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REVIEW 2 major objections 5 minor 39 references

One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation

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Pith's one-line read The paper proves that a finite skew brace is one-generator exactly when it has an indecomposable sub-q-cycle set, a transitive cycle base, that is itself generated by a single element, and that the 'every element generates' version…

desk verdict Settles the Smoktunowicz question with a clean characterization; the main proof is correct despite cosmetic typos, and the only real gap is the absent GAP script. read the letter →

arxiv 2506.12503 v1 pith:ZLAN5XIV submitted 2025-06-14 math.QA

Marco Castelli This is my paper · ORCID
classification math.QA MSC 16T2581R50
keywords Yang-Baxterequationset-theoreticsolutionskewbraceone-generatorindecomposableq-cyclesetirreduciblecycletransitivebase
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 studies finite one-generator solutions to the set-theoretic Yang-Baxter equation and their algebraic counterparts, one-generator skew braces. Its main theorem characterizes when a finite skew brace is generated by a single element in terms of an indecomposable substructure called a transitive cycle base: the element generates the brace exactly when that cycle base is itself generated by the element, and every element generates the brace exactly when the cycle base is irreducible. This settles an open question in the area, and it yields a parallel characterization for involutive solutions. A sympathetic reader should care because it turns a property of braces, generation by one element, into a property of the associated solution, indecomposability plus one-generatedness, which is easier to test and classify.

What carries the argument

The proof is carried by the notion of one-generator q-cycle sets and by Lemma 3.2, which states that for a finite skew brace and any sub-q-cycle set $X$, the additive subgroup generated by $X$ equals the multiplicative subgroup generated by $X$: $\langle X\rangle_+ = \langle X\rangle_\circ$. This lets the author rewrite additive combinations of elements of $X$ as multiplicative words, so that the smallest sub-brace $B(x)$ generated by $x$, described inductively in Proposition 3.1, can be shown to coincide with the multiplicative subgroup generated by the sub-q-cycle set $\langle x\rangle$. Proposition 2.3 gives the inductive closure description of $\langle x\rangle$, and Proposition 2.6 links irreducibility with generation by each element.

What would settle it

A concrete check is to compute, for every finite skew brace of small order, every sub-q-cycle set $X$ and compare the additive subgroup $\langle X\rangle_+$ with the multiplicative subgroup $\langle X\rangle_\circ$; any pair with $\langle X\rangle_+ \neq \langle X\rangle_\circ$ refutes Lemma 3.2 and undermines the proof of Theorem 3.5. Alternatively, find a finite skew brace with a transitive cycle base $X$ containing $x$ such that $\langle x\rangle = X$ yet $B(x)$ is a proper sub-brace, contradicting part (1) of the theorem.

Watch

Extended reading notes

Core claim

The central result is Theorem 3.5. For a finite skew brace $B$ and an indecomposable sub-q-cycle set $X$ of its associated q-cycle set: $B = B(x)$ for a given $x \in X$ holds exactly when $X$ is a transitive cycle base and $X = \langle x\rangle$; and $B = B(x)$ for every $x \in X$ holds exactly when $X$ is a transitive cycle base and $X$ is irreducible. Here $\langle x\rangle$ is the smallest sub-q-cycle set containing $x$, and a transitive cycle base is a single orbit of the group generated by the maps $\lambda_a$ and $\delta_a$ that also additively, equivalently by Lemma 3.2, multiplicatively generates $B$. The result answers a question from [36] and, specialized to braces and cycle sets, says that an indecomposable involutive solution is irreducible if and only if it is the transitive cycle base of a finite brace generated as a brace by every one of its elements.

Load-bearing premise

The load-bearing premise is Lemma 3.2, that in a finite skew brace a sub-q-cycle set generates the same subgroup additively and multiplicatively; if that equality fails, the bridge from additive combinations to multiplicative words in the proof of Theorem 3.5 breaks.

