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Compatible anti-pre-Lie algebras

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Pith's one-line read Compatible anti-pre-Lie algebras are pairs of anti-pre-Lie operations whose linear combinations stay anti-pre-Lie; the paper proves they are exactly compatible Lie-admissible algebras with negative left multiplications as a…

desk verdict A natural new notion and a plausible 2D classification, but the central anti-O-operator theorem is not proven: the key displayed equality in Proposition 3.3 is algebraically false. read the letter →

arxiv 2412.16273 v1 pith:UPKQNNM7 submitted 2024-12-20 math.RA

Zafar Normatov This is my paper · ORCID
classification math.RA MSC 16P1017A30
keywords compatibleanti-pre-LiealgebrasLieanti-O-operatorscommutative2-cocyclesLie-admissibleanti-Rota-Baxteroperators2-dimensionalclassification
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 introduces compatible anti-pre-Lie algebras: two anti-pre-Lie operations on the same vector space such that every linear combination of the two products is again anti-pre-Lie, generalizing the familiar compatibility idea from Lie, associative, and pre-Lie algebras. The central claim is a bridge to representation theory: a pair of anti-pre-Lie operations is compatible exactly when its commutators form a compatible Lie algebra and the negative left multiplication operators form a representation of that compatible Lie algebra. The paper then shows that strong anti-O-operators on compatible Lie algebras produce such structures, that nondegenerate commutative 2-cocycles induce them, and that an invertible anti-O-operator is equivalent to the existence of such a structure. For dimension two over the complex numbers, it obtains the complete classification: exactly 45 non-isomorphic families. If correct, this organizes the known anti-pre-Lie examples into a compatible-algebra framework and gives a new supply of examples from Lie-algebra data.

What carries the argument

The load-bearing objects are the pair of negative left multiplication operators $(-L_\circ,-L_*)$ viewed as a representation of the sub-adjacent compatible Lie algebra, together with the compatibility identities (2.8)--(2.9) that make the pair an anti-pre-Lie pair. The representation condition supplies equations (2.4)--(2.6); Proposition 2.12 says these are exactly equivalent to compatibility, so all later constructions, anti-O-operators, cocycles, and invariant forms, work by producing such a representation. For the classification, the machinery is the cohomology-style set $Z^2(A,A)$ of bilinear maps satisfying the compatibility conditions with a fixed anti-pre-Lie algebra $(A,\circ)$, together with the action of the automorphism group $\mathrm{Aut}(A)$; partitioning $Z^2(A,A)$ into orbits yields the second operation $*$ up to isomorphism.

What would settle it

Recompute the spaces of compatible second operations for each of the nine listed two-dimensional anti-pre-Lie algebras and the automorphism-group orbits on those spaces; if this yields exactly the 45 families $CA_1$--$CA_{45}$ with the stated parameter identifications, the classification stands, otherwise it fails. A simpler consistency check is to test that every listed family satisfies identities (2.8)--(2.9) and that families with different stated parameters are non-isomorphic.

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Extended reading notes

Core claim

The paper's discovery is that compatibility of anti-pre-Lie algebras is not an extra ad hoc condition but a representation-theoretic one. Work over a fixed vector space $A$ with two bilinear operations $\circ$ and $*$, each making $A$ anti-pre-Lie. Then $(A,\circ,*)$ is a compatible anti-pre-Lie algebra if and only if $A$ is compatible Lie admissible, meaning the commutators $[x,y]_1=x\circ y-y\circ x$ and $[x,y]_2=x*y-y*x$ form a compatible pair of Lie brackets, and the pair $(-L_\circ,-L_*)$ is a representation of the sub-adjacent compatible Lie algebra. From this characterization the paper derives that strong anti-O-operators on compatible Lie algebras construct compatible anti-pre-Lie algebras on representation spaces, and that nondegenerate commutative 2-cocycles on compatible Lie algebras induce such structures. The paper also proves the converse existence statement: a compatible Lie algebra carries a compatible anti-pre-Lie structure exactly when it admits an invertible anti-O-operator. The final section carries out the orbit-by-orbit classification and lists 45 isomorphism classes of two-dimensional complex compatible anti-pre-Lie algebras.

