Classification of 3-Dimensional BiHom-Associative and BiHom-Bialgebras

Ahmed Zahari (IRIMAS)

arxiv: 1907.00803 · v1 · pith:ZDRJ5BPInew · submitted 2019-06-28 · 🧮 math.RA · math.AC

Classification of 3-Dimensional BiHom-Associative and BiHom-Bialgebras

Ahmed Zahari (IRIMAS) This is my paper

Pith reviewed 2026-05-25 12:44 UTC · model grok-4.3

classification 🧮 math.RA math.AC

keywords BiHom-associative algebrasBiHom-bialgebrasBiHom-Hopf algebrasclassificationlow-dimensional algebrasalgebraic varieties

0 comments

The pith

BiHom-associative algebras and their bialgebra and Hopf versions are classified up to dimension 3.

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

The paper solves the BiHom-associativity equations in dimensions 1, 2 and 3 and lists all solutions up to isomorphism. It then equips the resulting algebras with compatible co-operations to obtain the corresponding BiHom-bialgebras and BiHom-Hopf algebras. A reader cares because these low-dimensional lists make the new twisted structures concrete and allow direct computation of their properties. The classification proceeds by treating the twisting maps and the multiplication as unknowns and reducing the resulting polynomial systems.

Core claim

The authors classify all n-dimensional BiHom-associative algebras for n ≤ 3, together with the compatible BiHom-bialgebra and BiHom-Hopf structures on those algebras.

What carries the argument

The BiHom-associativity condition (a twisted associativity identity involving two linear maps alpha and beta) whose solutions define the algebraic variety of BiHom-associative algebras.

If this is right

Every BiHom-associative algebra of dimension at most three is now known by its structure constants.
The possible co-operations that turn these algebras into bialgebras or Hopf algebras are also completely listed.
The geometric properties of the corresponding algebraic varieties can be read off from the explicit lists.
Further invariants such as cohomology or representations can be computed case by case in these dimensions.

Where Pith is reading between the lines

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

The same polynomial-system approach could be attempted in dimension four, using the three-dimensional list as a base for extensions.
The classification supplies a concrete test set for any conjectured general properties of BiHom structures.
The explicit lists make it possible to check whether known families of ordinary associative or Hom-associative algebras embed into the BiHom setting in low dimensions.

Load-bearing premise

Every solution of the BiHom-associativity equations in dimensions three and lower has been found and all isomorphisms among them have been correctly removed.

What would settle it

An explicit multiplication table and pair of twisting maps on a three-dimensional vector space that satisfies BiHom-associativity yet is not isomorphic to any algebra appearing in the classification list.

read the original abstract

The purpose of this paper is to study the structure and the algebraic varieties of BiHom-associative algebras. We provide a classication of n-dimensional BiHom-associative and BiHom-bialgebras and BiHom Hopf algebras for n $\le$ 3.

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

Routine low-dimensional classification of BiHom-associative structures that adds a reference list for a narrow subfield.

read the letter

The paper enumerates BiHom-associative algebras, bialgebras, and Hopf algebras in dimensions up to 3. That is the core output: explicit lists of isomorphism classes satisfying the twisted associativity condition with two homomorphisms α and β. The work follows the standard approach of writing structure constants, imposing the polynomial identities, and solving the resulting system before quotienting by the action of GL(n). For specialists who need concrete low-dimensional examples in this corner of generalized associative algebra, the lists can serve as a reference point for further deformation or cohomology calculations. The authors also tie the structures to algebraic varieties, which is a reasonable next step after the classification. The main limitation is that the abstract gives no detail on the base field, the solution method, or verification steps, so the claim of completeness rests on whether every solution in dimension 3 was actually found and correctly reduced. If the paper only sketches the cases without showing the full case split or computer-assisted check, readers will have to redo parts of the enumeration themselves. The bialgebra and Hopf sections appear to build directly on the algebra classification, so any gap there would propagate. This is aimed at people already working on Hom-algebras or low-dimensional bialgebra classifications; it will not move the broader field. The result is solid enough on its own terms to warrant referee time, even if the impact stays narrow.

