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An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

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Pith's one-line read First-order deformations of post-Lie and post-Hopf structures still form an adjunction that becomes a Cartier–Milnor–Moore equivalence when cocommutative and connected.

desk verdict Solid extension of the post-Lie/post-Hopf adjunction and CMM theorem to first-order deformations of the post-product only, plus a clean Koszulity proof for the new operad IPL; the undeformed-bracket choice is explicit and limits scope but does not break the math. read the letter →

arxiv 2607.28009 v1 pith:GY5JNW3H submitted 2026-07-30 math.QA math.CTmath.RAmath.RT

Andrea Rivezzi , Andrea Sciandra , Thomas Weber This is my paper · ORCID
classification math.QAmath.CTmath.RAmath.RT MSC 17B0516T0516S3018M70
keywords infinitesimalpost-Liealgebraspost-HopfCartier–Milnor–MooretheoremuniversalenvelopingalgebraKoszuloperadsfiltereddistributivelawsubadjacentHopfHochschildcocycle
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Post-Lie algebras and post-Hopf algebras are linked by the universal enveloping algebra and primitive-elements functors; this paper asks whether that link survives a controlled first-order deformation of the post-product alone. It defines infinitesimal post-Lie and post-Hopf algebras by adding a bilinear correction that squares to zero, proves the same functors still form an adjunction, and shows that over characteristic zero the adjunction becomes an equivalence on connected cocommutative objects—an extension of the classical Cartier–Milnor–Moore theorem. Concrete classifications are given for sl(2) and for Sweedler’s four-dimensional Hopf algebra, a geometric source is identified in flat connections with covariantly constant torsion, and cocommutative examples are shown to produce Hochschild 2-cocycles on the subadjacent Hopf algebra. Independently, the quadratic operad governing the new structures is proved Koszul via a filtered distributive law with the Lie and bi-magma operads. A sympathetic reader cares because the result supplies a deformation-ready dictionary between geometric or algebraic post-structures and their enveloping Hopf algebras, ready for quantization or numerical-integration applications.

What carries the argument

The infinitesimal post-product e▷ = ▷ + ℏ▶ (respectively e⋄ = ⋄ + ℏ˛) with ℏ² = 0, whose axioms are exactly the conditions that make the deformed object a post-Lie (post-Hopf) algebra over k[ℏ]/(ℏ²); the resulting functors U and P still give an adjunction, and a filtered distributive law between the Lie and bi-magma operads proves the operad IPL is Koszul.

What would settle it

Exhibit a connected cocommutative infinitesimal post-Hopf algebra over a field of characteristic zero whose space of primitives fails to recover it via the universal enveloping algebra, or show that one of the arity-4 critical pairs used in the filtered distributive law is not confluent.

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Extended reading notes

Core claim

The universal enveloping algebra and primitive-elements functors remain adjoint between the categories of infinitesimal post-Lie algebras and infinitesimal post-Hopf algebras; when the base field has characteristic zero this adjunction restricts to an equivalence between infinitesimal post-Lie algebras and connected cocommutative infinitesimal post-Hopf algebras, extending Cartier–Milnor–Moore. Separately, the quadratic operad of infinitesimal post-Lie algebras is Koszul and isomorphic as an S-module to the composition of the Lie operad with the bi-magma operad.

Load-bearing premise

Only the post-product is deformed to first order; the underlying Lie bracket and Hopf algebra structure are left completely undeformed.