Load-bearing premise

The whole list of 45 families stands or falls on whether the previously published inventory of two-dimensional anti-pre-Lie algebras and the paper's unshown case-by-case calculations of compatible second operations and symmetry groups are complete.

Editorial extensions

If this is right

  • Every compatible anti-pre-Lie algebra has an underlying compatible Lie algebra defined by the two commutators, and the negative left multiplication operators form a representation of it; conversely, any compatible Lie-admissible algebra with that representation is a compatible anti-pre-Lie algebra.
  • Strong anti-O-operators on compatible Lie algebras yield compatible anti-pre-Lie structures on representation spaces, and an invertible anti-O-operator is equivalent to the existence of such a structure on the compatible Lie algebra itself.
  • A nondegenerate commutative 2-cocycle on a compatible Lie algebra induces a compatible anti-pre-Lie algebra whose operations are defined by the cocycle, and invariant symmetric bilinear forms on compatible anti-pre-Lie algebras are commutative 2-cocycles on the sub-adjacent compatible Lie algebra.
  • In any dimension, a pair of commutative operations forms a compatible anti-pre-Lie algebra exactly when it forms a compatible associative algebra.
  • In dimension two over $\mathbb{C}$, the isomorphism classes are exactly the 45 families $CA_1$--$CA_{45}$; in particular, the second operation in any such algebra is obtained from one of the nine anti-pre-Lie algebras by a compatible deformation parameterized by $Z^2(A,A)$ modulo automorphisms.

Reading between the lines

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

  • By extension, the representation-theoretic characterization suggests that compatible anti-pre-Lie structures on a fixed compatible Lie algebra are governed by strong anti-O-operators; classifying those operators or their cohomology would give higher-dimensional examples without repeating the orbit-by-orbit enumeration.
  • Because commutative compatible anti-pre-Lie algebras are just compatible associative algebras, the 45-family list contains a known associative sub-list; comparing the commutative cases with classifications of 2-dimensional compatible associative algebras could test the classification's internal consistency.
  • The construction from commutative 2-cocycles links these algebras to the classical Yang--Baxter circle of ideas in the compatible setting; one could look for integrable-system interpretations of the 45 families by asking which of them arise from a nondegenerate cocycle on their sub-adjacent compatible Lie algebra.
  • The orbit method used here is a general deformation pattern: fixing one operation and solving the compatibility equations for the second operation is equivalent to computing the space $Z^2(A,A)$; applying it to 3-dimensional anti-pre-Lie algebras from the literature would extend the classification, though with many more parameters.
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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

4 major / 5 minor

Summary. The paper defines compatible anti-pre-Lie algebras as pairs of anti-pre-Lie operations whose arbitrary linear combinations remain anti-pre-Lie, and relates them to compatible Lie-admissible algebras with a representation given by the negative left multiplications. It then aims to construct compatible anti-pre-Lie algebras from anti-O-operators on compatible Lie algebras and from nondegenerate commutative 2-cocycles, and it presents a classification of complex 2-dimensional compatible anti-pre-Lie algebras into 45 families. The early definitions and the representation-theoretic characterization in Section 2 are largely coherent, but the main bridge results in Sections 3 and 4 contain a central invalid proof and a missing definition, and the classification in Section 5 is not verifiable from the manuscript as written.

