Crossed modules of ternary Leibniz algebras

Ibrahima Bakayoko; Kol B\'eatrice Gamou

REVIEW 4 major objections 6 minor 37 references

Crossed modules of ternary Leibniz algebras

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Pith's one-line read The paper shows that every crossed module of triassociative algebras, and of Leibniz algebras, transfers to a crossed module of ternary Leibniz algebras by explicit action maps.

desk verdict Salvageable idea, but the central transfer formulas have a load-bearing type error and the proofs are missing, so the paper as submitted should be rejected. read the letter →

arxiv 2501.18414 v1 pith:53M7BOIK submitted 2025-01-30 math.RA

Kol Béatrice Gamou , Ibrahima Bakayoko This is my paper · ORCID

classification math.RA MSC 17A4018G4517A32

keywords LeibnizalgebrasternarytriassociativecrossedmodulesactionsRota-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 aims to extend the notion of crossed module—a model for homotopy 2-types—from triassociative and Leibniz algebras to ternary Leibniz algebras. It proves that the transfer functor $T$, defined on a triassociative algebra by the ternary bracket $[x,y,z] = x \dashv (y \perp z - z \perp y) - (y \perp z - z \perp y) \vdash x$, and on a Leibniz algebra by $[x,[y,z]]$, carries actions and crossed modules along with the algebras themselves. The central result gives six explicit bilinear maps (22)–(27) that turn an action of $B$ on $A$ into an action of $T(B)$ on $T(A)$, so that $(T(A),T(B),\phi)$ is a crossed module of ternary Leibniz algebras whenever $(A,B,\phi)$ is a crossed module of triassociative algebras. The same transfer works for crossed modules of Leibniz algebras. If the paper is right, this gives a systematic way to build higher-arity Leibniz structures from classical ones, with consequences for homotopy invariants and deformation theory.

What carries the argument

The central object is the functor $T$ between algebraic categories. On a triassociative algebra $(A,\dashv,\perp,\vdash)$ it puts the ternary bracket $[x,y,z] = x \dashv (y \perp z - z \perp y) - (y \perp z - z \perp y) \vdash x$; on a Leibniz algebra $(L,[-,-])$ it puts $\{x,y,z\} = [x,[y,z]]$. The work of this functor is to push the crossed-module action through: the six maps $m_1,\dots,m'_3$ in (22)–(27) are defined so that each ternary Leibniz action axiom becomes a consequence of the six triassociative action equalities from Definition 2.7.

What would settle it

Take the two-dimensional triassociative algebra from Example 2.3, with basis $\{e_1,e_2\}$, all three products equal, $e_1 \dashv e_1 = e_2 \dashv e_1 = 0$, $e_1 \dashv e_2 = a e_1$, $e_2 \dashv e_2 = e_2$, and compute the bracket $[x,y,z] = x \dashv (y \perp z - z \perp y) - (y \perp z - z \perp y) \vdash x$ on all triples of basis elements. If any of the 32 instances of the identity $[[x,y,z],t,u] = [x,y,[z,t,u]] + [x,[y,t,u],z] + [[x,t,u],y,z]$ fails, Lemma 3.1 is false and Theorem 3.1 collapses; if all pass, the load-bearing bracket is confirmed on this example.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of transfer theorems. Theorem 3.1 states that any crossed module of triassociative algebras yields, through the bracket $[x,y,z] = x \dashv (y \perp z - z \perp y) - (y \perp z - z \perp y) \vdash x$ and the six maps (22)–(27), a crossed module of ternary Leibniz algebras, with morphisms of triassociative crossed modules inducing morphisms of the ternary ones. Theorem 3.3 states the analogous statement for crossed modules of Leibniz algebras via the bracket $\{x,y,z\} = [x,[y,z]]$. The construction is explicit: every action map for the source category is repackaged, using the middle product $\perp$ and the two-sided actions, into the six ternary action maps required by Definition 2.18.