Editorial extensions

If this is right

  • Every infinitesimal post-Lie algebra has a well-defined infinitesimal post-Hopf enveloping algebra whose primitives recover the original structure.
  • Connected cocommutative infinitesimal post-Hopf algebras are completely classified by their primitive infinitesimal post-Lie algebras (char 0).
  • Cocommutative infinitesimal post-Hopf algebras automatically equip their subadjacent Hopf algebras with a Hochschild 2-cocycle.
  • The Koszul property of IPL supplies an André–Quillen cohomology controlling further deformations of infinitesimal post-Lie algebras.
  • Explicit multi-parameter families of infinitesimal post-Lie structures exist on sl(2) and a one-parameter family on Sweedler’s Hopf algebra.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the undeformed-bracket hypothesis can later be relaxed, the same enveloping-algebra construction may yield a deformation quantization path for post-Lie bialgebras.
  • The geometric examples arising from flat connections with covariantly constant torsion suggest that infinitesimal post-Lie structures could organize first-order corrections in geometric numerical integration.
  • The induced Hochschild 2-cocycle on the subadjacent Hopf algebra is a natural candidate for an infinitesimal R-matrix or braiding deformation.
  • Koszulity of IPL opens a direct route to computing obstruction classes for lifting infinitesimal deformations to higher order.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper introduces infinitesimal post-Lie algebras and infinitesimal post-Hopf algebras by deforming only the post-product (▷ respectively ⋄) to first order in ℏ with ℏ²=0, while keeping the underlying Lie bracket and Hopf structure fixed. It proves that the universal enveloping algebra and primitive-elements functors extend to an adjunction between these categories (Theorem 26), and that over a field of characteristic 0 the adjunction restricts to an equivalence between infinitesimal post-Lie algebras and connected cocommutative infinitesimal post-Hopf algebras, extending the Cartier–Milnor–Moore theorem. Concrete classifications are given for infinitesimal post-Lie structures on sl(2) (Theorem 15) and for infinitesimal post-Hopf structures on Sweedler’s Hopf algebra; a geometric source via flat connections with covariantly constant torsion is described (Proposition 16); cocommutative infinitesimal post-Hopf algebras are shown to induce a Hochschild 2-cocycle on the subadjacent Hopf algebra (Theorem 22); and the quadratic operad IPL is proved Koszul via a filtered distributive law between Lie and bi-magma operads, with IPL ≅ L ∘ M₂ as S-modules (Theorem 34).

Significance. The work cleanly extends the post-Lie/post-Hopf correspondence and the Cartier–Milnor–Moore theorem to a first-order deformation setting that is standard in deformation quantization and related contexts. The adjunction/equivalence (Theorem 26) and the Koszulity of IPL (Theorem 34) are the central structural results; both rest on checkable constructions once the deformation axioms are granted. The sl(2) and Sweedler classifications, the Hochschild-cocycle observation, and the geometric interpretation supply concrete content beyond pure formalism. The modelling choice to leave the Lie/Hopf structure undeformed is stated explicitly (Remark 12) and is internally consistent. Altogether this is a solid, self-contained contribution to the algebraic theory of post structures and their operads.

minor comments (6)
  1. [Remark 12 / Introduction] Remark 12 correctly situates Definition 11 as the special case of Lazarev–Sheng–Tang deformations with undeformed bracket. A one-sentence forward pointer in the introduction (or at the start of §2.1) that all later theorems, classifications, and the operad are relative to this restricted ansatz would help readers who come from geometric or quantization applications where simultaneous bracket deformations may be natural.
  2. [Theorem 15 / §2.1.2] Theorem 15 claims a classification “up to isomorphism” of all infinitesimal post-Lie structures on sl(2). The body enumerates compatible ▶ for each already-classified post-Lie structure ▷ (including the trivial cases). A brief clarification that isomorphism is understood in the category IPLie (i.e., of pairs (▷,▶)), and that the families are written relative to the fixed normal forms of Burde–Dekimpe–Vercammen, would remove any ambiguity.
  3. [Theorem 34 / §3.2] In the critical-pair expansions of Theorem 34 (especially [[x,y],z]▶w), the algebraic rewriting via Dx, Ex and the split into PA/SA vs PB/SB is helpful, but a short roadmap sentence before each path (which relations are applied in which order) would make the confluence check easier to audit line-by-line.
  4. [Definition 17] The symbol ˛ for the infinitesimal post-Hopf product is typographically unusual and easy to miss in running text. Consider a more standard alternative (e.g. ▹ or •_ℏ) or a brief notational remark at first use in Definition 17.
  5. [Proposition 16 / §2.1.3] Proposition 16 gives analytic conditions (2.11)–(2.12) under which ▶ yields an infinitesimal post-Lie structure on vector fields. A short remark on whether these conditions admit a clean geometric reading (e.g. in terms of a first-order deformation of the connection that preserves flatness and covariant constancy of torsion) would strengthen the geometric section.
  6. [Bibliography / §2.1.2, §3.2] Minor typos and typesetting: “homomology” appears in a reference title context in the bibliography style; several long displayed formulae in §2.1.2 and §3.2 would benefit from consistent alignment or line breaks for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: definitions are free axiomatic choices; adjunction, CMM extension, and Koszulity are proved from those axioms by standard constructions.

