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arxiv: 2605.25576 · v1 · pith:LY2STJFCnew · submitted 2026-05-25 · 🧮 math.RT · math.KT· math.RA

Factorizations, classifying complements problem and deformation maps for Lie-Yamaguti algebras

Apurba Das This is my paper

Pith reviewed 2026-06-29 19:46 UTC · model grok-4.3

classification 🧮 math.RT math.KTmath.RA
keywords Lie-Yamaguti algebrasdeformation mapscomplements classificationbicrossed productscohomologyL-infinity algebrasfactorization problem
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The pith

Deformation maps classify all complements to a subalgebra in a Lie-Yamaguti algebra and unify several standard operators.

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

The paper examines factorizations of Lie-Yamaguti algebras through bicrossed products. Given an inclusion of one such algebra inside a larger one together with one strong complement, every other complement arises from the fixed one by a deformation map. These maps generalize homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators. The work defines a single cohomology theory that recovers all the earlier ones and builds an L-infinity algebra whose Maurer-Cartan elements control linear deformations of the map.

Core claim

Given an inclusion g ⊂ E of Lie-Yamaguti algebras and a strong g-complement h, any other g-complement in E is isomorphic to h by some deformation map r: h → g. A deformation map generalizes homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators on Lie-Yamaguti algebras. We define the cohomology of a deformation map unifying the cohomologies of all the operators mentioned above and construct the governing L∞-algebra of r that controls the linear deformations of r.

What carries the argument

The deformation map r: h → g from a fixed strong complement to the subalgebra, which supplies the isomorphism between any two complements and extends the listed classical operators.

If this is right

  • All g-complements are classified up to isomorphism once a single strong complement and its deformation maps are known.
  • The new cohomology recovers and unifies the separate cohomology theories for homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators.
  • Linear deformations of any deformation map are parametrized by the Maurer-Cartan elements of its governing L∞-algebra.
  • Factorizations of Lie-Yamaguti algebras correspond exactly to bicrossed-product constructions.

Where Pith is reading between the lines

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

  • The same deformation-map technique may classify complements in other varieties of non-associative algebras.
  • The L∞-algebra construction supplies a deformation-theoretic setting that could be compared with existing theories for Lie algebras or Lie triple systems.
  • Maurer-Cartan solutions might produce new invariants or obstructions for extensions of Lie-Yamaguti algebras.

Load-bearing premise

At least one strong g-complement exists so that the bicrossed-product construction and the definition of deformation maps can proceed.

What would settle it

An explicit inclusion g ⊂ E together with two strong g-complements that cannot be connected by any deformation map would refute the classification statement.

read the original abstract

A Lie-Yamaguti algebra is a non-associative algebraic structure that generalizes both Lie algebras and Lie triple systems. We first consider the factorization problem for Lie-Yamaguti algebras that essentially related to the bicrossed product of Lie-Yamaguti algebras. Next, given an inclusion $\mathfrak{g} \subset E$ of Lie-Yamaguti algebras and a strong $\mathfrak{g}$-complement $\mathfrak{h}$, we describe and classify all $\mathfrak{g}$-complements in $E$. In particular, we show that any other $\mathfrak{g}$-complement in $E$ is isomorphic to $\mathfrak{h}$ by some deformation map $r: \mathfrak{h} \rightarrow \mathfrak{g}$. Despite this importance, it turns out that a deformation map generalizes homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators on Lie-Yamaguti algebras. We define the cohomology of a deformation map unifying the cohomologies of all the operators mentioned above. Finally, we provide a Maurer-Cartan characterization and construct the governing $L_\infty$-algebra of a deformation map $r$ that controls the linear deformations of $r$.

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 / 3 minor

Summary. The paper examines the factorization problem for Lie-Yamaguti algebras in connection with bicrossed products. Given an inclusion g ⊂ E of Lie-Yamaguti algebras together with a fixed strong g-complement h, it classifies all g-complements in E by proving that each is isomorphic to h via a deformation map r : h → g. The manuscript shows that deformation maps generalize homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators; it introduces a cohomology theory for deformation maps that unifies the corresponding cohomologies and constructs the governing L∞-algebra whose Maurer-Cartan elements control the linear deformations of r.