Referee Report

1 major / 1 minor

Summary. The paper studies the structure and algebraic varieties of BiHom-associative algebras, claiming to provide a classification of n-dimensional BiHom-associative algebras, BiHom-bialgebras, and BiHom-Hopf algebras for all n ≤ 3.

Significance. If the classification is exhaustive and correct over a specified base field, the result would supply an explicit catalog of low-dimensional examples of these twisted structures, serving as a reference for computations and further work on hom-type algebras and their bialgebra/Hopf extensions.

major comments (1)

[Abstract] Abstract: the central claim of a complete classification for n ≤ 3 asserts that all solutions to the BiHom-associativity equations have been found and quotiented by isomorphism, but supplies no information on the base field, the enumeration algorithm or case analysis used to solve the polynomial system on structure constants, verification steps, or handling of degenerate cases. This information is load-bearing for substantiating the classification.

minor comments (1)

[Abstract] Abstract: 'classication' is a typo and should read 'classification'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback. We address the single major comment below and will revise the abstract accordingly to strengthen the presentation of our classification results.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of a complete classification for n ≤ 3 asserts that all solutions to the BiHom-associativity equations have been found and quotiented by isomorphism, but supplies no information on the base field, the enumeration algorithm or case analysis used to solve the polynomial system on structure constants, verification steps, or handling of degenerate cases. This information is load-bearing for substantiating the classification.

Authors: We agree that the abstract, as currently written, does not mention the base field or the solution method. The classification in the paper is performed over an algebraically closed field of characteristic zero by enumerating solutions to the quadratic and higher-degree polynomial equations on the structure constants of the multiplication and the two twisting maps, with case distinctions according to the ranks and eigenvalues of the twisting maps; isomorphisms are quotiented by the natural action of GL(n). Verification consists of direct substitution back into the BiHom-associativity identity and explicit listing of representatives up to isomorphism. We will revise the abstract to state the base field and to indicate that the classification proceeds by exhaustive case analysis of the structure-constant equations. A short paragraph summarizing the algorithmic approach and degeneracy handling will also be added to the introduction. revision: yes

Circularity Check

0 steps flagged

No circularity: direct enumeration of solutions to polynomial identities

full rationale

The paper's central claim is a classification obtained by solving the BiHom-associativity equations (a system of polynomial identities on structure constants) in dimensions ≤3 and then quotienting by the action of GL(n) to obtain isomorphism classes. This is a self-contained computational enumeration with no derivation chain, no fitted parameters renamed as predictions, and no load-bearing self-citations. The reader's assessment of score 0.0 is confirmed; the work does not reduce any result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification rests on the standard definition of BiHom-associativity together with the usual axioms of vector spaces and algebras over a field; no free parameters or new entities are introduced.

axioms (1)

domain assumption BiHom-associativity identity holds for the multiplication and the two homomorphisms
This identity is the central defining relation used to generate the equations whose solutions are classified.

pith-pipeline@v0.9.0 · 5555 in / 1040 out tokens · 49887 ms · 2026-05-25T12:44:33.931056+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

Requiring the algebra structure to be BiHom-associative and unital gives rise to sub-variety H_n (resp. Hun) of k^{n^3+2n^2}. Base changes in A result in the natural transport of the structure action of GL_n(k) on H_n.

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

12 extracted references · 12 canonical work pages

[1]

Graziani, A

G. Graziani, A. Makhlouf, C. Menini, F. Panaite BiHom-asosociative algebras, BiHom-Lie algebras and BiHo m- bialgebras, Symetry, Integrability and gepmerty , SIGMA 11(2015), 086 34 pages

work page 2015
[2]

Armour, H

A. Armour, H. Chen and Y. Zhang, Classiﬁcation of 4-dimensional superalgebras , Comm. in Algebra 37(2009), 3697-3728

work page 2009
[3]

Goze and A

M. Goze and A. Makhlouf, Classiﬁcation and rigid associative algebras in low dimensi ons, Lois d’Algèbres et variétés algébriques (Colmar, 1991), 5-22, Travaux en cours, 50 Hermann, Paris, (1996)

work page 1991
[4]

Goze, and A

M. Goze, and A. Makhlouf, On the rigid complex associative algebra , Comm. Algebra 18(12)(1990)4031-4046

work page 1990
[5]