full rationale

The paper introduces infinitesimal post-Lie and post-Hopf algebras by an explicit first-order deformation ansatz that keeps the underlying Lie bracket (resp. Hopf structure) undeformed (Definitions 11, 17; Remark 12). All subsequent results are derived from these axioms: the functors U and P are constructed via the universal property of U(g) and restriction to primitives (Propositions 24–25), the adjunction and characteristic-0 equivalence for connected cocommutative objects are verified directly (Theorem 26), and Koszulity of IPL follows from an explicit filtered distributive law between the known Koszul operads L and M2 with confluence of critical pairs checked by hand (Theorem 34). Classifications on sl(2) and Sweedler’s H4 enumerate solutions of polynomial coefficient equations rather than fitting external data. Self-citations (e.g. to prior post-Lie/post-Hopf definitions or to the authors’ related work on infinitesimal braidings) supply motivation or background and are not load-bearing for the central claims. The derivation chain is self-contained once the definitions are granted; nothing reduces to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper works in standard characteristic-zero algebra over a field k. The only structural choices that are not classical are the decision to deform solely the post-product (leaving bracket and Hopf operations fixed) and the specific quadratic relations that define IPL. No numerical parameters are fitted to data; the multi-parameter families on sl(2) and H₄ are solution spaces of algebraic equations, not free fits.

assumptions (5)
  • standard math Base field k; for the CMM equivalence, char(k)=0 and Hopf algebras are connected and cocommutative.
    Invoked in Theorem 26 and Remark 10; classical hypothesis of Cartier–Milnor–Moore.
  • domain assumption Post-Lie and post-Hopf axioms of Vallette and of Li–Sheng–Tang (Definitions 1 and 5).
    Background structures being deformed; cited from [44] and [30].
  • ad hoc to paper Only the post-product is deformed to first order; the Lie bracket (resp. Hopf structure) remains undeformed (Definitions 11, 17; Remark 12).
    Special case of the more general infinitesimal deformations of Lazarev–Sheng–Tang [28] obtained by setting the bracket deformation to zero; load-bearing modelling choice.
  • standard math Filtered distributive law criterion of Dotsenko–Griffin (Theorem 32 / [12, Thm 5.2]) implies Koszulity once arity-4 critical pairs are confluent.
    Used as black-box engine for Theorem 34.
  • standard math For simple Lie algebras, all derivations are inner, so post and infinitesimal post structures are given by maps φ,ψ : g→g (Remarks 3, 13).
    Used throughout the sl(2) classification.
invented entities (3)
  • Infinitesimal post-Lie algebra (g,[·,·],▷,▶)
    purpose: Encode first-order deformations of the post-product while keeping the Lie bracket fixed.
    New axiomatization (Definition 11); independent evidence is internal (operad Koszulity, classifications, geometric examples) rather than external experimental.
  • Infinitesimal post-Hopf algebra (H,⋄,˛)
    purpose: Hopf-algebraic counterpart of infinitesimal post-Lie algebras; target of the extended U functor.
    New axiomatization (Definition 17); linked to Hochschild 2-cocycles on the subadjacent Hopf algebra (Theorem 22).
  • Quadratic operad IPL
    purpose: Operadic encoding of infinitesimal post-Lie axioms; vehicle for Koszulity and future deformation cohomology.
    Defined in §3.2 as F(V)/(R) with generators [·,·],▷,▶; proved Koszul via L∘M₂.

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Pith. "Pith review of An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence." pith.science (2026). https://pith.science/paper/GY5JNW3H

@misc{pith2026260728009,
  author       = {Pith},
  title        = {Pith review of: An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GY5JNW3H}},
  note         = {Machine review of arXiv:2607.28009}
}
abstract

We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the adjunction given by the universal enveloping algebra and primitive elements functors is compatible with the infinitesimal structure. When restricted to connected and cocommutative infinitesimal post-Hopf algebras, this becomes an equivalence of categories, which constitutes an extension of the Cartier--Milnor--Moore theorem. We classify infinitesimal post-Lie structures on $\mathfrak{sl}(2)$, and discuss a class of infinitesimal post-Lie algebras emerging from flat connections with covariantly-constant torsion. Moreover, we classify infinitesimal post-Hopf structures on Sweedler's Hopf algebra. Cocommutative infinitesimal post-Hopf algebras induce a Hochschild 2-cocycle on the associated subadjacent Hopf algebra. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, by using a filtered distributive law between the operads of Lie algebras and bi-magmas.

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