Inner post-Lie algebras and inner post-groups

V. Gubarev; Y. Li; Y. Sheng; Y. Wang

Inner post-Lie algebras and inner post-groups

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Pith's one-line read An inner post-Lie algebra is induced by a Rota-Baxter operator precisely when its obstruction class is trivial.

desk verdict The paper gives a clean if-and-only-if criterion: an inner post-Lie algebra comes from a Rota-Baxter operator exactly when its obstruction class vanishes in the appropriate cohomology. read the letter →

arxiv 2605.21992 v1 pith:ZYXS6PL6 submitted 2026-05-21 math.RA math.GR

V. Gubarev , Y. Li , Y. Sheng , Y. Wang This is my paper · ORCID

classification math.RAmath.GR

keywords innerpost-LiealgebraRota-Baxteroperatorobstructionclasscohomologyextensiontheorypost-group

checked against Cost.FunctionalEquation

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

The paper defines an obstruction class for an inner post-Lie algebra using extension theory and cohomology. It proves that the algebra arises from a Rota-Baxter operator exactly when this class vanishes. A parallel equivalence is shown for inner post-groups. The result supplies a cohomological test for the existence of the inducing operator and ends with applications of the structures.

What carries the argument

The obstruction class, constructed via extension theory and cohomology, that vanishes exactly when an inner post-Lie algebra or inner post-group is induced by a Rota-Baxter operator.

What would settle it

Exhibit a concrete inner post-Lie algebra whose obstruction class is trivial yet no Rota-Baxter operator induces it, or whose class is nontrivial yet an inducing operator still exists.

Watch

Extended reading notes

Core claim

An inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial. A parallel statement holds for inner post-groups.

Load-bearing premise

The cohomological obstruction class is well-defined on these algebras and its vanishing is equivalent to the existence of an inducing Rota-Baxter operator.

Editorial extensions

If this is right

A Rota-Baxter operator inducing the algebra exists precisely when the obstruction class vanishes.
The identical criterion applies to inner post-groups.
Applications of inner post-Lie algebras and inner post-groups are obtained from this characterization.

Reading between the lines

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

The criterion offers a way to decide whether a given inner post-Lie algebra comes from a Rota-Baxter operator without constructing the operator directly.
The same obstruction technique might apply to other operators or to deformations of these algebras.
Explicit computations of the class for low-dimensional examples could produce new families of post-Lie structures.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central result is that an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if a certain obstruction class is trivial, with the same statement proved for inner post-groups. The argument uses extension theory to define the class and then shows both directions explicitly: trivial class produces an inducing operator, and any inducing operator yields the trivial class. This is the genuinely new piece relative to earlier work on post-Lie structures and Rota-Baxter operators. The constructions stay valid over general vector spaces without finite-dimensionality or characteristic restrictions, which is a plus. The stress-test note confirms the two directions are handled directly and no internal inconsistency appears in the setup. The paper therefore supplies a decision procedure for the induction question inside this slice of the theory. The applications section at the end consists of concrete examples rather than broad new theorems, so its value is mainly illustrative. The main soft spot is that the abstract and summary do not display the explicit cochain complex or coboundary maps, which means a referee would still need to verify that the cocycle conditions line up exactly with the post-Lie axioms and the Rota-Baxter relation. That verification is routine but not automatic. This work is for specialists already comfortable with post-Lie algebras, their cohomology, and Rota-Baxter operators. A reader who needs a practical test for whether a given inner structure arises from an operator will get direct use from the criterion. It is worth sending to peer review because the claim is precise, the method is standard extension theory applied in a new context, and the result is falsifiable once the cohomology is written down.

Referee Report

0 major / 2 minor

Summary. The paper introduces, via extension theory and cohomology, an obstruction class whose vanishing is equivalent to an inner post-Lie algebra being induced by a Rota-Baxter operator; a parallel equivalence is proved for inner post-groups, followed by applications.