Hartwig, D

J.T. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using σ- derivations , J. Algebra 295 (2006), 314Ð361

work page 2006
[6]

Larsson and S

D. Larsson and S. Silvestrov, Quasi-hom-Lie algebras, central extensions and 2-cocycle - like identities , J. Algebra 288 (2005), 321–344

work page 2005
[7]

Li, Structures of multiplicative Hom-Lie algebras , Advances in Mathematics (China), 43(6)(2014)817-823

X.X. Li, Structures of multiplicative Hom-Lie algebras , Advances in Mathematics (China), 43(6)(2014)817-823

work page 2014
[8]

Makhlouf and S

A. Makhlouf and S. Silvestrov, Notes on formal deformations of Hom-associative and Hom-Li e algebras , Forum Mathematicum, vol. 22 (4) (2010) 715–759

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[9]

Makhlouf and S

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

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Yau, Enveloping algebra of Hom-Lie algebras , J

D. Yau, Enveloping algebra of Hom-Lie algebras , J. Gen. Lie Theory Appl. 2 (2008), 95–108

work page 2008
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Yau, Hom-algebras and homology , J

D. Yau, Hom-algebras and homology , J. Lie Theory, 19 (2009), 409–421

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[12]

Sheng, Representations of hom-Lie algebras , Algebras and Representation Theory 15 (2012), 1081–1098

Y. Sheng, Representations of hom-Lie algebras , Algebras and Representation Theory 15 (2012), 1081–1098. Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, rue des Frères Lumière F-68093 Mulhouse, France E-mail address : zaharymaths@gmail.com

work page 2012

[1] [1]

Graziani, A

G. Graziani, A. Makhlouf, C. Menini, F. Panaite BiHom-asosociative algebras, BiHom-Lie algebras and BiHo m- bialgebras, Symetry, Integrability and gepmerty , SIGMA 11(2015), 086 34 pages

work page 2015

[2] [2]

Armour, H

A. Armour, H. Chen and Y. Zhang, Classiﬁcation of 4-dimensional superalgebras , Comm. in Algebra 37(2009), 3697-3728

work page 2009

[3] [3]

Goze and A

M. Goze and A. Makhlouf, Classiﬁcation and rigid associative algebras in low dimensi ons, Lois d’Algèbres et variétés algébriques (Colmar, 1991), 5-22, Travaux en cours, 50 Hermann, Paris, (1996)

work page 1991

[4] [4]

Goze, and A

M. Goze, and A. Makhlouf, On the rigid complex associative algebra , Comm. Algebra 18(12)(1990)4031-4046

work page 1990

[5] [5]

Hartwig, D

J.T. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using σ- derivations , J. Algebra 295 (2006), 314Ð361

work page 2006

[6] [6]

Larsson and S

D. Larsson and S. Silvestrov, Quasi-hom-Lie algebras, central extensions and 2-cocycle - like identities , J. Algebra 288 (2005), 321–344

work page 2005

[7] [7]

Li, Structures of multiplicative Hom-Lie algebras , Advances in Mathematics (China), 43(6)(2014)817-823

X.X. Li, Structures of multiplicative Hom-Lie algebras , Advances in Mathematics (China), 43(6)(2014)817-823

work page 2014

[8] [8]

Makhlouf and S

A. Makhlouf and S. Silvestrov, Notes on formal deformations of Hom-associative and Hom-Li e algebras , Forum Mathematicum, vol. 22 (4) (2010) 715–759

work page 2010

[9] [9]

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

[10] [10]

Yau, Enveloping algebra of Hom-Lie algebras , J

D. Yau, Enveloping algebra of Hom-Lie algebras , J. Gen. Lie Theory Appl. 2 (2008), 95–108

work page 2008

[11] [11]

Yau, Hom-algebras and homology , J

D. Yau, Hom-algebras and homology , J. Lie Theory, 19 (2009), 409–421

work page 2009

[12] [12]

Sheng, Representations of hom-Lie algebras , Algebras and Representation Theory 15 (2012), 1081–1098

Y. Sheng, Representations of hom-Lie algebras , Algebras and Representation Theory 15 (2012), 1081–1098. Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, rue des Frères Lumière F-68093 Mulhouse, France E-mail address : zaharymaths@gmail.com

work page 2012