Pith. sign in

REVIEW 7 minor 30 references

Post-Poisson algebras, extended $\mathcal{O}$-operators and extended Poisson Yang-Baxter equations

T0 review · 0 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper establishes a one-to-one correspondence between solutions of an extended Poisson Yang-Baxter equation with invariant symmetric part and extended O-operators, unifying the classical O-operator description of skew-symmetric…

desk verdict Workmanlike extension of O-operator/Yang-Baxter machinery to Poisson algebras; the central equivalence is real, the hypotheses are explicit, and the paper deserves refereeing. read the letter →

arxiv 2608.06736 v1 pith:WWBMAEBH submitted 2026-08-07 math.RA math-phmath.MP

Yuanchang Lin , Dilei Lu This is my paper · ORCID
classification math.RAmath-phmath.MP MSC 16T2517A3017A6017B3817B6217B63
keywords Poissonalgebraspost-PoissonextendedO-operatorsYang-BaxterequationbialgebrasRota-Baxteroperators
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

This paper establishes that the Poisson Yang-Baxter equation, which normally only sees skew-symmetric tensors, can be extended so that solutions with a nonzero $(\mathrm{ad},L)$-invariant symmetric part are governed by a new kind of operator, the extended $\mathcal{O}$-operator. The central equivalence says that a tensor $r=\Theta+\Lambda$ solves the extended Poisson Yang-Baxter equation of mass $(\kappa+1)/4$ exactly when its skew-symmetric part $\Lambda_+$ is an extended $\mathcal{O}$-operator of weight $0$ with symmetric extension $\Theta_+$ of mass $(\kappa,0)$. This turns a nonlinear tensor equation into a linear-algebraic operator condition and folds the classical $\mathcal{O}$-operator characterization of skew-symmetric solutions into a wider framework. The paper also shows that ordinary $\mathcal{O}$-operators of weight $\lambda$ build post-Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces under symmetry conditions on the extension map.

What carries the argument

The paper's load-bearing object is the extended $\mathcal{O}$-operator: a pair $(T,S)$ of linear maps $V\to A$, where $S$ is called the extension and is required to be balanced, $A$-invariant, and equivalence-compatible, while $T$ satisfies the deformed $\mathcal{O}$-operator identities with mass parameters $(\kappa,\mu)$. The companion object is the extended Poisson Yang-Baxter equation (EPYBE) of mass $\epsilon$, defined by modifying the classical $C(r)=A(r)=0$ equations with a term $\epsilon[r_{13}+r_{31},r_{23}+r_{32}]$ and $\epsilon(r_{13}+r_{31})\cdot(r_{23}+r_{32})$. The argument is carried by the symmetrizer-antisymmetrizer decomposition: writing $r=\Theta+\Lambda$, the operator $T$ corresponds to the skew-symmetric part and $S$ to the symmetric part, and the identities for an extended $\mathcal{O}$-operator are exactly the tensor identities of the EPYBE after this decomposition. Under the hypotheses that $S$ is balanced and $A$-invariant and that the equivalence condition holds with $\mu=\lambda$, the intermediate Lemma 2.26 and Theorem 2.28 convert the extension terms into cancellations that make $(V,\{\cdot,\cdot\}_T,\circ_T)$ a Poisson algebra.

What would settle it

Take the paper's 3-dimensional example, choose an extension map $S$ that is balanced and $A$-invariant but fails the equivalence condition with $\mu=\lambda$, and compute whether the operations (15)--(16) satisfy the Jacobi identity and Leibniz rule; any counterexample would show the matching condition is necessary. To test Theorem 3.17 directly, search in a quadratic Poisson algebra for an $r=\Theta+\Lambda$ with invariant $\Theta$ that satisfies the EPYBE of mass $(\kappa+1)/4$ while $\Lambda_+$ fails the two extended operator identities (25)--(26).