Load-bearing premise

The whole construction depends on one borrowed formula: the bracket $[x,y,z] = x \dashv (y \perp z - z \perp y) - (y \perp z - z \perp y) \vdash x$ must satisfy the ternary Leibniz identity inside every triassociative algebra, and if it fails, the induced maps (22)–(27) need not define an action or a crossed module.

Editorial extensions

If this is right

Any crossed module of triassociative algebras, for instance an ideal inclusion, becomes a crossed module of ternary Leibniz algebras once the six maps (22)–(27) are installed.
Any crossed module of Leibniz algebras becomes a ternary Leibniz crossed module without changing the underlying map $\phi$, only reinterpreting the bracket as $[x,[y,z]]$.
The transfer is functorial: a morphism of crossed modules in the source category is automatically a morphism between the induced ternary crossed modules.
The semi-direct product respects the transfer: $T(A \rtimes B) = T(A) \rtimes T(B)$, so extensions built from actions survive the passage to ternary structures.
Rota-Baxter deformations of Leibniz crossed modules also induce ternary Leibniz crossed modules, giving a second family of constructions.

Reading between the lines

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

Because $T$ is functorial on crossed modules, it plausibly induces a functor between the associated 2-categories of crossed modules; this would make $T$ a map of homotopy-theoretic models, though the paper does not spell that out.
The two alternative brackets in Remark 3.1 suggest that the transfer is not unique; one could test whether those brackets also lift actions and crossed modules, and whether the resulting ternary crossed modules are isomorphic.
A computational sanity check of the ternary Leibniz identity for (22)–(27) on a small example, such as the two-dimensional algebra in Example 2.3, would give independent evidence that no hidden case distinction is missing.
Averaging operators and Nijenhuis operators, which the paper uses to build new triassociative algebras, might similarly lift to operators on the induced ternary Leibniz crossed modules; the paper does not claim this.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The long story short: the paper's main transfer theorem is stated with a type error in the action formulas, so as it stands it doesn't prove what it claims. The underlying idea is a natural extension of Casas's T functor, but the execution is too careless to be useful.

What's genuinely new: the explicit six action maps (22)-(27) and the transfer statements for crossed modules, which I don't think appear in the cited literature. The paper also collects known constructions and examples in one place. If someone repairs the formulas and supplies the missing verifications, the transfer result would be a reasonable small contribution to the crossed-module program.

The soft spots are substantial. The big one: equations (26)-(27) use the middle product ⊥ on the left, while Lemma 3.1 forces the left product ⊣. This is not a cosmetic typo—checking the identity crossed module against the Peiffer condition fails exactly because of this mismatch. So Theorem 3.1(b) is false as stated. Theorem 3.3 has its own issue: the maps m2, m'2, and m'3 are defined self-referentially, and the defining equations don't type-check. Definition 2.18's list of action axioms is also full of broken equations, which makes it hard to even parse what an action is supposed to be. And all the central proofs are dismissed as 'long but straightforward.' No comparison is made with the Leibniz n-algebra crossed module results in [36], which is a gap given the T functor already appears there. The self-citation is not the problem; the problem is that the load-bearing formulas don't check out.

This paper is for someone working in the niche of crossed modules for non-associative algebras who might mine it for a repairable construction. It shouldn't have been submitted in this shape. A serious revision with full computations, or at least a formal verification of the transferred action axioms, could make it publishable.

I would not send this to peer review as is. If the authors fix the formulas and actually prove the action axioms, then a short paper in a specialized journal would be reasonable. Right now, the internal contradictions put it below the bar.

Referee Report

4 major / 6 minor

Summary. The paper proposes functorial constructions that send crossed modules of triassociative algebras and crossed modules of Leibniz algebras to crossed modules of ternary Leibniz algebras. The main mechanism is the known bracket [x,y,z] = x ⊣ (y ⊥ z - z ⊥ y) - (y ⊥ z - z ⊥ y) ⊢ x of Lemma 3.1, applied to semidirect products, together with an action transfer stated in Theorems 3.1 and 3.3. The paper also states Rota-Baxter deformations of such crossed modules in Theorems 3.4 and 3.5. The central claims are that the six maps (22)-(27) define an action and a crossed module, that T(A⋊B) = T(A)⋊T(B), and that analogous transfers work for Leibniz algebras and for Rota-Baxter operators.

