\rhd Z. \tag{PL2} \end{align}

2. Trace and Tracial Structure

Let $\End_A(L)$ be the algebra of $A$-linear endomorphisms of $L$. The connection $\nabla$ defines an extension on $\End_A(L)$ by:

$$(X\rhd\Phi)(Y) = \nabla_X\bigl(\Phi(Y)\bigr) - \Phi(\nabla_XY)$$

for $X\in L$, $\Phi\in\End_A(L)$, $Y\in L$. Define the elementary endomorphism $\delta: L\to \End_A(L)$ as:

$(\delta X)(Y) = Y\rhd X.$

Let $E\ell_A(L)$ be the subalgebra of $\End_A(L)$ generated by all iterated compositions of $\nabla_X$ and $\delta_X$. A trace on $(L,A)$ is an $A$-linear map

$\tr: E\ell_A(L) \longrightarrow A$

such that for all $\Phi,\Psi\in E\ell_A(L)$ and all $X\in L$:

$\tr(\Phi \circ \Psi) = \tr(\Psi \circ \Phi), \qquad \tr\bigl(\nabla_X\Phi\bigr) = \rho(X)\bigl(\tr(\Phi)\bigr).$

A post-Lie-Rinehart algebra equipped with such a trace is called tracial. The divergence operator is $\Div := \tr \circ \delta: L \to A$.

3. Free Tracial Post-Lie-Rinehart Algebras and Planar Aromatic Trees

Let $\mathcal{C}$ be a set of generators. The construction of the free tracial post-Lie-Rinehart algebra involves the following combinatorial objects:

• Planar rooted trees $PT$: Each vertex is labeled by an element of $\mathcal{C}$; $\Bbbk \langle PT \rangle$ carries the free post-Lie algebra structure.
• Planar aromas $PA$: Connected directed graphs with each vertex having exactly one outgoing edge and a planar embedding; $S(\Bbbk\langle PA\rangle)$ is their symmetric algebra.
• Planar aromatic trees: Elements $\alpha\,t$ with $\alpha\in S(PA)$ a (possibly empty) product of aromas, and $t \in PT$ a planar rooted tree. The free $S(PA)$-module is:

$$\mathcal{APT} = S\bigl(\Bbbk\langle PA\rangle\bigr) \otimes_{S(\Bbbk\langle PA\rangle)} \Bbbk\langle PT\rangle.$$

The operations essential for the post-Lie-Rinehart structure are:

• Lie–Rinehart bracket:

$$[\alpha_1 t_1, \alpha_2 t_2] = \alpha_1\alpha_2 [t_1,t_2]_{\mathrm{FreeLie}(PT)}$$

where $[\ ,\ ]_{\mathrm{FreeLie}(PT)}$ is the usual free Lie bracket.

• Anchor:

$$\rho(t_1)(A) = \sum_{\text{vertices }v\in A} (\text{add a leftmost edge at }v\text{ to root of }t_1)$$

extending to general elements via (PL2).

• Post-Lie connection:

$(\alpha_1 t_1)\rhd(\alpha_2 t_2) = \alpha_1\,\rho(t_1)(\alpha_2)\, t_2 + \alpha_1\alpha_2 (t_1 \graft t_2)$

with $t_1\graft t_2$ classical left-grafting.

One verifies all post-Lie-Rinehart axioms hold for $\mathcal{APT}$.

4. Structure of Endomorphisms and Trace Map

The algebra $E\ell_{S(PA)}(\mathcal{APT})$ is isomorphic to linear combinations of marked planar aromatic trees: each marked tree consists of a planar aromatic tree with a distinguished vertex and choice of an insertion slot. The composition corresponds to grafting operations.

On this algebra, the projection

$$\tau : \{\text{marked aromatic trees}\} \longrightarrow S(PA)$$

forgets the mark, converting it into an aroma in the central cycle, and:

• $\tau$ vanishes on commutators of marked trees,
• $\tau(\Phi\circ\Psi) = \tau(\Psi\circ\Phi)$,
• $$\tau\bigl(X\rhd\Phi\bigr) = \rho(X)\bigl(\tau(\Phi)\bigr)$$.

The induced trace $\tr:E\ell(\mathcal{APT})\to S(PA)$ makes $\mathcal{APT}$ a tracial post-Lie-Rinehart algebra.

5. Universal Property

The free tracial post-Lie-Rinehart algebra $(\mathcal{APT}_{\mathcal{C}}, S(PA_{\mathcal{C}}), \rhd, [\,,\,], \tr)$ satisfies the following universal property: for any tracial post-Lie-Rinehart algebra $(L,A,\rhd,[\,,\,],\rho,\tr)$ and any map $\iota:\mathcal{C}\to L$, there exist unique morphisms making the following diagram commute and preserving all algebraic structures:

$$\zeta: \mathcal{APT}_{\mathcal{C}} \to L, \qquad \gamma: S(PA_{\mathcal{C}}) \to A, \qquad \beta: E\ell_{S(PA_{\mathcal{C}})}(\mathcal{APT}_{\mathcal{C}}) \to E\ell_A(L)$$

with $\zeta$ $A$-linear over $\gamma$, $\zeta$ preserving brackets and post-Lie product, $\gamma$ an algebra map, $\beta$ compatible with composition, and $\gamma\circ\tr = \tr\circ\beta$. The proof proceeds by extending these maps from the free post-Lie part, using the marking and grafting structures, and ensuring compatibility via the trace quotient, in a unique manner (Rahm, 26 Jan 2026).

6. Corollaries and Examples

• Reduction to Pre-Lie–Rinehart: If the Lie bracket on $L$ vanishes, the structure reduces to a pre-Lie–Rinehart algebra, and all aromatic components vanish, yielding the non-aromatic rooted tree context underlying standard B-series and aromatic B-series.
• Classical Post-Lie Algebroids: Setting $A=C^\infty(M)$ for a smooth manifold $M$, and $L=\Gamma(E)$, sections of a vector bundle $E\to M$ with a post-Lie algebroid structure, recovers the Lie–Butcher series framework, with the universal enveloping algebra structure reflecting flatness and constant torsion.
• Divergence-Free and Volume-Preserving Methods: The trace $\tr$ encodes volume-preservation in the context of numerical integrators. Pullback of the universal series via $\zeta$ and the enveloping algebra construction recovers all Lie–Butcher and aromatic B-series, encompassing structure-preserving discrete flows.

7. Significance and Connections

Tracial post-Lie-Rinehart algebras synthesize concepts from Lie theory, connection theory, and combinatorics of trees/aromas, with repercussions in numerical geometry (Lie–Butcher theory), universal algebra, and invariant discretization. The explicit description of the free object via planar aromatic trees provides a universal representation, through which all volume-preserving and connection-preserving universal series must factor. This framework generalizes classical results for pre-Lie and post-Lie algebroids, and connects diagrammatically motivated objects (aromatic trees) with algebraic structure, supporting applications in geometry and computational mathematics (Rahm, 26 Jan 2026).