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arxiv: 2606.19244 · v1 · pith:36TBPA4Cnew · submitted 2026-06-17 · 🧮 math.RA · math.CO

On restricted Rota-Baxter Lie algebras of arbitrary weight

Yunnan Li , Ke Ou This is my paper

Pith reviewed 2026-06-26 18:26 UTC · model grok-4.3

classification 🧮 math.RA math.CO
keywords restricted Rota-Baxter Lie algebrapost-Lie algebrareplication propertygraph subalgebraRota-Baxter operatorprime characteristicp-envelope
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The pith

Restricted Rota-Baxter Lie algebras of arbitrary weight generate restricted post-Lie algebras via splitting and exhibit a replication property.

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

The paper introduces the definition of restricted Rota-Baxter Lie algebras of arbitrary weight, equipped with an intrinsic graph subalgebra characterization. It proves that these algebras yield restricted post-Lie algebras through the splitting property and that they satisfy a replication property. Two constructions are supplied in prime characteristic, one obtained from Rota-Baxter associative algebras of arbitrary weight and the other from Rota-Baxter Lie algebras of weight 1, while the Rota-Baxter p-envelopes of such structures are also studied.

Core claim

Restricted Rota-Baxter Lie algebras of arbitrary weight are introduced with an intrinsic graph subalgebra characterization. Via the splitting property, they give rise to restricted post-Lie algebras and possess a novel replication property. Two natural constructions exist in prime characteristic: one arising from Rota-Baxter associative algebras of arbitrary weight, and the other from Rota-Baxter Lie algebras of weight 1. The Rota-Baxter p-envelopes of a Rota-Baxter Lie algebra are also examined.

What carries the argument

The restricted Rota-Baxter operator of arbitrary weight whose graph is required to be a subalgebra of the direct sum Lie algebra.

If this is right

  • They produce restricted post-Lie algebras through the splitting property.
  • They satisfy a replication property that generates further examples from given ones.
  • One construction arises directly from any Rota-Baxter associative algebra of arbitrary weight in prime characteristic.
  • A second construction arises from any Rota-Baxter Lie algebra of weight 1 in prime characteristic.
  • Their Rota-Baxter p-envelopes can be formed and examined.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The replication property supplies a systematic way to enlarge families of examples once one instance is known.
  • The graph-subalgebra condition may serve as a template for defining restricted versions of other operator algebras.
  • The prime-characteristic constructions suggest that analogous statements could be tested when the base field has characteristic zero.

Load-bearing premise

The proposed definition of a restricted Rota-Baxter Lie algebra of arbitrary weight is consistent with the graph subalgebra characterization and the splitting and replication properties follow directly from it without additional restrictions on the weight or characteristic.

What would settle it

An explicit Rota-Baxter Lie algebra whose graph is a subalgebra yet whose splitting fails to produce a restricted post-Lie algebra, or whose replication map does not preserve the restricted structure.

read the original abstract

Recently, Ehret and Gilliers introduced the notion of a (trivially) restricted post-Lie algebra, recovering the concepts of a restricted Lie algebra and a restricted pre-Lie algebra. In this paper, we specifically introduce restricted Rota-Baxter Lie algebras of arbitrary weight with an intrinsic graph subalgebra characterization. We show that, via the splitting property, they give rise to restricted post-Lie algebras, and furthermore possess a novel replication property. We then present two natural constructions of such restricted Rota-Baxter structures in prime characteristic: one arising from Rota-Baxter associative algebras of arbitrary weight, and the other from Rota-Baxter Lie algebras of weight $1$. The Rota-Baxter $p$-envelopes of a Rota-Baxter Lie algebra are also examined.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper introduces the notion of restricted Rota-Baxter Lie algebras of arbitrary weight, equipped with an intrinsic graph subalgebra characterization. It shows that these structures give rise to restricted post-Lie algebras via the splitting property and possess a novel replication property. Two natural constructions are presented in prime characteristic—one from Rota-Baxter associative algebras of arbitrary weight and one from weight-1 Rota-Baxter Lie algebras—along with an examination of the Rota-Baxter p-envelopes of a Rota-Baxter Lie algebra.