Significance. The notion of compatible anti-pre-Lie algebras is a natural extension of both compatible Lie algebras and anti-pre-Lie algebras, and Proposition 2.12, if fully established, gives a clean representation-theoretic description. The explicit classification of 2-dimensional examples would be useful for testing conjectures about higher-dimensional structures. The paper also correctly draws on existing results of Liu–Bai and Wu–Bai rather than inventing ad hoc machinery. However, the manuscript's main contribution beyond the definition is not currently supported: the anti-O-operator construction depends on an undefined notion of 'strong' and on a false displayed identity, and the cocycle construction inherits that failure. The classification section, although potentially valuable, is presented as a collection of asserted computations without enough detail to check completeness.

major comments (4)
  1. [Section 3, Definition 3.2 and Proposition 3.3] The manuscript never defines what 'strong' means for an anti-O-operator on a compatible Lie algebra, so Proposition 3.3's statement 'T is strong' is not well-posed. More seriously, the proof's key displayed reduction is false. The cyclic sum in (2.9) is S = ρ([T(v),T(u)]_2)w + μ([T(v),T(u)]_1)w + cyclic, which equals −(ρ([T(u),T(v)]_2)w + μ([T(u),T(v)]_1)w + cyclic). The expression written as 1/(k1k2) times a sum containing k1^2ρ([T(u),T(v)]_1)w and k2^2μ([T(u),T(v)]_2)w is not equal to S, and the division by k1k2 makes the subsequent claim 'this is also true for k1k2=0' meaningless. Consequently Proposition 3.3, and with it Corollaries 3.4–3.6, is not established.
  2. [Section 3, Proposition 3.5] The proof that an invertible anti-O-operator is strong is circular. After showing that (V,∘,∗) is compatible Lie-admissible, it concludes 'Then T is strong due to Proposition 3.3.' But Proposition 3.3 is precisely the statement that the remaining anti-pre-Lie identity (2.9) is equivalent to strongness, so invoking it here assumes what is to be proved. No independent verification of the cross strongness condition is supplied, and Corollary 3.6 depends on this proposition.
  3. [Section 4, Theorem 4.3] The proof invokes 'Corollary 2.18', which does not exist in the manuscript; the intended reference is presumably Corollary 3.6. Since Corollary 3.6 rests on the unproved Propositions 3.3 and 3.5, the construction of a compatible anti-pre-Lie algebra from a nondegenerate commutative 2-cocycle is not supported as written.
  4. [Section 5, Lemmas 5.2–5.3 and Theorem 5.4] The classification is not sufficiently supported. Lemma 5.2 lists the spaces Z^2(A,A) and Lemma 5.3 lists the automorphism groups without derivation, and the proof of Theorem 5.4 is a sequence of parameter-normalization assertions with the orbit computations omitted. The completeness claim that the 45 algebras are 'one and only one' cannot be checked from the manuscript. Because the method depends on the completeness of these sets, any omission in Lemma 5.2 or Lemma 5.3 would propagate through the entire list.
minor comments (5)
  1. [Throughout] There are many typographical and grammatical errors ('Firtsly', 'classsification', 'repectively'); these should be corrected in a revision.
  2. [Remark 2.10] The remark labels an equality as following from (2.10), but the equivalence between (2.9) and (2.12) actually uses (2.8); the labeling should be corrected.
  3. [Theorem 4.3] The reference to 'Corollary 2.18' should be replaced by the correct corollary once the numbering is fixed.
  4. [Proposition 4.7] In the displayed verification of the 2-cocycle property, the signs of the terms involving ⟨b∗,x∘z⟩ and ⟨a∗,y∘z⟩ are opposite to what follows from the definition of the dual representation; as written the computation has a sign error.
  5. [Theorem 5.4] The families CA35 and CA38 contain a parameter λ in their multiplication tables, but λ is not declared in the theorem statement; the parameter ranges should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's results are derived from defining identities and independent prior classifications, with no fitted input presented as a prediction.

full rationale

The paper introduces compatible anti-pre-Lie algebras and derives its structural results from the defining identities together with independent prior theorems from [16] and [24]. Proposition 2.9 expands the compatibility condition directly; Proposition 2.11 and Proposition 2.12 use the representation characterization of anti-pre-Lie algebras from [16] and the compatible Lie algebra criterion from [24]. Neither of these citations is a self-citation, and neither imports a conclusion that is equivalent to the paper's own definitions. The classification in Section 5 is a standard orbit computation: Z^2(A,A) is computed from the compatibility identities, the automorphism groups are listed, and orbit representatives are selected, so the 45 families are outputs of the computation rather than fitted inputs. The base list of 2-dimensional anti-pre-Lie algebras is taken from [16] as an external, independent classification. There are genuine expositional and correctness gaps in the paper, notably that 'strong' for compatible anti-O-operators is not explicitly defined in Definition 3.2/3.1 and the proof of Proposition 3.3 contains an algebraically questionable reduction with a division by k1k2 and a reference to a nonexistent Corollary 2.18. However, these are errors in the derivation, not circularity: the alleged equivalence is a theorem to be proved, not a condition that has been assumed as its own conclusion. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or a definition that presumes the target result. Therefore the circularity score is 0.