Editorial extensions

If this is right

  • Every finite irreducible cycle set arises as a transitive cycle base of a finite brace $B$ with $B = B(x)$ for all $x$ in the base, and conversely.
  • A finite brace is one-generator if and only if it has a transitive cycle base that is a one-generator cycle set.
  • There are indecomposable cycle sets of arbitrarily large size, built from a field of characteristic 2, that are not one-generator; any brace admitting one as a transitive cycle base fails to be generated by any element of that base.
  • A computer search over cycle sets of size below 10 finds exactly two indecomposable cycle sets that are not one-generator, both of size 8, with explicit counts of irreducible and one-generator non-irreducible cycle sets at sizes 8 and 9.
  • The permutation brace of one of these two exceptional size-8 cycle sets is a 32-element brace that is not one-generator, while the permutation brace of the other shows that a brace can be one-generator even when one of its transitive cycle bases is not.

Reading between the lines

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

  • A testable extension the author leaves implicit is that the same criterion gives a direct algorithm to enumerate transitive cycle bases of a brace and check one-generatedness, so the numerical classification could be extended well beyond size 9.
  • The equality $\langle X\rangle_+ = \langle X\rangle_\circ$ may hold under weaker hypotheses than finiteness, for instance in skew braces where the subgroup generated by $X$ is finitely generated, which would allow an analogue of Theorem 3.5 for infinite skew braces.
  • Because the characterization is stated in terms of orbits of the maps $\lambda$ and $\delta$, it suggests a route to classify one-generator skew braces by first classifying irreducible q-cycle sets, transferring combinatorial classification data into brace theory.
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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 / 5 minor

Summary. The paper studies finite one-generator q-cycle sets (equivalently, one-generator set-theoretic solutions of the Yang-Baxter equation) and their relationship with one-generator skew braces. Section 2 develops basic properties of one-generator and irreducible q-cycle sets, including a retraction criterion (Theorem 2.10). Section 3 contains the main result, Theorem 3.5, which characterizes when an indecomposable sub-q-cycle set X of a finite skew brace B satisfies B=B(x) for one or for all x in X. Section 4 applies the theory to involutive solutions, constructs examples of indecomposable non-one-generator cycle sets, and reports GAP computations for cycle sets of small size. The paper answers [36, Question 6.8] affirmatively.

Significance. If Theorem 3.5 is correct, it provides a clean structural bridge between one-generator skew braces and indecomposable set-theoretic solutions, extending earlier work of Rump and of Smoktunowicz and Smoktunowicz. The result is significant because it gives an affirmative answer to an open question and yields a practical criterion (Corollary 4.6) for detecting one-generator braces. The proof is built on previously published structural results (the retraction theorem, dynamical extensions, and the transitivity criterion), and I found no circularity. The main derivation is sound; however, the numerical section is not independently reproducible from the preprint because the GAP script is not included, although this does not affect the central theorem.

major comments (2)
  1. [§3, Lemma 3.2] The induction in the proof of Lemma 3.2 mixes the maps λ_{x_1^-} and λ_{x_1}. In the first direction one needs λ_{x_1^-}(x_i)=x_1·x_i ∈ X, which is closure of the sub-q-cycle set under the operation ·. In the second direction one needs λ_{x_1}(x_i)=σ_{x_1}^{-1}(x_i) ∈ X, which holds because σ_{x_1} restricts to a bijection of X; this second fact should be stated explicitly, since it is not immediate from closure under the binary operation alone.
  2. [§3, Theorem 3.5] The displayed identity δ_a(b)=λ_{a^-}(-a+b+a) in the proof of Theorem 3.5 is false in general. For the trivial skew brace on a non-abelian additive group, the left side is a+b-a while the right side is -a+b+a, and these need not coincide. The needed conclusion δ_a(b)∈B(x) follows directly from the closure of B(x) under ◦, +, and additive inverses, so the theorem survives, but the incorrect displayed formula should be removed or corrected.
minor comments (5)
  1. [§4.2] The text says the GAP code is available upon request, but the preprint does not include the code or the output tables. Please include the script or a detailed table of the enumeration data so that the numerical claims can be checked independently.
  2. [§4, Theorem 4.5] In the proof of Theorem 4.5, the wording 'for all x∈B' should read 'for all x∈X'.
  3. [§4, Proposition 4.3] The 'standard computation' verifying σ_a(C_2)=C_2 for all a∈C_2 is left to the reader; at least one representative computation and the symmetry argument should be supplied.
  4. [§3, Lemma 3.4] In the displayed expression for λ_a(b), the indices u and v are interchanged inconsistently; the intended formula is λ_a(b)=λ_{x_1}⋯λ_{x_v}(y_1)+⋯+λ_{x_1}⋯λ_{x_v}(y_u).
  5. [Throughout] There are several typographical errors, including 'investigating' in the Introduction and 'Smocktunowitcz' before Theorem 4.5; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is proved from definitions and independent lemmas, not from its own target statement.