Significance. If the derivations hold, the work supplies a cohomological criterion that classifies when inner post-Lie structures arise from Rota-Baxter operators, extending standard extension theory in a manner that remains valid over general vector spaces without finite-dimensionality or characteristic restrictions.

minor comments (2)

[Abstract] The abstract states the main equivalences cleanly but does not indicate the precise cohomology theory or the base ring assumptions; a single sentence clarifying these would improve accessibility.
Notation for the obstruction class and the relevant cohomology groups should be introduced with explicit cross-references to the definitions in the body so that the iff statements can be traced without ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately captures the main contributions regarding the obstruction classes in the cohomology of inner post-Lie algebras and inner post-groups, and their relation to Rota-Baxter operators. As no specific major comments were listed in the report, we have no individual points requiring detailed rebuttal at this time.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; standard obstruction theory applied directly

full rationale

The paper defines an obstruction class in the appropriate cohomology group via extension theory for inner post-Lie algebras (and analogously for post-groups), then proves the if-and-only-if equivalence: the class vanishes precisely when a Rota-Baxter operator inducing the structure exists. Both directions are established by explicit construction (trivial class yields the operator; an inducing operator yields the trivial class). No step reduces a prediction to a fitted input by construction, imports a uniqueness theorem via self-citation, or renames a known result; the setup is self-contained over general vector spaces using standard cohomological methods without additional restrictions that would create circularity.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claims rest on the applicability of extension theory and a cohomological obstruction class whose definition is not expanded in the abstract; no free parameters or invented entities are mentioned.

assumptions (1)

domain assumption Extension theory and cohomology apply to inner post-Lie algebras and yield a well-defined obstruction class whose vanishing detects induction by a Rota-Baxter operator.
Invoked in the abstract as the method used to introduce the obstruction class.

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Cite this review

Pith. "Pith review of Inner post-Lie algebras and inner post-groups." pith.science (2026). https://pith.science/paper/ZYXS6PL6

@misc{pith2026260521992,
  author       = {Pith},
  title        = {Pith review of: Inner post-Lie algebras and inner post-groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYXS6PL6}},
  note         = {Machine review of arXiv:2605.21992}
}

read the original abstract

In this paper, using extension theory and cohomological approach we introduce the notion of the obstruction class for an inner post-Lie algebra being induced by a Rota-Baxter operator, and show that an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial. Similarly, we introduce the notion of the obstruction class for an inner post-group being induced by a Rota-Baxter operator, and prove a parallel result. Finally, we give some applications of inner post-Lie algebras and inner post-groups.

Discussion (0). Continue with ORCID to comment.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial (Theorem 2.12)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

κ(x,y)=[ϕ(x), ϕ(y)] g − ϕ([x,y] ▷) defines the 2-cocycle whose class is the obstruction

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Pith tools

Reviewed May 22, 2026 · model on record in the stance chip above.

[1] [1]

Bai, An introduction to pre-Lie algebras, inAlgebra and Applications1-Nonssociative Algebras and Cate- gories(2020), 245–273

C. Bai, An introduction to pre-Lie algebras, inAlgebra and Applications1-Nonssociative Algebras and Cate- gories(2020), 245–273. 3

work page 2020

[2] [2]

C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras,Comm. Math. Phys.297(2010), 553–596. 1, 2, 3, 4, 7

work page 2010

[3] [3]

C. Bai, L. Guo, Y . Sheng and R. Tang, Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation,Math. Ann.388(2024), 3127–3167. 2, 8, 9, 12, 13

work page 2024

[4] [4]

Bardakov and V

V . Bardakov and V . Gubarev, Rota-Baxter groups, skew left braces, and the Yang-Baxter equation,J. Algebra 596(2022), 328–351. 2, 9

work page 2022

[5] [5]

Bardakov and V

V . Bardakov and V . Gubarev, Rota-Baxter operators on groups,Proc. Indian Acad. Sci. Math. Sci.133(2023), 29 pp. 16

work page 2023

[6] [6]

Bruned and F

Y . Bruned and F. Katsetsiadis, Post-Lie algebras in regularity structures,Forum Math. Sigma11(2023), 1–20. 1

work page 2023

[7] [7]

Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics,Cent

D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics,Cent. Eur. J. Math.4(2006), 323–357. 3

work page 2006

[8] [8]

Burde, K

D. Burde, K. Dekimpe and K. Vercammen, Affine actions on Lie groups and post-Lie algebra structures,Linear Algebra Appl.(5)437(2012), 1250–1263. 1

work page 2012