Watch

Extended reading notes

Core claim

The central discovery is a correspondence at the level of tensors and operators on Poisson algebras. For a Poisson algebra $A$, decompose an element $r\in A\otimes A$ into its symmetric and skew-symmetric parts, $r=\Theta+\Lambda$, and suppose $\Theta$ is $(\mathrm{ad},L)$-invariant. Then $r$ solves the extended Poisson Yang-Baxter equation of mass $(\kappa+1)/4$, namely $C(r)=\frac{\kappa+1}{4}[r_{13}+r_{31},r_{23}+r_{32}]$ and $A(r)=\frac{\kappa+1}{4}(r_{13}+r_{31})\cdot(r_{23}+r_{32})$, if and only if the induced map $\Lambda_+:A^*\to A$ is an extended $\mathcal{O}$-operator of weight $0$ with extension $\Theta_+$ of mass $(\kappa,0)$. This generalizes the classical result that skew-symmetric solutions of the Poisson Yang-Baxter equation correspond to ordinary $\mathcal{O}$-operators; the symmetric part is no longer required to vanish. On the structural side, the paper proves that when the extension map $S$ is balanced, $A$-invariant, and satisfies an equivalence condition with $\mu=\lambda$, any extended $\mathcal{O}$-operator $T$ with extension $S$ makes $(V,\{\cdot,\cdot\}_T,\circ_T)$ into a Poisson algebra.

Load-bearing premise

The load-bearing premise is that the correction map $S$ is highly symmetric: swapping its two inputs must flip the sign in the bracket part and keep the product part unchanged, it must commute with the algebra's actions as a module homomorphism, and an equivalence condition tied to the weight must hold exactly; without these, the paper does not prove that the induced operations form a Poisson algebra or that the Yang-Baxter correspondence holds.

Editorial extensions

If this is right

  • Solutions of the EPYBE with $(\mathrm{ad},L)$-invariant symmetric part are reduced from quadratic tensor equations to one operator condition on $\Lambda_+$.
  • The classical $\mathcal{O}$-operator characterization of skew-symmetric solutions of the PYBE is recovered as the special case $\kappa=-1$; the new theorem is a strict generalization.
  • On quadratic Poisson algebras, the nondegenerate bilinear form converts extended $\mathcal{O}$-operators on the adjoint module into solutions of the EPYBE, and taking $S=0$ recovers the Rota-Baxter description of skew-symmetric PYBE solutions.
  • In semi-direct product Poisson algebras, extended $\mathcal{O}$-operators on a module correspond to EPYBE solutions in the larger algebra, and Rota-Baxter operators of nonzero weight produce pairs of PYBE solutions.
  • Because ordinary $\mathcal{O}$-operators of weight $\lambda$ build post-Poisson algebras, the paper connects the extended operator/EPYBE correspondence to the operadic trisuccessor picture of post-Poisson algebras.

Reading between the lines

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

  • If the $(\mathrm{ad},L)$-invariance assumption on $\Theta$ could be removed, the correspondence would cover solutions whose symmetric part is arbitrary; the paper only treats the invariant case, so finding a counterexample without it would clarify how load-bearing that hypothesis is.
  • The mass parameter $(\kappa+1)/4$ interpolates between the classical PYBE and deformed equations, so varying $\kappa$ suggests a family of Yang-Baxter-type equations whose classical limit is the standard one; quantifying this deformation on concrete examples is a natural next step.
  • Since the extension $S$ is no longer required to vanish, the theory gives a way to measure the failure of a map $T$ to be an ordinary $\mathcal{O}$-operator; one could try to associate this 'defect' with geometric or integrability data on Poisson manifolds.
  • The same symmetrizer-antisymmetrizer mechanism likely transfers to other algebras with both a Lie-type and an associative-type operation, such as pre-Lie or Novikov settings, where extended operators and extended Yang-Baxter equations already exist; establishing the analogous correspondence there is a testable extension.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 7 minor

Summary. This paper introduces extended O-operators and generalized/extended Poisson Yang-Baxter equations (GPYBE/EPYBE) for Poisson algebras. It proves that ordinary O-operators of weight λ produce post-Poisson algebra structures (Prop. 2.17), that extended O-operators, under the balanced, A-invariant, and equivalence conditions (22)-(24), induce Poisson algebra structures on the module space (Thm. 2.28), and that equivalent characterizations hold through the symmetrizer-antisymmetrizer decomposition (Thm. 2.33). The central result, Theorem 3.17, states that for r=Θ+Λ with Θ (ad,L)-invariant, r solves the EPYBE of mass (κ+1)/4 if and only if Λ_+ is an extended O-operator of weight 0 with extension Θ_+ of mass (κ,0). This is applied to quadratic and semi-direct product Poisson algebras, recovering known O-operator/PYBE characterizations in the skew-symmetric case.