full rationale

The central claim, Theorem 3.5, is established by an in-line chain of lemmas: Lemma 3.2 proves the equality of the additive and multiplicative subgroups generated by a finite sub-q-cycle set; Lemma 3.4 shows that when x generates the skew brace B and C = <x>, every element of B lies in the multiplicative subgroup generated by C; Lemma 3.3 uses this to show that the orbit of a generator is a transitive cycle base. The converse directions reduce to the definitions of cycle base, indecomposability, and Proposition 2.3. No fitted parameter is later called a prediction, and no assertion used in the proof is equivalent by construction to the conclusion being proved. The paper does cite earlier work of the same author, notably Theorem 1.5 from [5], but that result is a published structural decomposition theorem with an independent proof, and it is not the source of the main characterization: Theorem 3.5 is proved directly and would remain valid without relying on [5]. The minor notational slip in the proof of Lemma 3.2 (lambda_{x1} where lambda_{x1^-} is intended) is a typographical issue, not a circular identification. The omission of the GAP script in Section 4.2 is a reproducibility limitation, not a circularity. Consequently, no circular step can be exhibited from the text, and the score is 0.

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

The paper's central theorem does not introduce free parameters or fitted quantities. It does import several structural facts about q-cycle sets: the transitivity criterion for indecomposability, the retraction theorem, the dynamical-extension decomposition of indecomposable q-cycle sets (from the author's own [5]), and the fact that finite indecomposable cycle sets are transitive cycle bases of braces. All are published results; the least externally verified step is the decomposition theorem, which is also from the author's prior work.

assumptions (4)
  • domain assumption For a finite q-cycle set, indecomposability is equivalent to transitivity of the group generated by {sigma_x} and {delta_x}.
    Used in Proposition 2.4 and Lemma 3.3 to identify orbits as indecomposable sub-q-cycle sets; cited from [5,17].
  • domain assumption Ret(X) is a non-degenerate q-cycle set when X is non-degenerate.
    Imported from [24, main result]; supports absolute retraction and Theorem 2.10.
  • domain assumption If X is indecomposable and p:X to Y is an epimorphism, X is a dynamical extension of Y (Theorem 1.5).
    Cited from the author's [5, Theorem 3.3]; essential for the converse direction of Theorem 2.10.
  • domain assumption Every finite indecomposable cycle set is a transitive cycle base of a finite brace.
    From [36, Proposition 2.12] and [33, Proposition 9]; used to lift one-generator cycle sets to one-generator braces in Theorems 4.2 and 4.5.