Significance. If the central claims hold, the work provides a definitional extension of Rota-Baxter Lie algebras to the restricted setting of arbitrary weight, with explicit links to restricted post-Lie algebras and a replication property. The two constructions in prime characteristic and the study of p-envelopes supply concrete realizations that may support further developments in characteristic-p algebra. The graph-subalgebra characterization is presented as intrinsic, which strengthens the framework if verified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the contributions, and recommendation to accept the manuscript. The referee's assessment aligns with the scope and results presented.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines restricted Rota-Baxter Lie algebras of arbitrary weight via a new intrinsic graph subalgebra characterization, then derives the splitting property (yielding restricted post-Lie algebras) and replication property directly from those axioms. The two constructions in prime characteristic are obtained by extending prior Rota-Baxter objects, without any reduction of the central claims to fitted parameters, self-citations, or definitional loops. No load-bearing self-citation chains or ansatz smuggling appear; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the consistency of the newly introduced definition of restricted Rota-Baxter Lie algebra and on standard properties of Rota-Baxter operators and restricted Lie algebras in prime characteristic. No free parameters are visible from the abstract. The new algebra itself is an invented entity whose independent evidence would be the explicit constructions and the splitting/replication theorems.

axioms (2)
  • standard math Standard axioms of Lie algebras and Rota-Baxter operators hold in the restricted setting.
    Invoked implicitly when extending the Ehret-Gilliers restricted post-Lie notion.
  • ad hoc to paper The graph subalgebra characterization is intrinsic to the restricted Rota-Baxter structure.
    Stated as part of the definition in the abstract; no external justification supplied.
invented entities (1)
  • Restricted Rota-Baxter Lie algebra of arbitrary weight no independent evidence
    purpose: New algebraic structure that admits graph subalgebra description, splitting to post-Lie, and replication.
    Introduced in the paper; independent evidence would consist of the two constructions and the stated properties.

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discussion (0)

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Works this paper leans on

33 extracted references

  1. [1]

    Aguiar, pre-Poisson algebras,Lett

    M. Aguiar, pre-Poisson algebras,Lett. Math. Phys.54(2000), 263–277. 2

    2000

  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. 2, 10, 11, 14, 20

    2010

  3. [3]

    Y . Bao, Y . Ye and J. J. Zhang, Restricted Poisson algebras,Pacific J. Math.289(2017), 1–34. 2

    2017

  4. [4]

    Bezrukavnikov and D

    R. Bezrukavnikov and D. Kaledin, Fedosov quantization in positive characteristic,J. Amer. Math. Soc.21 (2008), 409–438. 2 24 YUNNAN LI AND KE OU

    2008

  5. [5]

    Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups,Comm

    M. Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups,Comm. Math. Phys.135(1990), 201–216. 2

    1990

  6. [6]

    Bruned and F

    Y . Bruned and F. Katsetsiadis, Post-Lie algebras in regularity structures,Forum Math. Sigma11(2023), Paper No. e98. 2, 10

    2023

  7. [7]

    Burde, Left-symmetric structures on simple modular Lie algebras,J

    D. Burde, Left-symmetric structures on simple modular Lie algebras,J. Algebra169(1994), 112–138. 2

    1994

  8. [8]

    Dokas, Pre-Lie algebras in positive characteristic,J

    I. Dokas, Pre-Lie algebras in positive characteristic,J. Lie Theory23(2013), 937–952. 2, 7, 20

    2013

  9. [9]

    Dokas, Cohomology of restricted Lie-Rinehart algebras and the Brauer group,Adv

    I. Dokas, Cohomology of restricted Lie-Rinehart algebras and the Brauer group,Adv. Math.231(2012), 2573–

    2012

  10. [10]

    Dzhumadil’daev, Jacobson formula for right-symmetric algebras in characteristicp,Comm

    A. Dzhumadil’daev, Jacobson formula for right-symmetric algebras in characteristicp,Comm. Algebra29 (2001), 3759–3771. 2, 7, 20

    2001

  11. [11]

    Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation,Lett

    K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation,Lett. Math. Phys.61(2002), 139–147. 20

    2002

  12. [12]

    Ebrahimi-Fard, L

    K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable renormalization I: the ladder case,J. Math. Phys.45 (2004), 3758–3769. 13

    2004

  13. [13]

    Ebrahimi-Fard, A

    K. Ebrahimi-Fard, A. Lundervold and H. Munthe-Kaas, On the Lie enveloping algebra of a post-Lie algebra, J. Lie Theory25(2015), 1139–1165. 2, 5

    2015

  14. [14]

    Ebrahimi-Fard, I

    K. Ebrahimi-Fard, I. Mencattini and H. Munthe-Kaas, Post-Lie algebras and factorization theorems,J. Geom. Phys.119(2017), 19–33. 6

