Primes in coverings of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation. (English) Zbl 07789451
During the talk “Skew Braces and the Yang-Baxter Equation” held in Oberwolfach in the period 26 February–4 March 2023, Okniński raised the question of whether the primes dividing the size \(n\) of a finite indecomposable solution are related to the primes dividing the order of the associated permutation group. In the paper [Adv. Math. 430, Article ID 109221, 26 p. (2023; Zbl 1531.16029)], F. Cedó and J. Okniński proved that both prime sets are equal if \(n\) is square-free.
In this paper, the author proves that surjective morphisms of solutions admit a canonical factorization into a covering and a morphism given by an ideal of a brace. The problem of the existence of solutions with non-equality of the prime sets is reduced to the retractability problem. In addition, it is shown that the case of non-equality is possible, and an example is constructed.
In this paper, the author proves that surjective morphisms of solutions admit a canonical factorization into a covering and a morphism given by an ideal of a brace. The problem of the existence of solutions with non-equality of the prime sets is reduced to the retractability problem. In addition, it is shown that the case of non-equality is possible, and an example is constructed.
Reviewer: Marzia Mazzotta (Lecce)
MSC:
| 16T25 | Yang-Baxter equations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 08A05 | Structure theory of algebraic structures |
Citations:
Zbl 1531.16029References:
| [1] | I. Angiono, C. Galindo, L. Vendramin: Hopf braces and Yang-Baxter opera-tors, Proc. Amer. Math. Soc. 145 (2017), 1981-1995. · Zbl 1392.16032 |
| [2] | D. Bachiller: Counterexample to a conjecture about braces, J. Algebra 453 (2016), 160-176. · Zbl 1338.16022 |
| [3] | D.Bachiller: Extensions, matched products, and simple braces, J. Pure Appl. Algebra 222 (2018), 1670-1691. · Zbl 1437.20031 |
| [4] | F. Catino, I. Colazzo, P. Stefanelli: On regular subgroups of the affine group, Bull. Aust. Math. Soc. 91 (2015), 76-85. · Zbl 1314.20001 |
| [5] | F. Catino, R. Rizzo: Regular subgroups of the affine group and radical circle algebras, Bull. Aust. Math. Soc. 79 (2009), 103-107. · Zbl 1184.20001 |
| [6] | F. Cedó, E. Jespers, J. Okniński: Retractability of set theoretic solutions of the Yang-Baxter equation, Adv. Math. 224 (2010), 2472-2484. · Zbl 1192.81202 |
| [7] | F. Cedó, E. Jespers,Á. del Río: Involutive Yang-Baxter groups, Trans. Amer. Math. Soc. 362 (2010), 2541-2558. · Zbl 1188.81115 |
| [8] | F. Cedó, J. Okniński: Indecomposable solutions of the Yang-Baxter equation of square-free cardinality, arXiv:2212.06753v1[math.QA] |
| [9] | L. N. Childs: Fixed-point free endomorphisms and Hopf Galois structures, Proc. Amer. Math. Soc. 141 (2013), 1255-1265. · Zbl 1269.12003 |
| [10] | F. Chouraqui: Garside groups and Yang-Baxter equation, Comm. Algebra 38 (2010), 4441-4460. · Zbl 1216.16023 |
| [11] | F. Chouraqui, E. Godelle: Finite quotients of groups of I-type, Adv. Math. 258 (2014), 46-68. · Zbl 1344.20051 |
| [12] | P. Dehornoy: Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs, Adv. Math. 282 (2015), 93-127. · Zbl 1326.20039 |
| [13] | V. G. Drinfeld: On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer-Verlag, Berlin, 1992, 1-8. · Zbl 0765.17014 |
| [14] | P. Etingof, T. Schedler, A. Soloviev: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209. · Zbl 0969.81030 |
| [15] | M. A. Farinati, J. García Galofre: A differential bialgebra associated to a set theoretical solution of the Yang-Baxter equation, J. Pure Appl. Algebra 220 (2016), 3454-3475. · Zbl 1347.16036 |
| [16] | S. C. Featherstonhaugh, A. Caranti, L. N. Childs: Abelian Hopf Galois struc-tures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364 (2012), 3675-3684. · Zbl 1287.12002 |
| [17] | T. Gateva-Ivanova: Noetherian properties of skew-polynomial rings with binomial relations, Trans. Amer. Math. Soc. 343 (1994), 203-219. · Zbl 0807.16026 |
| [18] | T. Gateva-Ivanova: Skew Polynomial Rings with Binomial Relations, J. Algebra 185 (1996), 710-753. · Zbl 0863.16016 |
| [19] | T. Gateva-Ivanova, M. Van den Bergh: Semigroups of I-type, J. Algebra 206 (1998), 97-112. · Zbl 0944.20049 |
| [20] | L. Guarnieri, L. Vendramin: Skew braces and the Yang-Baxter equation, Math. Comp. 86 (2017), 2519-2534. · Zbl 1371.16037 |
| [21] | N. Jacobson: Structure of rings, Amer. Math. Soc. Colloq. Publ. 37, 1964. |
| [22] | V. Lebed, L. Vendramin: Cohomology and extensions of braces, Pacific J. Math. 284 (2016), 191-212. · Zbl 1357.20009 |
| [23] | V. Lebed, L. Vendramin: Homology of left non-degenerate set-theoretic so-lutions to the Yang-Baxter equation, Adv. Math. 304 (2017), 1219-1261. · Zbl 1356.16027 |
| [24] | J.-H. Lu, M. Yan, Y.-C. Zhu: On the set-theoretical Yang-Baxter equation, Duke Math. J. 104 (2000), 1-18. · Zbl 0960.16043 |
| [25] | J. Okniński: Talk on the meeting “Skew Braces and the Yang-Baxter Equa-tion”, Oberwolfach, 26 February -4 March, 2023. |
| [26] | W. Rump: A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005), 40-55. · Zbl 1074.81036 |
| [27] | W. Rump: Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), 153-170. · Zbl 1115.16022 |
| [28] | W. Rump: Construction of finite braces, Ann. Combinatorics 23 (2019), 391-416. · Zbl 1458.20025 |
| [29] | W. Rump: Classification of indecomposable involutive set-theoretic solu-tions to the Yang-Baxter equation, Forum Math. 32 (2020), 891-903. · Zbl 1446.16041 |
| [30] | W. Rump: One-generator braces and indecomposable set-theoretic solutions to the Yang-Baxter equation, Proc. Edinb. Math. Soc. 63 (2020), 676-696. · Zbl 1473.17091 |
| [31] | J. Tate, M. Van den Bergh: Homological properties of Sklyanin Algebras, Invent. Math. 124 (1996), 619-647. · Zbl 0876.17010 |
| [32] | A. Weinstein, P. Xu: Classical solutions of the quantum Yang-Baxter equa-tion, Comm. Math. Phys. 148 (1992), 309-343. · Zbl 0849.17015 |
| [33] | Institute for Algebra and Number Theory, University of Stuttgart Pfaffenwaldring 57, D-70550 Stuttgart, Germany e-mail: rump@mathematik.uni-stuttgart.de |
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