Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation

References

[1] I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang–Baxter operators, Proc. Amer. Math. Soc. 145 (2017), no. 5, 1981–1995. 10.1090/proc/13395Suche in Google Scholar

[2] D. Bachiller, F. Cedó and E. Jespers, Solutions of the Yang–Baxter equation associated with a left brace, J. Algebra 463 (2016), 80–102. 10.1016/j.jalgebra.2016.05.024Suche in Google Scholar

[3] R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193–228. 10.1142/9789812798336_0003Suche in Google Scholar

[4] N. Ben David and Y. Ginosar, On groups of I-type and involutive Yang–Baxter groups, J. Algebra 458 (2016), 197–206. 10.1016/j.jalgebra.2016.03.025Suche in Google Scholar

[5] J. S. Carter, M. Elhamdadi and M. Saito, Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles, Fund. Math. 184 (2004), 31–54. 10.4064/fm184-0-3Suche in Google Scholar

[6] M. Castelli, F. Catino and G. Pinto, Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation, J. Pure Appl. Algebra 223 (2019), no. 10, 4477–4493. 10.1016/j.jpaa.2019.01.017Suche in Google Scholar

[7] F. Catino, I. Colazzo and P. Stefanelli, On regular subgroups of the affine group, Bull. Aust. Math. Soc. 91 (2015), no. 1, 76–85. 10.1017/S000497271400077XSuche in Google Scholar

[8] F. Catino, I. Colazzo and P. Stefanelli, Regular subgroups of the affine group and asymmetric product of radical braces, J. Algebra 455 (2016), 164–182. 10.1016/j.jalgebra.2016.01.038Suche in Google Scholar

[9] F. Catino and R. Rizzo, Regular subgroups of the affine group and radical circle algebras, Bull. Aust. Math. Soc. 79 (2009), no. 1, 103–107. 10.1017/S0004972708001068Suche in Google Scholar

[10] F. Cedó, E. Jespers and A. del Río, Involutive Yang–Baxter groups, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2541–2558. 10.1090/S0002-9947-09-04927-7Suche in Google Scholar

[11] F. Cedó, E. Jespers and J. Okniński, Retractability of set theoretic solutions of the Yang–Baxter equation, Adv. Math. 224 (2010), no. 6, 2472–2484. 10.1016/j.aim.2010.02.001Suche in Google Scholar

[12] L. N. Childs, Fixed-point free endomorphisms and Hopf Galois structures, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1255–1265. 10.1090/S0002-9939-2012-11418-2Suche in Google Scholar

[13] F. Chouraqui and E. Godelle, Finite quotients of groups of I-type, Adv. Math. 258 (2014), 46–68. 10.1016/j.aim.2014.02.009Suche in Google Scholar

[14] M. R. Darnel, Theory of Lattice-Ordered Groups, Monogr. Textb. Pure Appl. Math. 187, Marcel Dekker, New York, 1995. Suche in Google Scholar

[15] P. Dehornoy, Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs, Adv. Math. 282 (2015), 93–127. 10.1016/j.aim.2015.05.008Suche in Google Scholar

[16] V. G. Drinfel’d, On some unsolved problems in quantum group theory, Quantum Groups (Leningrad 1990), Lecture Notes in Math. 1510, Springer, Berlin (1992), 1–8. 10.1007/BFb0101175Suche in Google Scholar

[17] M. Eisermann, Quandle coverings and their Galois correspondence, Fund. Math. 225 (2014), no. 1, 103–168. 10.4064/fm225-1-7Suche in Google Scholar

[18] P. Etingof and M. Graña, On rack cohomology, J. Pure Appl. Algebra 177 (2003), no. 1, 49–59. 10.1016/S0022-4049(02)00159-7Suche in Google Scholar

[19] P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999), no. 2, 169–209. 10.1215/S0012-7094-99-10007-XSuche in Google Scholar

[20] P. Etingof, A. Soloviev and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang–Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001), no. 2, 709–719. 10.1006/jabr.2001.8842Suche in Google Scholar

[21] M. A. Farinati and J. García Galofre, A differential bialgebra associated to a set theoretical solution of the Yang–Baxter equation, J. Pure Appl. Algebra 220 (2016), no. 10, 3454–3475. 10.1016/j.jpaa.2016.04.010Suche in Google Scholar

[22] S. C. Featherstonhaugh, A. Caranti and L. N. Childs, Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3675–3684. 10.1090/S0002-9947-2012-05503-6Suche in Google Scholar