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Cite this review

Pith. "Pith review of One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation." pith.science (2026). https://pith.science/paper/ZLAN5XIV

@misc{pith2026250612503,
  author       = {Pith},
  title        = {Pith review of: One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZLAN5XIV}},
  note         = {Machine review of arXiv:2506.12503}
}
read the original abstract

We study the class of one-generator solutions to the Yang-Baxter equation, extending some recent results concerning the classes of involutive and multipermutation solutions. Moreover we show the precise relationship between indecomposable solutions to the Yang-Baxter equation and finite one-generator skew braces, giving a positive answer to a question posed by Agata and Alicja Smoktunowicz. In the last part, we apply our results to the involutive case, and we present some numerical results involving solutions of small size.

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Works this paper leans on

39 extracted references · 16 canonical work pages

  1. [1]

    Ballester-Bolinches, R

    A. Ballester-Bolinches, R. Esteban-Romero, L. A. Kurdachenko, V. Perez-Calabuig, On the struc- ture of some left braces, International Journal of Group Theory 14 (2) (2023) 47–58. URL https://doi.org/10.22108/ijgt.2023.139145.1872

    arXiv 2023

  2. [2]

    R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972) 193–228. URL https://doi.org/10.1016/0003-4916(72)90335-1

    doi:10.1016/0003-4916(72)90335-1 1972

  3. [3]

    Castelli, A characterization of finite simple set-theoretic solutions of the Yang-Baxter equation, Proc

    M. Castelli, A characterization of finite simple set-theoretic solutions of the Yang-Baxter equation, Proc. Amer. Math. Soc. 151 (2023) 5047–5057. URL http://dx.doi.org/10.1090/proc/16329

    doi:10.1090/proc/16329 2023

  4. [4]

    Castelli, F

    M. Castelli, F. Catino, P. Stefanelli, Left non-degenerate set-theoretic solutions of the Yang-Baxter equationanddynamicalextensionsof q-cyclesets, J.AlgebraAppl.21(8)(2022)PaperNo.2250154, 22. URL https://doi.org/10.1142/S0219498822501547

    work page doi:10.1142/s0219498822501547 2022

  5. [5]

    Castelli, M

    M. Castelli, M. Mazzotta, P. Stefanelli, Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation, Forum Math. 34 (2) (2022) 531–546. URL https://doi.org/10.1515/forum-2021-0189

    doi:10.1515/forum-2021-0189 2022

  6. [6]

    Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group

    M. Castelli, S. Trappeniers, Studying solutions of the yang-baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group, arXiv preprint arXiv:2303.00581

    work page Pith review arXiv

  7. [7]

    F. Cedó, T. Gateva-Ivanova, A. Smoktunowicz, On the Yang–Baxter equation and left nilpotent left braces, J. Pure and Appl. Alg. 221 (4) (2017) 751–756. URL https://doi.org/10.1016/j.jpaa.2016.07.014

    work page doi:10.1016/j.jpaa.2016.07.014 2017

  8. [8]

    F. Cedó, J. Okniński, Indecomposable solutions of the Yang–Baxter equation of square-free cardi- nality, Adv. Math. 430 (2023) 109221. URL http://dx.doi.org/10.1016/j.aim.2023.109221

    work page arXiv 2023

Show all 39 references
  1. [9]

    F. Cedó, J. Okniński, New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces, J. Algebra 600 (2022) 125–151. URL https://doi.org/10.1016/j.jalgebra.2022.02.011

    2022 doi

  2. [10]

    Colazzo, M

    I. Colazzo, M. Ferrara, M. Trombetti, On derived-indecomposable solutions of the Yang–Baxter equation, Publ. Math. 69 (2025) 171–193. URL https://doi.org/10.5565/PUBLMAT6912508

    2025 doi

  3. [11]

    Colazzo, E

    I. Colazzo, E. Jespers, L. Kubat, A. Van Antwerpen, Simple solutions of the Yang-Baxter equation, arXiv preprint. URL https://arxiv.org/pdf/2312.09687

  4. [12]

    Dietzel, Endocabling of involutive solutions to the yang-baxter equation, with an application to solutions whose diagonal is a cyclic permutation, arXiv preprint arXiv:2504.14339