Significance. If the transfer theorems were correct, the paper would provide a useful bridge between crossed modules of triassociative algebras, Leibniz algebras, and ternary Leibniz algebras, complementing known constructions in the literature. The paper does not contain machine-checked proofs or reproducible code; all central verifications are delegated to 'straightforward' computation. Because several displayed formulas are type-incorrect or internally inconsistent, the significance of the claimed results cannot be assessed from the submitted manuscript.

major comments (4)

[Theorem 3.1, Eqs. (26)-(27)] The displayed maps m'_2 and m'_3 use the middle product in the first term, writing y ⊥ (µ⊥_2(c,z) - µ⊥_1(z,c)). Lemma 3.1, however, defines the ternary bracket with the left product in that position: [x,y,z] = x ⊣ (y ⊥ z - z ⊥ y) - (y ⊥ z - z ⊥ y) ⊢ x. Expanding [(y,0),(0,c),(z,0)] in T(A⋊B) therefore gives y ⊣ (µ⊥_2(c,z)-µ⊥_1(z,c)) - (...) ⊢ y, not y ⊥ (...). For the crossed module (A,A,id) with the canonical self-action, Condition 2 of Definition 2.19 forces m'_2(y,c,z) = [y,c,z]_{T(A)} = y ⊣ (c⊥z-z⊥c) - (c⊥z-z⊥c) ⊢ y. The printed formula differs by y⊥(...) - y⊣(...), which is not an identity in the free triassociative algebra. Thus Theorem 3.1(b) and Theorem 3.2 are false as stated; the defect is not merely a missing proof. If the intended operation in (26)-(27) is ⊣ rather than ⊥, the theorem may be repairable, but the typeset claim is incorrect.
[Theorem 3.3] The formulas defining the six action maps are type-incorrect. For instance, m2(p1,l,p2) is defined as µ2(p2, m2(l,p2)), which refers to m2 on the right-hand side before m2 is defined and with an inadmissible number of arguments; m3(p1,p2,l) is defined as µ2(p1, m2(p2,l)), again using the yet-undefined ternary map m2 as a binary map; similarly m'_2(l1,p,l2) = µ1(l1, m2(p,l2)) and m'_3(l1,l2,p) = µ1(l1, m1(l2,p)) use binary applications of ternary maps. Consequently the induced action of T(P) on T(L) is not actually defined, and part (2) of the theorem is unsubstantiated.
[Definition 2.18] The action axioms for ternary Leibniz algebras are presented in a badly corrupted form. Equations (7)-(21) contain unmatched parentheses, missing commas, and variable mismatches, e.g. equation (9) contains '[p1,p3,p4)' with a closing parenthesis instead of a bracket, equation (12) contains '[l1, m1(l2,p2,p3), l2]' where the middle term has the wrong type, and equation (13) contains 'l[p2,p3,p4]' with no comma. Since every subsequent theorem relies on this definition, the paper does not provide a usable and checkable notion of action.
[Proofs of Theorems 3.1, 3.3, 3.4, 3.5] The main results are all asserted with 'the proof is long but straightforward' or 'it comes from direct computation' without any verification of the action axioms or the crossed-module conditions. These verifications are the core content of the paper, not routine decoration. In view of the concrete type errors in the displayed formulas, such delegation is insufficient and prevents the reader from distinguishing a repairable typo from an incorrect construction.

minor comments (6)

