Post-Hopf Algebroids

• Post-Hopf algebroids are generalized algebraic structures that extend pre-Hopf algebroids by enforcing post-Lie associativity and convolution-inverse properties.
• They merge features from Hopf algebroids and post-Lie systems, enabling the modeling of flows, parallel transport, and curvature in geometric numerical integration.
• Applications include action and universal enveloping constructions, free object generation, and advanced backward error analysis on manifolds.

A post-Hopf algebroid is a generalized algebraic structure that combines features from Hopf algebroids and post-Lie algebraic systems, developed to provide a universal algebraic framework for encoding the algebraic structures underlying Lie–Butcher and aromatic S-series in geometric numerical integration on manifolds. Formally, it extends the notion of a pre-Hopf algebroid by enforcing post-Lie associativity and a convolution-inverse condition, yielding new algebraic symmetries essential for understanding flows, parallel transport, and curvature in the algebraic analysis of numerical integrators on manifolds (Laurent et al., 26 Dec 2025).

1. Classical and Pre-Hopf Algebroids

Consider a commutative unital algebra $R$ over a field of characteristic zero. An $R$-bialgebroid $H$ in the sense of Lu consists of an $R$-ring, two $R$-algebra maps (source and target) $\iota\colon R\to H$, an $R$-bilinear coassociative coproduct $\Delta\colon H\to H\otimes_R H$ (restricted to the Takeuchi submodule), and an $R$-linear counit $\varepsilon\colon H\to R$ satisfying compatibilities that extend the classical Hopf algebra axioms. If, in addition, there is an antipode $S\colon H\to H$ satisfying the antipode axioms, this structure is called a Hopf algebroid.

A pre-Hopf algebroid is a bialgebroid $(H, \iota, \Delta, \varepsilon)$ equipped with an $R$-linear pre-operation

$\rhd\colon H\otimes_R H\longrightarrow H$

subject to:

1. $$\Delta(h\rhd k) = h_{(1)}\rhd k_{(1)} \otimes h_{(2)}\rhd k_{(2)}$$
2. $h\rhd (kk') = (h_{(1)}\rhd k)(h_{(2)}\rhd k')$
3. $h\rhd\iota(r) = \iota(r)\varepsilon(h)$

These axioms arise in the combinatorial study of exotic aromatic S-series, encoding grafting operations on trees relevant to the structure of formal series solutions in geometric integration.

2. Formalism of Post-Hopf Algebroids

A post-Hopf algebroid generalizes pre-Hopf algebroids and incorporates the essential features of post-Lie algebras at the bialgebroid level. Concretely, starting from a cocommutative Hopf $R$-algebra $(H, m, 1, \iota, \Delta, \varepsilon, S)$, a post-Hopf algebroid is defined by an $R$-linear product: $\rhd\colon H\otimes_R H\to H$ satisfying:

• Coproduct and unit compatibility:

$$\Delta(x\rhd y) = (x_{(1)}\rhd y_{(1)})\otimes (x_{(2)}\rhd y_{(2)})$$, $x\rhd 1_H = \iota(\varepsilon(x))$, $1_H\rhd x = x$.

• $R$-linearity:

$(r x)\rhd y = r(x\rhd y)$.

• Leibniz-type rule:

$x\rhd (y z) = (x_{(1)}\rhd y)(x_{(2)}\rhd z)$.

• Post-Lie associator law:

$x\rhd (y\rhd z) = (x_{(1)}(x_{(2)}\rhd y))\rhd z$.

A structure $(H, \iota, \Delta, \varepsilon, S, \rhd)$ satisfying these is called a weak post-Hopf algebroid. If, further, the Grossman–Larson product

$x *_\rhd y := x_{(1)}(x_{(2)} \rhd y)$

admits an antipode-like anti-automorphism $\theta$ satisfying

$$\Delta(\theta(x)) = \theta(x_{(1)}) \otimes \theta(x_{(2)}), \quad x_{(1)} *_\rhd \theta(x_{(2)}) = \theta(x_{(1)}) *_\rhd x_{(2)} = \iota(\varepsilon(x))$$

then $(H, *_\rhd, \Delta, \varepsilon, \iota, \theta)$ forms a (Grossman–Larson) Hopf algebroid (Laurent et al., 26 Dec 2025).