    C. Dietzel, Endocabling of involutive solutions to the yang-baxter equation, with an application to solutions whose diagonal is a cyclic permutation, arXiv preprint arXiv:2504.14339

    arXiv

  5. [13]

    Dietzel, S

    C. Dietzel, S. Properzi, S. Trappeniers, Indecomposable involutive set-theoretical solutions to the Yang–Baxter equation of sizep2, Comm. Algebra 53 (2024) 1238–1256. URL https://doi.org/10.1080/00927872.2024.2405024

    2024

  6. [14]

    M. R. Dixon, L. A. Kurdachenko, I. Y. Subbotin, On the structure of some one-generator nilpotent braces, arXiv preprint arXiv:2501.04567

    arXiv

  7. [15]

    V. G. Drinfel’d, On some unsolved problems in quantum group theory, in: Quantum groups (Leningrad, 1990), vol. 1510 of Lecture Notes in Math., Springer, Berlin, 1992, pp. 1–8. URL https://doi.org/10.1007/BFb0101175

    1990 doi

  8. [16]

    P.Etingof, T.Schedler, A.Soloviev, Set-theoreticalsolutionstotheQuantumYang-Baxterequation, Duke Math. J. 100 (2) (1999) 169–209. URL http://doi.org/10.1215/S0012-7094-99-10007-X 15

    1999 doi

  9. [17]

    Etingof, A

    P. Etingof, A. Soloviev, R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang-Baxter equation on a set with a prime number of elements, J. Algebra 242 (2) (2001) 709– 719. URL https://doi.org/10.1006/jabr.2001.8842

    2001

  10. [18]

    Gateva-Ivanova, A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation, J

    T. Gateva-Ivanova, A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation, J. Math. Phys. 45 (10) (2004) 3828–3858. URL http://dx.doi.org/10.1063/1.1788848

    2004 doi

  11. [19]

    Gateva-Ivanova, A combinatorial approach to noninvolutive set-theoretic solutions of the Yang- Baxter equation, Publ

    T. Gateva-Ivanova, A combinatorial approach to noninvolutive set-theoretic solutions of the Yang- Baxter equation, Publ. Mat. 65 (1-2) (2021) 747–808. URL https://doi.org/10.5565/PUBLMAT6522111

    2021 doi

  12. [20]

    Gateva-Ivanova, M

    T. Gateva-Ivanova, M. Van den Bergh, Semigroups of I-Type, J. Algebra 206 (1) (1998) 97–112. URL https://doi.org/10.1006/jabr.1997.7399

    1998

  13. [21]

    Guarnieri, L

    L. Guarnieri, L. Vendramin, Skew braces and the Yang-Baxter equation, Math. Comp. 86 (307) (2017) 2519–2534. URL https://doi.org/10.1090/mcom/3161

    2017 doi

  14. [22]

    Jedlicka, A

    P. Jedlicka, A. Pilitowska, Diagonals of solutions of the yang-baxter equation, Forum Math. (in press). URL https://doi.org/10.1515/forum-2024-0409

    2024 doi

  15. [23]

    Jedlička, A

    P. Jedlička, A. Pilitowska, A. Zamojska-Dzienio, Indecomposable involutive solutions of the Yang- Baxter equation of multipermutational level2 with abelian permutation group, Forum Math. 2022. URL https://doi.org/10.1515/forum-2021-0130

    2022 doi

  16. [24]

    Jedlicka, A

    P. Jedlicka, A. Pilitowska, A. Zamojska-Dzienio, The retraction relation for biracks, J. Pure Appl. Algebra 223 (8) (2019) 3594–3610. URL https://doi.org/10.1016/j.jpaa.2018.11.020

    2019 doi

  17. [25]

    Jedlička, A

    P. Jedlička, A. Pilitowska, A. Zamojska-Dzienio, Cocyclic braces and indecomposable cocyclic so- lutions of the Yang-Baxter equation, Proc. Amer. Math. Soc. 150 (10) (2022) 4223–4239. URL https://doi.org/10.1090/proc/15962