[Definition 2.3] The first line states 'Let (A, ⊣, ⊥, ⊣) be an triassociative algebra'; the fourth operation should be ⊢.
[Proposition 2.1] In the proof, the same symbols x,y,z are used both for elements of A and for their classes in A/I, and the final equality omits the class notation; the displayed chain is not a well-formed proof as written.
[Proposition 2.2] The second and third multiplication formulas contain 'bA ⊥B b2' and 'bA ⊢B b2', which appear to be typographical errors for 'b1 ⊥B b2' and 'b1 ⊢B b2'.
[Example 2.4] The example refers to a 'morphism of triassociative dialgebras'; the terminology should be 'triassociative algebras'.
[Theorem 3.4] The notation mR_2 is used in the definitions of m2, m3, m'_2, and m'_3, but mR_2 is never defined; presumably a definition or a reference to µR_2 is intended.
[Throughout] There are numerous typographical and grammatical errors, including 'satisfyied', 'propreties', 'connexion', 'communt fact', 'echanging', and 'associate trialgebra'; these should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transfer constructions are explicit formulas built from external lemmas, not re-derivations of their own inputs.

full rationale

The paper's central claims (Theorems 3.1 and 3.3) are constructive: given a crossed module of triassociative algebras or Leibniz algebras, six explicit action maps are written down in formulas (22)-(27) and in Theorem 3.3, and the crossed-module verification is asserted as a direct computation. The key transfer mechanism is Lemma 3.1, attributed to Casas [32], an external source; Lemma 2.1 is also attributed to [32] (and [27]) and does not assume any crossed-module conclusion. The self-citations [27], [29], and [15] concern operator and Rota-Baxter constructions used in supporting results (Theorems 2.1, 3.4, 3.5), not in a way that makes the main transfer theorems depend on their own output. No parameter is fitted to a subset of 'data' and later renamed a prediction; no uniqueness theorem from the authors' prior work is invoked to force a choice. The repeated statement that proofs are 'long and tedious, but straightforward' (Theorems 3.1, 3.3, 3.5) is an omitted-proof rigor concern, not circularity. The skeptic's objection to equations (26)-(27) — that m'_2 is written with a middle product where expansion of T(A⋊B) through Lemma 3.1 would give a left product — is a possible correctness or typographical defect, but a false or unverified theorem is not thereby equivalent to its assumptions. No circular step can be exhibited from the paper's own equations or citation chain, so the circularity score is 0.

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

No numerical parameters are fitted. The central load rests on the cited functor T from Leibniz to ternary Leibniz algebras and on the correctness of the action axioms from [27]; the paper adds no new free entities.

assumptions (5)

standard math The right Leibniz identity in Definition 2.10 and the ternary Leibniz identity in Definition 2.14 are the working definitions of the algebras.
The paper uses these identities throughout; if a different convention were intended, Lemma 2.1 and Lemma 3.1 would need rechecking.
domain assumption Any triassociative algebra satisfies the 11 identities in Definition 2.1.
The construction of the ternary bracket in Lemma 3.1 uses these identities implicitly through the cited reference [32].
domain assumption The 30-identity action of ternary Leibniz algebras in Definition 2.18, taken from the authors' earlier paper [27], is the correct notion of action.
The main theorems assert that certain maps form such an action; a misstated axiom set would invalidate the transfer results.
standard math Lemma 3.1: the bracket x ⊣ (y ⊥ z - z ⊥ y) - (y ⊥ z - z ⊥ y) ⊢ x makes any triassociative algebra into a ternary Leibniz algebra.
Cited from Casas [32] without proof; it is the base functor for the triassociative-to-ternary transfer.
ad hoc to paper The printed formulas in Theorem 3.3 can be repaired to type-correct definitions; as written, m2 is defined in terms of itself.
The displayed formula m2(p1,l,p2) = mu2(p2, m2(l,p2)) uses two arguments and is circular; a charitable reading assumes a typographical error.

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Cite this review

Pith. "Pith review of Crossed modules of ternary Leibniz algebras." pith.science (2026). https://pith.science/paper/53M7BOIK

@misc{pith2026250118414,
  author       = {Pith},
  title        = {Pith review of: Crossed modules of ternary Leibniz algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53M7BOIK}},
  note         = {Machine review of arXiv:2501.18414}
}

read the original abstract

The aim of this paper is to construct triassociative algebras (from operators), new actions and crossed modules from a given one, and to make the connexion between these notions on Leibniz algebras or triassociative algebras and the corresponding notions on ternary Leibniz algebras.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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