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Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations

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Pith's one-line read This paper introduces embedding tensors on 3-Leibniz algebras and shows they are in two-way correspondence with 3-tri-Leibniz algebras, then classifies linear deformations of such tensors through a first cohomology group.

desk verdict The structural dictionary between embedding tensors and 3-tri-Leibniz algebras is largely fine, but the deformation-cohomology bijection in Theorem 5.6 is false, and the zero embedding tensor kills the headline result. read the letter →

arxiv 2502.04023 v1 pith:3CECANXO submitted 2025-02-06 math.RA

Wen Teng , Shuangjian Guo This is my paper · ORCID
classification math.RA MSC 17A0117A6017A3217A40
keywords 3-Leibnizalgebraembeddingtensor3-tri-LeibnizaveragingoperatorlineardeformationfirstcohomologydialgebraNijenhuiselement
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 introduces embedding tensors—linear maps from a representation space into a 3-Leibniz algebra that intertwine the bracket with the representation—and studies the algebraic structures they create. Its central structural result is a two-way correspondence: an embedding tensor endows the representation space with a 3-tri-Leibniz algebra, and conversely every 3-tri-Leibniz algebra arises as the induced structure on a representation of a 3-Leibniz algebra. The authors also show that any 3-tri-Leibniz algebra embeds into an averaging 3-Leibniz algebra, and that homomorphic embedding tensors produce 3-tri-Leibniz dialgebras. Finally, following Gerstenhaber's deformation method, they define 1-cocycles and a first cohomology for embedding tensors and claim a bijection between equivalence classes of linear deformations and that cohomology.

What carries the argument

The central object is the embedding tensor T: V → g, a linear map from a representation (V; ρl, ρm, ρr) of a 3-Leibniz algebra (g, [·,·,·]) satisfying [Tu,Tv,Tw] = Tρl(Tu,Tv,w) = Tρm(Tu,v,Tw) = Tρr(u,Tv,Tw). The derived object is the 3-tri-Leibniz algebra, a space with three trilinear brackets [·,·,·]⊣, [·,·,·]⊥, [·,·,·]⊢ satisfying five identities; the induced brackets on V are [u,v,w]⊢ = ρl(Tu,Tv,w), [u,v,w]⊥ = ρm(Tu,v,Tw), and [u,v,w]⊣ = ρr(u,Tv,Tw). The correspondence is carried by the hemisemidirect product g ⋉_{l,m,r} V: T is an embedding tensor exactly when its graph is a subalgebra of this 3-tri-Leibniz algebra, equivalently when the map (x,u) ↦ (Tu,0) is a Nijenhuis operator. For deformations, the mechanism is the cochain complex whose 1-cocycles are maps T1 satisfying Eq. (5.1), with equivalence generated by δ(a,b)u = Tρl(a,b,u) − [a,b,Tu].

What would settle it

Take a non-abelian 3-Leibniz algebra (for instance $R^{4}$ with the vector-product bracket of Example 2.5) and set T = 0 with the adjoint representation. Then every linear map T1 is a 1-cocycle, but T + tT1 is an embedding tensor only when T1 is an averaging operator; a T1 that is not an averaging operator therefore has a cohomology class but no associated linear deformation, so the claimed bijection would fail.

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Extended reading notes

Core claim

The paper's central claim is that embedding tensors on 3-Leibniz algebras are governed by 3-tri-Leibniz algebras: the representation space of an embedding tensor carries a canonical 3-tri-Leibniz structure, and conversely every 3-tri-Leibniz algebra is induced by an embedding tensor on a 3-Leibniz algebra with respect to a representation. A further claim is that homomorphic embedding tensors induce 3-tri-Leibniz dialgebras, and that linear deformations of an embedding tensor can be studied through a first cohomology group, with Theorem 5.6 asserting a bijection between equivalence classes of linear deformations and HH1_T(V,g).

Load-bearing premise

The deformation classification rests on the unstated assumption that every 1-cocycle automatically satisfies the higher-order equations that make T + tT1 an embedding tensor for every t.

Editorial extensions

If this is right

  • Every 3-tri-Leibniz algebra is realized from an embedding tensor, so structural results for 3-tri-Leibniz algebras and for embedding tensors transfer across the correspondence.
  • Any 3-tri-Leibniz algebra embeds into an averaging 3-Leibniz algebra, meaning its three brackets can be realised as the image of a single averaging operator on an ambient algebra.
  • Homomorphic embedding tensors turn the representation space into a 3-tri-Leibniz dialgebra, connecting crossed modules of 3-Leibniz algebras to dialgebra structures.
  • If Theorem 5.6's bijection holds, equivalence classes of linear deformations of an embedding tensor are completely described by the first cohomology HH1_T(V,g), and Nijenhuis elements describe exactly the trivial deformations.