    2017

  15. [15]

    Ehret and N

    Q. Ehret and N. Gilliers, Jacobson identities for post-Lie algebras in positive characteristic, arXiv:2504.14540. J. Algebra Appl.(2026), online ready. 2, 3, 6, 7, 8, 9

    arXiv 2026

  16. [16]

    Foissy, Extension of the product of a post-Lie algebra and application to the SISO feedback transformation group,Abel Symp.13(2018), 369–399

    L. Foissy, Extension of the product of a post-Lie algebra and application to the SISO feedback transformation group,Abel Symp.13(2018), 369–399. 9

    2018

  17. [17]

    Goncharov, Rota-Baxter operators on cocommutative Hopf algebras,J

    M. Goncharov, Rota-Baxter operators on cocommutative Hopf algebras,J. Algebra582(2021), 39–56. 21

    2021

  18. [18]

    Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Mathematics4, Higher education press, China, 2012

    L. Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Mathematics4, Higher education press, China, 2012. 10, 11, 14

    2012

  19. [19]

    L. Guo, H. Lang and Y . Sheng, Integration and geometrization of Rota-Baxter Lie algebras,Adv. Math.387 (2021), 107834. 9

    2021

  20. [20]

    Jacobson, Abstract derivation and Lie algebras,Trans

    N. Jacobson, Abstract derivation and Lie algebras,Trans. Amer. Math. Soc.42(1937), 206–224. 1

    1937

  21. [21]

    Jacobson, Lie algebras, Republication of the 1962 original, Dover Publications, Inc., New York, 1979

    N. Jacobson, Lie algebras, Republication of the 1962 original, Dover Publications, Inc., New York, 1979. 3, 4, 6

    1962

  22. [22]

    J. D. Jacques and L. Zambotti, Post-Lie algebras of derivations and regularity structures, arXiv:2306.02484; J. Comb. Algebra, online first. 2, 10

    arXiv

  23. [23]

    B. A. Kupershmidt, What a classicalr-matrix really is,J. Nonlinear Math. Phys.6(1999), 448–488. 2

    1999

  24. [24]

    Li, On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra,J

    Y . Li, On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra,J. Algebra 706(2026), 1–25. 22

    2026

  25. [25]

    Y . Li, Y . Sheng and R. Tang, Post-Hopf algebras, relative Rota-Baxter operators and solutions to the Yang- Baxter equation,J. Noncommut. Geom.18(2024), 605–630. 5, 21

    2024

  26. [26]

    Loday and M

    J.-L. Loday and M. Ronco, Trialgebras and families of polytopes,Contemp. Math.346(2004), 369–398. 20

    2004

  27. [27]

    Montgomery, Hopf algebras and their actions on rings, Amer

    S. Montgomery, Hopf algebras and their actions on rings, Amer. Math. Soc., Regional Conf. Ser. in Math.,82,

  28. [28]

    H. Z. Munthe-Kaas and A. Lundervold, On post-Lie algebras, Lie-Butcher series and moving frames,Found. Comput. Math.13(4) (2013), 583–613. 2

    2013

  29. [29]

    Oudom and D

    J.-M. Oudom and D. Guin, On the Lie enveloping algebra of a pre-Lie algebra,J. K-Theory2(2008), 147–167. 2, 5

    2008

  30. [30]

    Rahm, An operadic approach to substitution in Lie-Butcher series,Forum Math

    L. Rahm, An operadic approach to substitution in Lie-Butcher series,Forum Math. Sigma10(2022), Paper No. e20. 2

    2022

  31. [31]

    M. A. Semenov-Tian-Shansky, What is a classicalr-matrix?Funct. Anal. Appl.17(1983), 259–272. 2

    1983

  32. [32]

    Strade and R

    H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Monogr. Textbooks Pure Appl. Math.,116, 1988. 4, 8, 13, 16 ON RESTRICTED ROTA-BAXTER LIE ALGEBRAS OF ARBITRARY WEIGHT 25

    1988

  33. [33]

    Vallette, Homology of generalized partition posets,J

    B. Vallette, Homology of generalized partition posets,J. Pure Appl. Algebra208(2007), 699–725. 2, 5 School ofMathematics andInformationScience, GuangzhouUniversity, Guangzhou510006, China Email address:ynli@gzhu.edu.cn School ofStatistics andMathematics, YunnanUniversity ofFinance andEconomics, Kunming650221, China Email address:keou@ynufe.edu.cn

    2007