Significance. If the stated theorems hold, the work supplies a unified algebraic framework that extends known classification and deformation results from Lie algebras and Lie triple systems to the broader setting of Lie-Yamaguti algebras. The explicit construction of the governing L∞-algebra is a concrete strength, as it furnishes a homotopical mechanism for controlling deformations once a strong complement is given.

minor comments (3)
  1. The precise definition of a 'strong g-complement' (used in the classification statement) should be recalled or cross-referenced at the beginning of the section that states the main isomorphism theorem, to make the hypothesis self-contained for readers.
  2. Notation for the bicrossed product and the action maps appearing in the deformation-map axioms should be introduced uniformly in a preliminary subsection, rather than piecemeal, to improve readability of the subsequent cohomology construction.
  3. The abstract claims that the L∞-algebra 'controls the linear deformations of r'; a brief sentence in the introduction indicating which Maurer-Cartan equation encodes these deformations would help orient the reader before the technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its unifying framework for deformation maps in Lie-Yamaguti algebras, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines Lie-Yamaguti algebras, bicrossed products, strong complements, and deformation maps explicitly from the standard non-associative identities. The classification result (any g-complement isomorphic to a fixed strong h via a deformation map r) is a direct consequence of these definitions and the bicrossed-product construction; it does not reduce any claimed prediction or uniqueness statement to a fitted parameter, self-citation chain, or renamed input. No load-bearing self-citations, imported uniqueness theorems, or ansatzes appear in the derivation chain. The constructions are self-contained against the algebraic axioms and therefore receive the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard axioms of Lie-Yamaguti algebras (already present in the literature) and on the existence of a strong complement; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Lie-Yamaguti algebra identities (bilinear operations satisfying the defining relations of a Lie-Yamaguti algebra)
    Invoked throughout to guarantee that bicrossed products and deformation maps are well-defined.

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Reference graph

Works this paper leans on

30 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    A. L. Agore, Classifying complements for associative algebras,Linear Algebra Appl.446 (2014), 345-355

    2014

  2. [2]

    A. L. Agore and G. Militaru, Classifying complements for Hopf algebras and Lie algebras,J. Algebra391 (2013), 193-208

    2013

  3. [3]

    A. L. Agore and G. Militaru, Unified products for Leibniz algebras. Applications,Linear Algebra Appl.439 (2013), 2609-2633

    2013

  4. [4]

    Benito, C

    P. Benito, C. Draper and A. Elduque, Lie-Yamaguti algebras related tog 2,J. Pure Appl. Algebra202 (2005), 22-54

    2005

  5. [5]

    Benito, A

    P. Benito, A. Elduque and F. Mart´ ın-Herce, Irreducible Lie-Yamaguti algebras,J. Pure Appl. Algebra213 (2009), 795-808

    2009

  6. [6]

    Das, Associative-Yamaguti algebras, arXiv:2509.03648 [math.RA]

    A. Das, Associative-Yamaguti algebras, arXiv:2509.03648 [math.RA]

    work page arXiv

  7. [7]

    Getzler, Lie theory of nilpotentL ∞-algebras,Ann

    E. Getzler, Lie theory of nilpotentL ∞-algebras,Ann. Math. (2)170 (2009), 271-301

    2009

  8. [8]

    Goswami, S

    S. Goswami, S. K. Mishra and G. Mukherjee, Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem,J. Algebra641 (2024), 268-306

    2024

  9. [9]

    Jacobson, Lie and Jordan triple systems,Amer

    N. Jacobson, Lie and Jordan triple systems,Amer. J. Math.71 (1949), 149-170

    1949

  10. [10]

    Deformation maps of quasi-twilled Lie algebras

    J. Jiang, Y. Sheng and R. Tang, Deformation maps of quasi-twilled Lie algebras, arXiv:2405.02532 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    Kikkawa, Remarks on solvability of Lie triple algebras,Mem