Reading between the lines

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

  • A reader's check of Theorem 5.6: the converse claim that any 1-cocycle gives a linear deformation needs the higher-order equations (5.2)–(5.3). With T=0 on a non-abelian 3-Leibniz algebra in its adjoint representation, every T1 is a 1-cocycle, but T+tT1 is an embedding tensor only if T1 is an averaging operator, so the bijection would need extra conditions.
  • By analogy with embedding tensors on Lie algebras and on 3-Lie algebras, one expects an L∞-algebra whose Maurer–Cartan elements are exactly embedding tensors; the paper's cohomology would then be the cohomology of that structure, and the full deformation equations (5.1)–(5.3) would be the Maurer–Cartan equation rather than just a linear cocycle condition.
  • The dialgebra section proves one direction (a homomorphic embedding tensor gives a 3-tri-Leibniz dialgebra); the natural converse question—whether every 3-tri-Leibniz dialgebra arises this way—is not addressed and would be a direct test of how sharp the construction is.
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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

2 major / 7 minor

Summary. The paper introduces 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras, proving structural results that connect embedding tensors with Nijenhuis operators, graph subalgebras, and induced 3-tri-Leibniz algebra structures. Section 4 introduces 3-tri-Leibniz dialgebras and shows that homomorphic embedding tensors induce them. Section 5 defines 1-cocycles and a first cohomology group HH^1_T(V,g) for an embedding tensor T, and claims in Theorem 5.6 that equivalence classes of linear deformations of T are in bijection with HH^1_T(V,g).

Significance. If the main results were correct, the paper would provide a uniform higher-arity analogue of known embedding-tensor constructions and a deformation classification. Some structural observations, such as the graph and Nijenhuis characterizations in Theorems 3.8 and 3.9 and the explicit examples in Section 2 and 3, are natural and potentially useful. However, the central deformation-theoretic claim, Theorem 5.6, is false: the stated cohomology group ignores the nonlinear equations (5.2)-(5.3) that a genuine deformation generator must satisfy. Since this theorem is the paper's principal new contribution, the paper cannot be accepted in its current form.

major comments (2)
  1. [Section 5 (Theorem 5.6, Definition 5.2)] The bijection stated in Theorem 5.6 is false. Take T=0 on a non-abelian 3-Leibniz algebra g with the adjoint representation. Then Eq. (5.1) is identically zero, so every T1 in Hom(g,g) is a 1-cocycle. Moreover, the equivalence relation in Definition 5.2 is trivial because delta(a,b)u = Tρl(a,b,u) - [a,b,Tu]_g = 0 when T=0, so HH^1_T(g,g)=Hom(g,g). But Tt=tT1 is an embedding tensor only if Eq. (5.3) holds, which for the adjoint representation is exactly the averaging-operator condition (3.2): [T1u,T1v,T1w]_g = T1[T1u,T1v,w]_g = T1[T1u,v,T1w]_g = T1[u,T1v,T1w]_g. A generic linear map is not an averaging operator, so a generic 1-cocycle does not generate a linear deformation. Thus the sentence immediately before Theorem 5.6, 'Conversely, any 1-cocycle T1 gives rise to the linear deformation T+tT1', is false, and the claimed bijection between HH^1_T(V,g) and equivalence classes of linear deformations collapses.
  2. [Section 3 (Theorem 3.12)] The proof of Theorem 3.12 asserts that the subspace I_g spanned by the differences of the three brackets is an ideal of g, and then asserts without proof that the maps ρl, ρm, ρr are well-defined on the quotient and form a representation. These facts are load-bearing for the claim that every 3-tri-Leibniz algebra is induced by an embedding tensor. The well-definedness of ρm and ρr with respect to representatives of the quotient requires checking membership in I_g for each slot separately, and the ideal property itself is not immediate from axioms (2.7)-(2.11). The theorem may be true, but as written its proof is incomplete and should be supplied explicitly.
minor comments (7)
  1. [Definition 3.7] The name 'Nejinhuis operator' is a typo and should read 'Nijenhuis operator'.
  2. [Definition 5.2] The phrase 'the set of theses 1-cocycles' should read 'the set of these 1-cocycles'.
  3. [Section 5, before Theorem 5.6] The text 'a homomorphism from from ~Tt to Tt' contains a duplicated 'from'.
  4. [Equations (5.1) and (5.2)] Writing each brace-enclosed triple of equations as three separate labeled equations would improve readability and avoid ambiguity about which right-hand side corresponds to which expression.
  5. [Proposition 5.3] The proof displays only the first-order terms and writes 'mod t^2'; since a linear deformation of a 3-tri-Leibniz algebra requires the deformed brackets to satisfy all axioms for all t, the argument should explicitly use that T_t is an embedding tensor for all t to conclude that the displayed ω define a genuine deformation.
  6. [Theorem 5.8] The proof establishes the embedding-tensor identity for |t| sufficiently small and then concludes that Tt is an embedding tensor for all t; this inference should be justified by noting that Eqs. (5.1)-(5.3) are polynomial identities in t, so validity on an open interval implies validity for all t.
  7. [Remark 3.13] The notation (x,x) for elements of g3Leib ⊕ g is confusing because x denotes both an element of g and its class in g3Leib; using different symbols for the two components would clarify the formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the structural theorems are direct verifications from the definitions, and the flawed deformation converse is a non-sequitur rather than a circular identification.