[23] T. Gateva-Ivanova, Noetherian properties of skew polynomial rings with binomial relations, Trans. Amer. Math. Soc. 343 (1994), no. 1, 203–219. 10.1090/S0002-9947-1994-1173854-3Suche in Google Scholar

[24] T. Gateva-Ivanova, Skew polynomial rings with binomial relations, J. Algebra 185 (1996), no. 3, 710–753. 10.1006/jabr.1996.0348Suche in Google Scholar

[25] T. Gateva-Ivanova, Quadratic algebras, Yang–Baxter equation, and Artin–Schelter regularity, Adv. Math. 230 (2012), no. 4–6, 2152–2175. 10.1016/j.aim.2012.04.016Suche in Google Scholar

[26] T. Gateva-Ivanova and M. Van den Bergh, Semigroups of I-type, J. Algebra 206 (1998), no. 1, 97–112. 10.1006/jabr.1997.7399Suche in Google Scholar

[27] L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comp. 86 (2017), no. 307, 2519–2534. 10.1090/mcom/3161Suche in Google Scholar

[28] P. Hall, On the Sylow systems of a soluble group, Proc. London Math. Soc. (2) 43 (1937), no. 4, 316–323. 10.1112/plms/s2-43.4.316Suche in Google Scholar

[29] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ. 37, American Mathematical Society, Providence, 1964. Suche in Google Scholar

[30] E. Jespers and J. Okniński, Monoids and groups of I-type, Algebr. Represent. Theory 8 (2005), no. 5, 709–729. 10.1007/s10468-005-0342-7Suche in Google Scholar

[31] V. Lebed, Plactic monoids: A braided approach, preprint (2016), https://arxiv.org/abs/1612.05768. 10.1016/j.jalgebra.2020.08.010Suche in Google Scholar

[32] V. Lebed, Cohomology of idempotent braidings with applications to factorizable monoids, Internat. J. Algebra Comput. 27 (2017), no. 4, 421–454. 10.1142/S0218196717500229Suche in Google Scholar

[33] V. Lebed and L. Vendramin, Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation, Adv. Math. 304 (2017), 1219–1261. 10.1016/j.aim.2016.09.024Suche in Google Scholar

[34] J.-H. Lu, M. Yan and Y.-C. Zhu, On the set-theoretical Yang–Baxter equation, Duke Math. J. 104 (2000), no. 1, 1–18. 10.1215/S0012-7094-00-10411-5Suche in Google Scholar

[35] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings. With the Cooperation of L. W. Small, revised ed., Grad. Texts in Math. 30, American Mathematical Society, Providence, 2001. 10.1090/gsm/030/02Suche in Google Scholar

[36] D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math. 80, Springer, New York, 1996. 10.1007/978-1-4419-8594-1Suche in Google Scholar

[37] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005), no. 1, 40–55. 10.1016/j.aim.2004.03.019Suche in Google Scholar

[38] W. Rump, Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307 (2007), no. 1, 153–170. 10.1016/j.jalgebra.2006.03.040Suche in Google Scholar

[39] W. Rump, Classification of cyclic braces, J. Pure Appl. Algebra 209 (2007), no. 3, 671–685. 10.1016/j.jpaa.2006.07.001Suche in Google Scholar

[40] W. Rump, Right l-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation, J. Algebra 439 (2015), 470–510. 10.1016/j.jalgebra.2015.04.045Suche in Google Scholar

[41] W. Rump, A covering theory for non-involutive set-theoretic solutions to the Yang–Baxter equation, J. Algebra 520 (2019), 136–170. 10.1016/j.jalgebra.2018.11.007Suche in Google Scholar

[42] W. Rump, Classification of cyclic braces, II, Trans. Amer. Math. Soc. 372 (2019), no. 1, 305–328. 10.1090/tran/7569Suche in Google Scholar

[43] W. Rump, Construction of finite braces, Ann. Comb. 23 (2019), no. 2, 391–416. 10.1007/s00026-019-00430-1Suche in Google Scholar

[44] A. Smoktunowicz and A. Smoktunowicz, Set-theoretic solutions of the Yang–Baxter equation and new classes of R-matrices, Linear Algebra Appl. 546 (2018), 86–114. 10.1016/j.laa.2018.02.001Suche in Google Scholar

[45] J. Tate and M. van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), no. 1–3, 619–647. 10.1007/s002220050065Suche in Google Scholar

[46] L. Vendramin, Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra 220 (2016), no. 5, 2064–2076. 10.1016/j.jpaa.2015.10.018Suche in Google Scholar

[47] 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. 10.1103/PhysRevLett.19.1312Suche in Google Scholar