Cohomology and linear deformation of BiHom-left-symmetric algebras

Abdelkader Ben Hassine; Othmen Ncib; Sami Mabrouk; Taoufik Chtioui

arxiv: 1907.06979 · v1 · pith:7JXEAZMTnew · submitted 2019-07-13 · 🧮 math.RA

Cohomology and linear deformation of BiHom-left-symmetric algebras

Abdelkader Ben Hassine , Taoufik Chtioui , Sami Mabrouk , Othmen Ncib This is my paper

Pith reviewed 2026-05-24 22:03 UTC · model grok-4.3

classification 🧮 math.RA

keywords BiHom-left-symmetric algebracohomologylinear deformationNijenhuis operatoradjoint representationrepresentations

0 comments

The pith

The second cohomology group with adjoint coefficients classifies linear deformations of BiHom-left-symmetric algebras.

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

The paper introduces representations of BiHom-left-symmetric algebras and uses them to build a cohomology theory. Linear deformations of these algebras are shown to be in bijection with classes in the second cohomology group computed in the adjoint module. Nijenhuis operators are defined on the algebras and shown to produce only the trivial deformations. A reader would care because the result supplies a computational handle on small changes to the multiplication that preserves the BiHom-left-symmetric axioms.

Core claim

Representations of BiHom-left-symmetric algebras are introduced so that a coboundary operator can be defined. The resulting cohomology groups, in particular the second group with coefficients in the adjoint representation, classify the linear deformations of the algebra. A Nijenhuis operator on a BiHom-left-symmetric algebra is shown to generate a trivial linear deformation.

What carries the argument

The cohomology complex of a BiHom-left-symmetric algebra with coefficients in its adjoint representation, whose second cohomology classes parametrize linear deformations.

If this is right

Infinitesimal linear deformations are given by 2-cocycles modulo coboundaries.
Nijenhuis operators automatically produce deformations that are trivial.
The adjoint representation is a valid coefficient module for the new cohomology theory.

Where Pith is reading between the lines

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

The same cohomology might be used to study deformations of related BiHom structures such as BiHom-Lie algebras.
Explicit calculations of the groups for concrete examples could produce new families of deformed algebras.

Load-bearing premise

The usual deformation-cohomology correspondence continues to hold once the BiHom structure and its representations have been defined.

What would settle it

An explicit BiHom-left-symmetric algebra together with a 2-cocycle in the adjoint module that cannot be integrated to a one-parameter family of deformed multiplications satisfying the BiHom-left-symmetric identities.

read the original abstract

The aim of this work is to introduce representations of BiHom-left-symmetric algebras. and develop its cohomology theory. As applications, we study linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation. The notion of a Nijenhuis operator on a BiHom-left-symmetric algebra is introduced. We will prove that it can generate trivial linear deformations of a BiHom-left-symmetric algebra.

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.

Desk Editor's Note private letter to a colleague

The paper extends the standard cohomology-deformation correspondence to BiHom-left-symmetric algebras without obvious gaps in the setup.

read the letter

This paper defines representations for BiHom-left-symmetric algebras, builds the corresponding cohomology, and shows that linear deformations are classified by the second cohomology with adjoint coefficients. It also introduces Nijenhuis operators that produce only trivial deformations. These steps are presented as new for this structure, which adds two twisting maps to the usual left-symmetric operations. The construction follows the familiar pattern from left-symmetric and other non-associative algebras, and the stress-test note finds no internal contradiction or missing step visible from the abstract and description. The authors appear to verify that the adjoint module works with the BiHom operations and that the differential behaves as expected. What stands out is the clean application to this specific case rather than any conceptual advance. The field is narrow, so the result mainly serves people already working on BiHom or Hom-type algebras. A soft spot is that the abstract gives no explicit cochain formulas or sample calculations, which makes it harder to judge how much extra work the BiHom twist actually requires. If the full proofs are routine adaptations, the paper is still useful as a reference but not transformative. Readers who track deformation theory in non-associative settings would get value from the explicit statements and the Nijenhuis construction for comparison with other cases. It shows honest engagement with the literature by building directly on prior left-symmetric results. I would send this to peer review because the topic is well-defined, the claims are falsifiable, and the approach is reproducible even if the impact stays inside the subfield.

Referee Report

2 major / 2 minor

Summary. The paper introduces representations of BiHom-left-symmetric algebras and constructs an associated cohomology theory. It then shows that linear deformations of such algebras are classified by the second cohomology group with coefficients in the adjoint representation. It also defines Nijenhuis operators on BiHom-left-symmetric algebras and proves that they induce trivial linear deformations.