    M. Kikkawa, Remarks on solvability of Lie triple algebras,Mem. Fac. Sci. Shimane Univ.13 (1979), 17–22

    1979

  12. [12]

    M. K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids, and multiplications on reductive homo- geneous spaces,Amer. J. Matth.123 (2001), 525-550

    2001

  13. [13]

    W. G. Lister, A structure theory of Lie triple systems,Trans. Amer. Math. Soc.72, no. 2 (1952), 217-242

    1952

  14. [14]

    Majid, Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction,J

    S. Majid, Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction,J. Algebra130 (1990), 17-64

    1990

  15. [15]

    Meyberg, Lectures on algebras and triple systems, Lecture notes, Charlottesville VA, University of Virginia (1972)

    K. Meyberg, Lectures on algebras and triple systems, Lecture notes, Charlottesville VA, University of Virginia (1972)

    1972

  16. [16]

    Mondal and R

    B. Mondal and R. Saha, Deformation cohomology of morphisms of Lie-Yamaguti algebras,J. Algebra658 (2024), 660-685

    2024

  17. [17]

    Nomizu, Invariant affine connections on homogeneous spaces,Amer

    K. Nomizu, Invariant affine connections on homogeneous spaces,Amer. J. Math.76 (1954), 33-65

    1954

  18. [18]

    A. A. Sagle, On simple algebras obtained from homogeneous general Lie triple systems,Pacific J. Math.15 (1965), 1397–1400

    1965

  19. [19]

    A. A. Sagle, A note on simple anti-commutative algebras obtained from reductive homogeneous spaces,Nagoya Math. J.31 (1968), 105-124

    1968

  20. [20]

    Sheng and J

    Y. Sheng and J. Zhao, Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras, Comm. Algebra50 (2022), 4056-4073

    2022

  21. [21]

    Smirnov, Imbedding of Lie triple systems into Lie algebras,J

    O. Smirnov, Imbedding of Lie triple systems into Lie algebras,J. Algebra341 (2011), 1-12

    2011

  22. [22]

    Sun and S

    Q. Sun and S. Chen, Representations and cohomologies of differential Lie-Yamaguti algebras with any weights, J. Lie Theory33 (2023), No. 2, 641-662

    2023

  23. [23]

    Takahashi, Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras,J

    N. Takahashi, Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras,J. Algebra 664 (2025), 452-483

    2025

  24. [24]

    Voronov, Higher derived brackets and homotopy algebras,J

    Th. Voronov, Higher derived brackets and homotopy algebras,J. Pure Appl. Algebra202 (2005), 133-153

    2005

  25. [25]

    Yamaguti, On the Lie triple system and its generalization,J

    K. Yamaguti, On the Lie triple system and its generalization,J. Sci. Hiroshima Univ. Ser. A21, no. 3 (1958), 155-160

    1958

  26. [26]

    Yamaguti, On cohomology groups of general Lie triple systems,Kumamoto J

    K. Yamaguti, On cohomology groups of general Lie triple systems,Kumamoto J. Sci. Ser. A8, no. 4 (1969), 135-146. F ACTORIZATIONS, CLASSIFYING COMPLEMENTS PROBLEM AND DEFORMATION MAPS 21

    1969

  27. [27]

    Zhang and J

    T. Zhang and J. Li, Deformations and extensions of Lie-Yamaguti algebras,Linear Multilin. Algebra63, no. 11 (2015), 2212-2231

    2015

  28. [28]

    Zhao and Y

    J. Zhao and Y. Qiao, Maurer-Cartan characterization,L ∞-algebras, and cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras, arXiv:2310.05360 [math.RA]

    work page arXiv

  29. [29]

    Zhao and Y

    J. Zhao and Y. Qiao, The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras,Sci. Sin Math.53 (2023), 1303-1324

    2023

  30. [30]

    J. Zhao, S. Xu and Y. Qiao, Post Lie-Yamaguti algebras, relative Rota-Baxter operators of nonzero weights, and their deformations, arXiv:2304.06324 [math.RA]. Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India. Email address:apurbadas348@gmail.com, apurbadas348@maths.iitkgp.ac.in

    work page arXiv