Significance. If correct, the paper provides a useful unifying framework: it extends the O-operator method from Lie and associative algebras to Poisson algebras, connects post-Poisson algebras with O-operators, and gives a tensor-form correspondence between extended O-operators and solutions of the EPYBE. The main theorems are proven by explicit computations and reduce in special cases (κ=-1 or S=0) to known characterizations of the PYBE, which is a good consistency check. The paper also contains concrete worked examples. The central correspondence is explicitly conditional on the (ad,L)-invariance of the symmetric part of r; this is a stated scope restriction rather than a hidden assumption. I found no load-bearing technical gap in the main derivation.

minor comments (7)
  1. [§3.3, Proposition 3.26] The statement contains a typo: “P_S = S I_B : P*→P” should read “P_S = S I_B : A*→A” (and similarly in the following line).
  2. [§3.4, Proposition 3.11, Eq. (61)] There is a duplicated plus sign: “ζ(T([w,v]_V))u+ +λζ(T([w,u]_V))v” should have a single plus.
  3. [§2.3, Definition 2.22] For μ=0 the “equivalent of mass μ” condition (24) is vacuous, and for κ=0 the “A-invariant of mass κ” condition (23) is vacuous; since the degenerate cases μ=0 and κ=0 are used in Theorem 3.17, a short remark making this explicit would prevent misreading.
  4. [§2.1, Lemma 2.26(i), Eq. (27)] In the displayed computation, the bracket “[v,ζ(S(u))w]” is the bracket on V, not on A; using “[,]_V” consistently throughout the proof would remove ambiguity.
  5. [§3.2, Remark 3.16] The remark proves the equivalence of the three forms of the bracket identity but only asserts the corresponding multiplication identity; adding the one-line computation for A(r) would make the remark self-contained.
  6. [§3.3, Corollary 3.27] The phrase “the 2-tensor form of r± = T I_B ± S I_B” is imprecise because T I_B and S I_B are linear maps, while r± denotes tensors; it would be clearer to say “the 2-tensor forms of T I_B±S I_B”.
  7. [§3.4, Corollary 3.32(ii)] The notation r_id is used before it is defined; it should be introduced explicitly, for example as r_id = Σ_i e_i⊗e_i for a basis (e_i) of A.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence is a tensor/operator dictionary with explicit hypotheses, not a definitional identity.

full rationale

The central equivalence Theorem 3.17 is not circular. Definition 2.23 (extended O-operator) and Definition 3.14 (EPYBE) are independent: one is an operator equation for a pair of linear maps T,S, and the other is a tensor equation for an element r. The proof of Theorem 3.17 exhibits the reduction at the level of 3-tensor pairings: pairing the EPYBE defect C(r) - ((kappa+1)/4)[r13+r31,r23+r32] with x* tensor y* tensor z* yields exactly the extended-O-operator defect [Lambda+(x*),Lambda+(y*)] - Lambda+(-ad*_{[,]}(Lambda+(x*))y* + ad*_{[,]}(Lambda+(y*))x*) - kappa[Theta+(x*),Theta+(y*)]. This is a coordinate translation, not an identity of definitions; the mass factor (kappa+1)/4 is a normalization determined by the computation. The only imported ingredient in that proof is Lemma 3.8, cited from the authors' prior work [17], which identifies (ad,L)-invariance of Theta with balanced/A-invariance of Theta_+. That lemma is a parameter-free technical equivalence whose assumptions do not include Theorem 3.17, and the paper's own Lemma 3.25 proves the analogous statement in the quadratic case, so the citation is independent support rather than a circular load-bearing premise. Theorem 2.28's hypotheses (22), (23), (24) are explicit and are exactly what the proof needs; no hidden parameter is fitted to make the theorem true. In the application, the balanced and A-invariant conditions on Theta_+ follow from the stated (ad,L)-invariance of Theta, and the equivalence condition (24) is vacuous because mu=0. No prediction is statistically forced, no uniqueness theorem is imported to forbid alternatives, and the known PYBE/O-operator characterization is credited to [16,17] in Remark 3.20 rather than being renamed. The restriction to solutions with (ad,L)-invariant symmetric part is stated as a hypothesis, not disguised. I therefore find no circular step.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central results rest on the standard definitions of module Poisson algebras and on the new balanced, A-invariant and equivalence conditions for the extension map. No hidden physical entities are introduced. The mass parameters are algebraic scalars, not fitted to any data.