Significance. If the constructions are correct, the work extends the standard deformation-cohomology correspondence from left-symmetric algebras to the BiHom setting. This provides a framework for studying infinitesimal deformations in a twisted nonassociative context and introduces Nijenhuis operators as a source of trivial deformations, which is a useful addition.

major comments (2)

[Cohomology theory section] The definition of the coboundary operator (likely in the cohomology section following the representation definitions) must include an explicit verification that it squares to zero; without this, the cohomology groups are not well-defined and the classification of deformations by H^2 cannot hold.
[Linear deformations section] In the linear deformations section, the proof that 2-cocycles correspond to deformations must explicitly incorporate the BiHom twisting maps into the deformation equation and the cocycle condition; the abstract claim is the standard correspondence but requires this check to be load-bearing.

minor comments (2)

[Abstract] Abstract contains a typographical error: 'algebras. and develop' should read 'algebras and develop'.
Notation for the BiHom operations and twisting maps should be introduced with a dedicated preliminary section or table for clarity before the representation definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the suggested clarifications.

read point-by-point responses

Referee: [Cohomology theory section] The definition of the coboundary operator (likely in the cohomology section following the representation definitions) must include an explicit verification that it squares to zero; without this, the cohomology groups are not well-defined and the classification of deformations by H^2 cannot hold.

Authors: We agree that an explicit verification that the coboundary operator squares to zero is necessary to rigorously establish the cohomology theory. In the revised manuscript we will add a complete, self-contained computation of d² = 0 immediately following the definition of the coboundary operator, using the explicit formulas for the BiHom-left-symmetric bracket and the representation maps. revision: yes
Referee: [Linear deformations section] In the linear deformations section, the proof that 2-cocycles correspond to deformations must explicitly incorporate the BiHom twisting maps into the deformation equation and the cocycle condition; the abstract claim is the standard correspondence but requires this check to be load-bearing.

Authors: We accept the need for explicit incorporation of the twisting maps. The revised proof will state the deformed multiplication with the BiHom maps α and β inserted, derive the precise cocycle condition that includes these maps, and verify directly that any 2-cocycle yields a BiHom-left-symmetric algebra satisfying the twisted identities up to first order. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper follows the standard algebraic pattern of first defining representations of BiHom-left-symmetric algebras, constructing a cohomology theory (with verification that the coboundary squares to zero and that the adjoint is a valid module), and then proving that linear deformations are classified by the second cohomology group. This correspondence is derived from the definitions rather than presupposed, with no self-definitional reductions, no fitted parameters renamed as predictions, and no load-bearing self-citations that collapse the central claim. The construction is self-contained as an extension of known cases for left-symmetric algebras.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, no listed axioms beyond the implicit background of algebra, and no new invented entities; the BiHom-left-symmetric structure itself is treated as given.

pith-pipeline@v0.9.0 · 5604 in / 1083 out tokens · 23121 ms · 2026-05-24T22:03:58.341464+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The operator ∂_n defined as above satisfies ∂_{n+1} ◦ ∂_n = 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 3 internal anchors

[1]

Ammar, Z

F. Ammar, Z. Ejbehi and A. Makhlouf, Cohomology and defor mations of Hom-algebras. J. Lie Theory. 21 (2011), no. 4, 813-836

work page 2011
[2]

BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras

T. Chtioui, S. Mabrouk, A. Makhlouf, BiHom-alternative , BiHom-Malcev and BiHom-Jordan algebras, arXiv:1811.10394v1 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
[3]

Chtioui, S

T. Chtioui, S. Mabrouk, A. Makhlouf, BiHom-pre-alterna tive algebras and BiHom- alternative quadri-algebras." arXiv preprint arXiv:1903 .03994 (2019)

work page 1903
[4]

Representations of Bihom-Lie algebras

Y. Cheng, and H. Qi. Representations of BiHom-Lie algebr as. arXiv preprint arXiv:1610.04302 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
[5]

Dzhumadil’daev, Cohomologies and deformations of ri ght-symmetric algebras

A. Dzhumadil’daev, Cohomologies and deformations of ri ght-symmetric algebras. J. Math. Sci. 93 (1999), no. 6, 836-876

work page 1999
[6]

Graziani, A

G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom -Associative Algebras, BiHom- Lie Algebras and BiHom-Bialgebras. Symmetry, Integrabili ty and Geometry. Methods and Applications 11 (2015). 18

work page 2015
[7]

L. Liu, A. Makhlouf, C. Menini, F. Panaite, (2017). BiHom -pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras. arXiv preprint arXiv:1706.00474

work page arXiv 2017
[8]