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

No data fitting or ad hoc constants appear; the classification parameters (α, β, γ, δ, λ) are isomorphism invariants rather than free inputs. The new structure 'compatible anti-pre-Lie algebra' is a definition, not an invented entity with independent evidence. The load-bearing inputs are prior classifications and representation-theoretic characterizations from [16] and [24].

assumptions (5)
  • domain assumption The list in Proposition 5.1 is a complete and correct classification of 2-dimensional complex anti-pre-Lie algebras.
    The classification in Theorem 5.4 starts from this list, which is cited from [16] and not re-derived.
  • domain assumption The characterizations of anti-pre-Lie algebras via the negative left multiplication representation and of strong anti-O-operators from [16] are correct and extend to the compatible setting.
    Used in Propositions 2.12, 3.3, and 3.5 to transfer single-structure results to pairs.
  • domain assumption The compatibility criteria for Lie algebras and representations from [24] are correct.
    Used throughout Section 2, for example in Proposition 2.11 and 2.12.
  • domain assumption The orbit method for classifying compatible algebras via Z²(A,A) and Aut(A) yields a complete classification.
    Section 5 adopts this method from [1] and [15] without proving its completeness.
  • standard math All vector spaces are finite-dimensional over C and standard linear algebra applies.
    Stated in the introduction and used throughout.

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Pith. "Pith review of Compatible anti-pre-Lie algebras." pith.science (2026). https://pith.science/paper/UPKQNNM7

@misc{pith2026241216273,
  author       = {Pith},
  title        = {Pith review of: Compatible anti-pre-Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPKQNNM7}},
  note         = {Machine review of arXiv:2412.16273}
}
abstract

In this paper, we introduce the notion of compatible anti-pre-Lie algebras and study relationship between them and the related structures such as anti-$\mathcal{O}$-operators, commutative $2$-cocycles on compatible Lie algebras. Moreover, we give the classification of $2$-dimensional compatible anti-pre-Lie algebras from the classification of anti-pre-Lie algebras of the same dimension.

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Works this paper leans on

24 extracted references · 23 canonical work pages

  1. [1]

    The Algebraic and Geometric Classification of Compatible Pre-Lie Algebras

    Abdelwahab H., Kaygorodov I., Makhlouf A. The algebraic and geometric classification of compatible pre-Lie algebras, http://arxiv.org/abs/2406.10947v1

    work page Pith review arXiv

  2. [2]

    , Classification of three dimensional anti-dendriform algebras, Comm

    Abdurasulov K., Adashev J., Normatov Z., Solijonova SH. , Classification of three dimensional anti-dendriform algebras, Comm. Alg., (2024), 1–17. https://doi.org/10.1 080/00927872.2024.2426037

    arXiv 2024

  3. [3]

    Bai C., An introduction to pre-Lie algebras, in: Algebra and Applications 1: Nonassociative Algebras and Cate- gories, Wiley Online Library, (2021), 245-273

    work page 2021

  4. [4]

    Bai c., Meng D., The classification of Novikov algebras in low dimensions, J. Phys. A: Math. Gen. 34 (2001) 1581-1594

    work page 2001

  5. [5]

    Beites P., Ouaridi A., Kaygorodov I., The algebraic and g eometric classification of transposed Poisson algebras, arXiv:2311.00459v1

    work page Pith review arXiv

  6. [6]