free parameters (1)
  • Scalar parameters κ, μ, λ = arbitrary elements of k (no fitted values)
    Definition 2.22 and 2.23 introduce κ, μ, λ to unify cases; Theorem 3.17 reparametrizes the EPYBE mass as (κ+1)/4. The theorems hold for all choices, so they are degrees of freedom in the definitions, not fitted constants.
assumptions (4)
  • domain assumption All algebras and modules are finite-dimensional over a field k of characteristic 0.
    Stated in the introduction; used throughout for duals, tensor products, and the semi-direct product constructions.
  • domain assumption Definition of A-module Poisson algebra (Definition 2.4) and the compatibility identities (1)-(3).
    The paper takes the notion of module Poisson algebra from [21] as background; all extended O-operator results are formulated relative to this structure.
  • domain assumption Prior results quoted from [21]: Proposition 2.15, Theorem 3.2, and Definition 3.6 of (ad,L)-invariance.
    These are used as black boxes; e.g., Proposition 2.17 is cited from [21], and Lemma 3.8 from [17, Lemma 2.14].
  • ad hoc to paper The balanced, A-invariant, and equivalence conditions (22)-(24) on the extension map S are assumed whenever S is used as an extension.
    These conditions are new in this paper and are load-bearing: Lemma 2.26 and Theorem 2.28 fail without them. They are not derived from more basic principles.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Post-Poisson algebras, extended $\mathcal{O}$-operators and extended Poisson Yang-Baxter equations." pith.science (2026). https://pith.science/paper/WWBMAEBH

@misc{pith2026260806736,
  author       = {Pith},
  title        = {Pith review of: Post-Poisson algebras, extended $\mathcalO$-operators and extended Poisson Yang-Baxter equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWBMAEBH}},
  note         = {Machine review of arXiv:2608.06736}
}
abstract

This paper introduces the extended $\mathcal{O}$-operators on Poisson algebras as a natural generalization of ordinary $\mathcal{O}$-operators, together with the extended Poisson Yang-Baxter equations. We show that $\mathcal{O}$-operators of weight $\lambda$ on Poisson algebras give rise to post-Poisson algebras, whose operad are the trisuccessor of the operad of Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces.Equivalent characterizations of extended $\mathcal{O}$-operators are obtained via the symmetrizer-antisymmetrizer decomposition. The generalized Poisson Yang-Baxter equations are also introduced, and their connections with coboundary Poisson bialgebras and extended $\mathcal{O}$-operators are established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended Poisson Yang-Baxter equations, which generalizes the notion of the Poisson Yang-Baxter equations. Finally, the relationships among extended $\mathcal{O}$-operators, the extended Poisson Yang-Baxter equations, and the Poisson Yang-Baxter equations are studied in the framework of quadratic Poisson algebras and semi-direct product Poisson algebras.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    V . I. Arnol’d,Mathematical methods of classical mechanics, Springer Science & Business Media, 2013. 2

    work page 2013

  2. [2]

    C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operations, Manin products and Rota-Baxter operators, Int. Math. Res. Not. 3 (2013), 485–524. 2

    work page 2013

  3. [3]

    C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and post-Lie algebras, Commun. Math. Phys. 297 (2010), 553–596. 1, 2, 5, 10

    work page 2010

  4. [4]

    C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation andO-operators, J. Math. Phys. 52 (2011). 2

    work page 2011

  5. [5]

    C. Bai, L. Guo and X. Ni,O-operators on associative algebras and associative Yang-Baxter equations, Pacific J. Math. 256 (2012), 257–289. 1, 2, 5, 10

    work page 2012

  6. [6]

    Chari and A

    V . Chari and A. Pressley,A guide to quantum groups, Cambridge University Press, 1995. 2

    work page 1995

  7. [7]

    S. Chen, C. Bai and L. Guo, Solving the Poisson Yang-Baxter equation via deformation-to-quasiclassical-limits, arXiv e-prints, arXiv:2404.11232v2, 2024. 2

    work page Pith review arXiv 2024

  8. [8]