full rationale

The paper's main structural results are self-contained computations from the stated definitions. Proposition 2.15 verifies the 3-tri-Leibniz identities for the hemisemidirect product using the representation axioms; Theorem 3.8 and Theorem 3.9 are direct equivalence proofs by expanding the relevant brackets and comparing with Eq. (3.1); Theorem 3.12 constructs a quotient and representation so that the quotient map is an embedding tensor, which is a realization theorem rather than a circular reduction. The deformation section defines 1-cocycles by Eq. (5.1), deformations by Eqs. (5.1)-(5.3), and equivalence by Eq. (5.13); these are distinct conditions. The sentence 'Conversely, any 1-cocycle T1 gives rise to the linear deformation T + tT1' is unjustified and, as the zero-embedding-tensor example shows, false in general; however, this is a correctness/rigor flaw, not circularity, because the paper does not define the deformation set to coincide with the cocycle set. The single self-citation [14] is motivational and not load-bearing. Overall, the derivation chain does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 4 invented entities

The paper introduces several new algebraic definitions and the theory rests on them. No free parameters are fitted. The main background is standard 3-Leibniz algebra theory. The ad hoc axioms are the new definitions and the unproved ideal/quotient claim in Theorem 3.12.

assumptions (5)
  • standard math The vector space K is a field of characteristic zero (throughout).
    States the base field assumption at the end of Section 1.
  • domain assumption 3-Leibniz algebras satisfy identity (2.1), taken from [1].
    Definition 2.1 imported from prior literature.
  • domain assumption A representation of a 3-Leibniz algebra is defined by Eqs. (2.2)-(2.6), from [1].
    Definition 2.7 imported from prior literature.
  • ad hoc to paper The subspace I_g in Theorem 3.12 is an ideal and the maps ρ_l, ρ_m, ρ_r on the quotient are well-defined.
    Asserted as 'easy to verify' without proof; the converse theorem depends on it.
  • ad hoc to paper The axioms (2.7)-(2.11) of a 3-tri-Leibniz algebra are consistent and capture the intended structure.
    New definition introduced in this paper; all subsequent results depend on this choice.
invented entities (4)
  • 3-tri-Leibniz algebra
    purpose: A ternary algebra with three brackets ⊢, ⊣, ⊥ that serves as the induced structure from an embedding tensor.
    Defined in Def. 2.10; no external confirmation beyond this paper.
  • 3-tri-Leibniz dialgebra
    purpose: A structure pairing a 3-Leibniz algebra with a 3-tri-Leibniz algebra, induced by homomorphic embedding tensors.
    Defined in Def. 4.5; no external confirmation.
  • Homomorphic embedding tensor
    purpose: An embedding tensor that is also a homomorphism, used to induce dialgebras.
    Defined in Def. 4.3; no external confirmation.
  • Nijenhuis element
    purpose: An element A=(a,b) in ∧²g used to construct trivial linear deformations of an embedding tensor.
    Defined in Def. 5.7; no external confirmation.

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Pith. "Pith review of Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations." pith.science (2026). https://pith.science/paper/3CECANXO

@misc{pith2026250204023,
  author       = {Pith},
  title        = {Pith review of: Embedding tensors on 3-Leibniz algebras and their derived algebraic structures and deformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CECANXO}},
  note         = {Machine review of arXiv:2502.04023}
}
read the original abstract

In this paper, first we introduce the notions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras. We show that an embedding tensor gives rise to a 3-tri-Leibniz algebra. Conversely, a 3-tri-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the quotient map is an embedding tensor. Furthermore, any 3-tri-Leibniz algebra can be embedded into an averaging 3-Leibniz algebra. Next, we introduce the notion of 3-tri-Leibniz dialgebras and demonstrate that homomorphic embedding tensors inherently induce 3-tri-Leibniz dialgebras. Finally, we study the linear deformations of embedding tensors by defining first cohomology.

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