Pre-Hopf algebroids lack the post-Lie associator and convolution-inverse; post-Hopf algebroids enforce both, providing additional structure necessary for encoding the geometry of flows and connections.

3. Examples: Action and Universal-Enveloping Post-Hopf Algebroids

Action Post-Hopf Algebroids

Given a cocommutative post-Hopf algebra $(A, m_A, 1_A, \Delta_A, \varepsilon_A, S_A, \rhd)$ and an algebra $R$ on which the associated Hopf algebra $$A_\rhd = (A, *_\rhd, 1_A, \Delta_A, \varepsilon_A, S_\rhd)$$ acts as a module-algebra, the smash-product $H=R\otimes A$ is a post-Hopf algebroid over $R$ with:

• Antipode: $S_H(f\otimes a)=f\otimes S_A(a)$.
• Post-operation: $$(f\otimes a)\;\overline\rhd\;(g\otimes b) = f\,(a_{(1)}\rightharpoonup g)\otimes (a_{(2)}\rhd b)$$ The Grossman–Larson Hopf algebroid coincides with the action Hopf algebroid $R\# A_\rhd$ (Laurent et al., 26 Dec 2025).

Universal Enveloping Case

For a post-Lie–Rinehart algebra $$(R, L, [-,-]_L, \rho: L_\rhd\to \operatorname{Der}(R), \rhd)$$, the universal enveloping algebra $U_R(L)$—the quotient of the tensor $R$-algebra by the standard relations—is equipped with a unique $R$-bilinear extension of the post-operation $\rhd$, recursively defined. The coproduct, counit, and antipode are given on generators by: $$\Delta(f) = f\otimes 1, \quad \Delta(x) = x\otimes 1 + 1\otimes x, \quad \varepsilon(f) = f, \; \varepsilon(x) = 0, \quad S(f) = f, \; S(x) = -x$$ This makes $U_R(L)$ a (weak) post-Hopf algebroid (Laurent et al., 26 Dec 2025).

4. Free Objects and Low-Dimensional Examples

Let $(V,\star)$ be a magma algebra with a linear map $f_V: V\to \Der(R)$. The free post-Lie algebra $\operatorname{PostLie}(V)$ is constructed, and $f_V$ extends to a Lie homomorphism $\rho_V$. The induced object

$$(R, R\otimes\operatorname{Lie}(V), [-,-], \overline{\rho}_V, \overline{\rhd}_V) = \operatorname{PostLR}(V)$$

is free in the category of post-Lie–Rinehart algebras with generators $(V, \star, f_V)$ (Laurent et al., 26 Dec 2025).

Special cases include:

• $R=k$ (a field): recovers the cocommutative post-Hopf algebra $(H, \rhd)$.
• $R\otimes A$ for a post-group algebra $A=k[G]$.
• $R = C^\infty(M)$, $L = \Gamma(A)$ with a Weitzenböck connection yielding a post-Lie–Rinehart algebra whose enveloping algebra has post-Hopf algebroid structure (Laurent et al., 26 Dec 2025).

5. Applications to Geometric Numerical Integration

The algebraic formalism of post-Hopf algebroids encodes fundamental aspects of geometric integration methods on manifolds:

• The Grossman–Larson product $*_\rhd$ models composition of flows, relating to Lie group and Butcher series methods.
• The post-operation $\rhd$ captures parallel transport and connection, governing the grafting of trees or forests and encoding curvature/torsion.
• Aromatic S-series, formal series indexed by non-planar trees, naturally admit pre- or post-Hopf algebroid structures, underpinning advanced analysis for volume-preserving and high-order stochastic integrators (Laurent et al., 26 Dec 2025).

This establishes post-Hopf algebroids as the algebraic foundation for high-order methods and backward error analysis in stochastic geometry.

6. Connections and Further Structures

Post-Hopf algebroids generalize classical Hopf and pre-Hopf algebroids and admit analogous categorical structures. BiGalois and Ehresmann-type Hopf algebroids extend Galois and monoidal equivalence theories to these settings, with quantum and twisted jet algebroids providing noncommutative generalizations (Han et al., 20 Oct 2025, Han et al., 3 Jul 2025). Post-Hopf algebroids, through their universal, action, and free-object constructions, provide new symmetry mechanisms essential for encoding the "post-classical" behaviors seen in the algebraic theory of geometric integration.