    P. A. Dirac, Lectures on quantum mechanics, Courier Corporation, 2013. 2

    work page 2013

Show all 30 references
  1. [9]

    Dotsenko and P

    V . Dotsenko and P. Tamaroff, Endofunctors and Poincaré-Birkhoff-Witt Theorems, Int. Math. Res. Not. 16 (2021), 12670–12690 2

    2021

  2. [10]

    Drinfeld,Quantum groups, Proc

    V . Drinfeld,Quantum groups, Proc. Int. Congr. Math., 1986, 798–820. 2

    1986

  3. [11]

    V . G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68–71. 2

    1983

  4. [12]

    Ginzburg and D

    V . Ginzburg and D. Kaledin, Poisson deformations of symplectic quotient singularities,Adv. Math. 186 (2004), 1–57. 2

    2004

  5. [13]

    Huebschmann, Poisson cohomology and quantization, J

    J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. 2

    1990

  6. [14]

    Kontsevich, Deformation quantization of Poisson manifolds, Lett

    M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157–216. 2

    2003

  7. [15]

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

    1999

  8. [16]

    Y . Lin, X. Liu and C. Bai, Differential antisymmetric infinitesimal bialgebras, coherent derivations and Poisson bialgebras, Symmetry Integrability Geom. Methods Appl. 19 (2023), 018. 24 30

    2023

  9. [17]

    Lin and D

    Y . Lin and D. Lu, Quasitriangular and factorizable Poisson bialgebras,Pacific J. Math. 343 (2026), 453–483. 2, 17, 18, 23, 25

    2026

  10. [18]

    J. Liu, C. Bai and Y . Sheng, Noncommutative Poisson bialgebras, J. Algebra 556 (2020), 35–66. 24

    2020

  11. [19]

    Loday, On the algebra of quasi-shuffles, Manuscripta Math

    J.-L. Loday, On the algebra of quasi-shuffles, Manuscripta Math. 123 (2007), 79–93. 6

    2007

  12. [20]

    Loday and M

    J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, Contemp. Math. 346 (2004), 369–398. 6

    2004

  13. [21]

    Ni and C

    X. Ni and C. Bai, Poisson bialgebras, J. Math. Phys. 54 (2013), 023515. 2, 4, 5, 7, 8, 17, 18, 22

    2013

  14. [22]

    Odzijewicz, Hamiltonian and quantum mechanics, Geom

    A. Odzijewicz, Hamiltonian and quantum mechanics, Geom. Topol. Monogr 17 (2011), 385–472. 2

    2011

  15. [23]

    Ospel, F

    C. Ospel, F. Panaite and P. Vanhaecke, Polarization and deformations of generalized dendriform algebras, J. Noncommut. Geom. 16 (2022), 561–594. 2

    2022

  16. [24]

    Polishchuk, Algebraic geometry of Poisson brackets, J

    A. Polishchuk, Algebraic geometry of Poisson brackets, J. Math. Sci. 84 (1997), 1413–1444. 2

    1997

  17. [25]

    Semenov-Tyan-Shanskii, What is a classicalr-matrix?, Funct

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

    1983

  18. [26]

    Vaisman, Lectures on the geometry of Poisson manifolds, Birkhäuser, 2012

    I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkhäuser, 2012. 2

    2012

  19. [27]

    Vallette, Homology of generalized partition posets, J

    B. Vallette, Homology of generalized partition posets, J. Pure Appl. Algebra 208 (2007), 699–725. 6

    2007

  20. [28]

    Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J

    B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. reine angew. Math. 620 (2008), 105–164. 2

    2008

  21. [29]

    Weinstein, Lectures on symplectic manifolds, American Mathematical Soc., 1977

    A. Weinstein, Lectures on symplectic manifolds, American Mathematical Soc., 1977. 2

    1977

  22. [30]

    Yu and Y

    J. Yu and Y . Hong, ExtendedO-operators, Novikov Yang-Baxter equations and post-Novikov algebras, J. Nonlinear Math. Phys. 33 (2026), 14. 2 School ofMathematics, NorthUniversity ofChina, Taiyuan030051, China Email address:linyuanchang@mail.nankai.edu.cn College ofAppliedScienc...

    2026

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.