    95 (1998), 397-411

    Burde D., Simple left-symmetric algebras with solvable Lie algebra, Manuscripta Math. 95 (1998), 397-411

    work page 1998

  7. [7]

    Cabrera Casado Yo., Siles Molina M., Velasco M., Classifi cation of three-dimensional evolution algebras, Linear Algebra Its Appl., 524 (2017), 68-108

    work page 2017

  8. [8]

    and Zusmanovich P., Commutative 2-co cycles on Lie algebras, J

    Dzhumadil’daev A. and Zusmanovich P., Commutative 2-co cycles on Lie algebras, J. Algebra 324 (2010), 732- 748

    work page 2010

Show all 24 references
  1. [9]

    Gao D., Liu G., Bai C., Anti-dendriform algebras, new spl itting of operations and Novikov type algebras, JACO, 59(3) (2024), 661-696

    2024

  2. [10]

    Gerstenhaber M., The cohomology structure of an associ ative ring. Ann. of Math., 78 (1963), 267-288

    1963

  3. [11]

    Golubchik I., Sokolov V., Compatible Lie brackets and i ntegrable equations of the principal chiral field model type , Funct. Anal. Appl., 36(3) (2002), 172–181

    2002

  4. [12]

    Golubchik I., Sokolov V., Compatible Lie brackets and t he Yang-Baxter equation, Theor. Math. Phys., 146(2) (2006), 159–169

    2006

  5. [13]

    Kobayashi Y., Shirayanagi K., Takahasi S., Tsukada M., A complete classification of three-dimensional algebras over R and C, AEJM, 14(8) (2021), 2150131

    2021

  6. [14]

    L., Domaines born /acute.ts1es homog‘enes et orbites de groupes de transformations affine s

    Koszul J. L., Domaines born /acute.ts1es homog‘enes et orbites de groupes de transformations affine s. Ann. of Math., 78 (1963), 267-288

    1963

  7. [15]

    Ladra M., Leite da Cunha B., Lopes S., A classification of nilpotent compatible Lie algebras, arXiv:2406.04036

    arXiv

  8. [16]

    Algebra, 609 (2022), 337-379

    Liu G., Bai C., Anti-pre-Lie algebras, Novikov algebra s and commutative 2-cocycles on Lie algebras, J. Algebra, 609 (2022), 337-379

    2022

  9. [17]

    Odesskii A., Sokolov V., Compatible Lie brackets relat ed to elliptic curve, J. Math. Phys., 47(1) (2006), 013506

    2006

  10. [18]

    Odesskii A., Sokolov V., Algebraic structures connect ed with pairs of compatible associative algebras, IMRN, (2006), 43734. J. Phys. A., 39(40) (2006), 12447–12456

    2006

  11. [19]

    Odesskii A., Sokolov V., Pairs of compatible associati ve algebras, classical Yang-Baxter equation and quiver rep re- sentations, Comm. Math. Phys., 278(1) (2008), 83–99. 20 NORMATOV Z

    2008

  12. [20]

    Petersson H., The classification of two-dimensional no nassociative algebras, Results Math., 37(1-2) (2000), 120 -154

    2000

  13. [21]

    Rikhsiboev I., Rakhimov I., Basri W., Classification of 3-dimensional complex diassociative algebras, MJMS, 4(2) (2010), 241-254

    2010

  14. [22]

    Rikhsiboev I., Rakhimov I., Basri W., The description o f dendriform algebra structures on two-dimensional comple x space, JPANTA, 4(1) (2010), 1-18

    2010

  15. [23]

    B., The theory of homogeneous convex cones

    Vinberg E. B., The theory of homogeneous convex cones. B ull. soc. Math. France. 89 (1961), 515-533

    1961

  16. [24]

    W u M., Bai C., Compatible Lie bialgebras. Comm. Theore. Phys. 63 (2015), 653-664. (Zafar Normatov) School of Mathematics, Jilin University, Changchun, 130012 , China, Institute of Mathematics, Uzbekistan Academy of Sciences, Un ivesity Street, 9, Olmazor district, Tashkent, 1...

    2015

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