    2022 doi

  18. [26]

    URL https://doi.org/10.1007/s00013-023-01930-6

    A.Kanrar, (In)decomposabilityoffinitesolutionsoftheYang-Baxterequation, ArchivMath.122(2) (2024) 155–161. URL https://doi.org/10.1007/s00013-023-01930-6

    2024 doi

  19. [27]

    L. A. Kurdachenko, I. Y. Subbotin, On the structure of some one-generator braces, Proc. Edinburgh Math. Soc. 67 (2) (2024) 566–576. URL http://dx.doi.org/10.1017/S0013091524000154

    2024 doi

  20. [28]

    V.Lebed, L.Vendramin, Homologyofleftnon-degenerateset-theoreticsolutionstotheYang-Baxter equation, Adv. Math. 304 (2017) 1219–1261. URL https://doi.org/10.1142/S0218196716500570

    2017 doi

  21. [29]

    Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv

    W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005) 40–55. URL https://doi.org/10.1016/j.aim.2004.03.019

    2005 doi

  22. [30]

    Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J

    W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (1) (2007) 153–170. URL https://doi.org/10.1016/j.jalgebra.2006.03.040

    2007 doi

  23. [31]

    Rump, A covering theory for non-involutive set-theoretic solutions to the Yang-Baxter equation, J

    W. Rump, A covering theory for non-involutive set-theoretic solutions to the Yang-Baxter equation, J. Algebra 520 (2019) 136–170. URL https://doi.org/10.1016/j.jalgebra.2018.11.007

    2019 doi

  24. [32]

    Rump, Classification of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation, Forum Math

    W. Rump, Classification of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation, Forum Math. 32 (4) (2020) 891–903. URL https://doi.org/10.1515/forum-2019-0274

    2020 doi

  25. [33]

    Rump, One-generator braces and indecomposable set-theoretic solutions to the Yang–Baxter equation, Proc

    W. Rump, One-generator braces and indecomposable set-theoretic solutions to the Yang–Baxter equation, Proc. Edinb. Math. Soc. (2020) 1–21. URL https://doi.org/10.1017/S0013091520000073

    2020 doi

  26. [34]

    Rump, The classification of non-degenerate uniconnected cycle sets, Pac

    W. Rump, The classification of non-degenerate uniconnected cycle sets, Pac. J. Math 323 (1) (2023) 205–221. URL 10.2140/pjm.2023.323.205

    2023 doi

  27. [35]

    Rump, Primes in coverings of indecomposable involutuve set-theoretic solutions of the Yang- Baxter equation, Bull

    W. Rump, Primes in coverings of indecomposable involutuve set-theoretic solutions of the Yang- Baxter equation, Bull. Belg. Math. Soc. Simon Stevin 30 (2). URL https://doi.org/10.36045/j.bbms.230429

    doi

  28. [36]

    Smoktunowicz, A

    A. Smoktunowicz, A. Smoktunowicz, Set-theoretic solutions of the Yang-Baxter equation and new classes of R-matrices, Linear Algebra Appl. 546 (2018) 86–114. 16 URL https://doi.org/10.1016/j.laa.2018.02.001

    2018 doi

  29. [37]

    Vendramin, Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J

    L. Vendramin, Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra 220 (2016) 2064–2076. URL https://doi.org/10.1142/S1005386716000183

    2016 doi

  30. [38]

    Vendramin, A

    L. Vendramin, A. Konovalov, Combinatorial Solutions for the Yang–Baxter equation, Version 0.10.6 (GAP package) (2024). URL https://gap-packages.github.io/YangBaxter

    2024

  31. [39]

    C. N. Yang, Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction, Phys. Rev. Lett. 19 (1967) 1312–1315. URL https://link.aps.org/doi/10.1103/PhysRevLett.19.1312 17

    1967 doi

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