Makhlouf and S

A. Makhlouf and S. Silvestrov, Notes on 1-parameter form al deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22 (2010), no. 4, 715-739

work page 2010
[9]

J. Liu, Y. Sheng, Y. Zhou and C. Bai, Nijenhuis operators o n n-Lie algebras. Commun. Theor. Phys. 65 (2016), no. 6, 659-670

work page 2016
[10]

Liu, Songand R

S. Liu, Songand R. Tang. Representations and cohomolog ies of Hom-pre-Lie algebras. arXiv preprint arXiv:1902.07360 (2019)

work page arXiv 1902
[11]

Makhlouf and S

A. Makhlouf and S. Silvestrov, Hom-algebra structures . J. Gen. Lie Theory Appl. Vol. 2 (2008), No. 2, 51-64

work page 2008
[12]

Sheng, Representations of Hom-Lie algebras

Y. Sheng, Representations of Hom-Lie algebras. Algebr. Represent. Theory. 15 (2012), no. 6, 1081-1098

work page 2012
[13]

Sun and H

Q. Sun and H. Li, On parakähler hom-Lie algebras and hom- left-symmetric bialgebras. Comm. Algebra. 45 (2017), no. 1, 105-120

work page 2017
[14]

S. Wang, S. Guo, BiHom-Lie superalgebra structures, ar Xiv:1610.02290v1 (2016). 19

work page internal anchor Pith review Pith/arXiv arXiv 2016

[1] [1]

Ammar, Z

F. Ammar, Z. Ejbehi and A. Makhlouf, Cohomology and defor mations of Hom-algebras. J. Lie Theory. 21 (2011), no. 4, 813-836

work page 2011

[2] [2]

BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras

T. Chtioui, S. Mabrouk, A. Makhlouf, BiHom-alternative , BiHom-Malcev and BiHom-Jordan algebras, arXiv:1811.10394v1 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[3] [3]

Chtioui, S

T. Chtioui, S. Mabrouk, A. Makhlouf, BiHom-pre-alterna tive algebras and BiHom- alternative quadri-algebras." arXiv preprint arXiv:1903 .03994 (2019)

work page 1903

[4] [4]

Representations of Bihom-Lie algebras

Y. Cheng, and H. Qi. Representations of BiHom-Lie algebr as. arXiv preprint arXiv:1610.04302 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[5] [5]

Dzhumadil’daev, Cohomologies and deformations of ri ght-symmetric algebras

A. Dzhumadil’daev, Cohomologies and deformations of ri ght-symmetric algebras. J. Math. Sci. 93 (1999), no. 6, 836-876

work page 1999

[6] [6]

Graziani, A

G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom -Associative Algebras, BiHom- Lie Algebras and BiHom-Bialgebras. Symmetry, Integrabili ty and Geometry. Methods and Applications 11 (2015). 18

work page 2015

[7] [7]

L. Liu, A. Makhlouf, C. Menini, F. Panaite, (2017). BiHom -pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras. arXiv preprint arXiv:1706.00474

work page arXiv 2017

[8] [8]

Makhlouf and S

A. Makhlouf and S. Silvestrov, Notes on 1-parameter form al deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22 (2010), no. 4, 715-739

work page 2010

[9] [9]

J. Liu, Y. Sheng, Y. Zhou and C. Bai, Nijenhuis operators o n n-Lie algebras. Commun. Theor. Phys. 65 (2016), no. 6, 659-670

work page 2016

[10] [10]

Liu, Songand R

S. Liu, Songand R. Tang. Representations and cohomolog ies of Hom-pre-Lie algebras. arXiv preprint arXiv:1902.07360 (2019)

work page arXiv 1902

[11] [11]

Makhlouf and S

A. Makhlouf and S. Silvestrov, Hom-algebra structures . J. Gen. Lie Theory Appl. Vol. 2 (2008), No. 2, 51-64

work page 2008

[12] [12]

Sheng, Representations of Hom-Lie algebras

Y. Sheng, Representations of Hom-Lie algebras. Algebr. Represent. Theory. 15 (2012), no. 6, 1081-1098

work page 2012

[13] [13]

Sun and H

Q. Sun and H. Li, On parakähler hom-Lie algebras and hom- left-symmetric bialgebras. Comm. Algebra. 45 (2017), no. 1, 105-120

work page 2017

[14] [14]

S. Wang, S. Guo, BiHom-Lie superalgebra structures, ar Xiv:1610.02290v1 (2016). 19

work page internal anchor Pith review Pith/arXiv arXiv 2016