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Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups. (English) Zbl 1437.16028

Matched pairs of groups were introduced and studied by M. Takeuchi [Commun. Algebra 9, 841–882 (1981; Zbl 0456.16011)] in connection with Hopf algebras, later also in the context of set-theoretic solutions to the Yang-Baxter equation [Banach Cent. Publ. 61, 305–331 (2003; Zbl 1066.16044)]. The structure group of an involutive solution carries a ring-like structure which makes it into a brace, generalizing the concept of radical ring. The paper under review studies braces as braided groups with an involutive braiding, which have also been called “symmetric groups”.
If the brace associated to a finite solution \(X\) is a radical ring \(A\) of nilpotency class \(m>1\), the multipermutation level of \(X\) is shown to satisfy the inequality \(m-1\le\operatorname{mpl}X\le m\), with \(\operatorname{mpl}X=m\) for a square-free solution.
Reviewer’s remark: Note that the connection between braided groups and (not necessarily involutive) solutions to the Yang-Baxter equation is fundamental in the concept of braided tensor category (see [A. Joyal and R. Street, Adv. Math. 102, No. 1, 20–78 (1993; Zbl 0817.18007)]).

MSC:

16T25 Yang-Baxter equations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16S37 Quadratic and Koszul algebras
20B35 Subgroups of symmetric groups
81R60 Noncommutative geometry in quantum theory
16W50 Graded rings and modules (associative rings and algebras)
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References:

[1] Angiono, I.; Galindo, C.; Vendramin, L., Hopf braces and Yang-Baxter operators, (Proceedings of the American Mathematical Society, vol. 145, (2017)), 1981-1995 · Zbl 1392.16032
[2] Bachiller, D.; Cedó, F., A family of solutions of the Yang-Baxter equation, J. Algebra, 412, 218-229, (2014) · Zbl 1303.16036
[3] Bachiller, D.; Cedó, F.; Jespers, E., Solutions of the Yang-Baxter equation associated with a left brace, J. Algebra, 463, 80-102, (2016) · Zbl 1348.16027
[4] Bachiller, D.; Cedó, F.; Jespers, E.; Okniński, J., A family of irretractable square-free solutions of the Yang-Baxter equation, Forum Math., vol. 29, 1291-1306, (2017), De Gruyter · Zbl 1394.16041
[5] Carter, J. S.; Elhamdadi, M.; Saito, M., Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles, Fund. Math., 184, 31-54, (2004) · Zbl 1067.57006
[6] Cedó, F.; Gateva-Ivanova, T.; Smoktunowicz, A., On the Yang-Baxter equation and left nilpotent left braces, J. Pure Appl. Algebra, 221, 751-756, (2017) · Zbl 1397.16033
[7] Cedóó, F.; Gateva-Ivanova, T.; Smoktunowicz, A., Braces and symmetric groups with special conditions, J. Pure Appl. Algebra, 222, 3877-3890, (2018) · Zbl 1427.16027
[8] Cedó, F.; Jespers, E.; Okniński, J., The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition, Proc. Amer. Math. Soc., 134, 653-663, (2006) · Zbl 1092.16014
[9] Cedó, F.; Jespers, E.; Okniński, J., Retractability of set theoretic solutions of the Yang Baxter equation, Adv. Math., 224, 2472-2484, (2010) · Zbl 1192.81202
[10] Cedó, F.; Jespers, E.; Okniński, J., Braces and the Yang-Baxter equation, Comm. Math. Phys., 327, 101-116, (2014) · Zbl 1287.81062
[11] Drinfeld, V., On some unsolved problems in quantum group theory, (Lecture Notes in Mathematics, vol. 1510, (1992)), 1-8 · Zbl 0765.17014
[12] Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 169-209, (1999) · Zbl 0969.81030
[13] Gateva-Ivanova, T., Noetherian properties of skew polynomial rings with binomial relations, Trans. Amer. Math. Soc., 343, 203-219, (1994) · Zbl 0807.16026
[14] Gateva-Ivanova, T., Skew polynomial rings with binomial relations, J. Algebra, 185, 710-753, (1996) · Zbl 0863.16016
[15] Gateva-Ivanova, T.; Jespers, E.; Okninski, J., Quadratic algebras of skew type and the underlying semigroups, J. Algebra, 270, 635-659, (2003) · Zbl 1054.16024
[16] Gateva-Ivanova, T., A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation, J. Math. Phys., 45, 3828-3858, (2004) · Zbl 1065.16037
[17] Gateva-Ivanova, T., Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity, Adv. in Math., 230, 2152-2175, (2012) · Zbl 1267.81209
[18] Gateva-Ivanova, T., Set-theoretic solutions of the Yang-Baxter equation, braces, and symmetric groups, (31 Aug. 2015), v2, pp. 1-47
[19] Gateva-Ivanova, T., A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation, (2018)
[20] Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of I-type, J. Algebra, 206, 97-112, (1998) · Zbl 0944.20049
[21] Gateva-Ivanova, T.; Cameron, P. J., Multipermutation solutions of the Yang-Baxter equation, Comm. Math. Phys., 309, 583-621, (2012) · Zbl 1247.81211
[22] Gateva-Ivanova, T.; Majid, S., Set theoretic solutions of the Yang-Baxter equations, graphs and computations, J. Symbolic Comput., 42, 1079-1112, (2007) · Zbl 1150.17014
[23] Gateva-Ivanova, T.; Majid, S., Matched pairs approach to set theoretic solutions of the Yang-Baxter equation, J. Algebra, 319, 1462-1529, (2008) · Zbl 1140.16016
[24] Gateva-Ivanova, T.; Majid, S., Quantum spaces associated to multipermutation solutions of level two, Algebr. Represent. Theory, 14, 341-376, (2011) · Zbl 1241.81106
[25] Jespers, E.; Okniński, J., Quadratic algebras of skew type satisfying the cyclic condition, Internat. J. Algebra Comput., 14, 479-498, (2004) · Zbl 1069.16027
[26] Kac, G. I.; Paljutkin, V. G., Finite ring groups, Trans. Amer. Math. Soc., 15, 251-294, (1966) · Zbl 0218.43005
[27] Lam, T. Y., A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, (2001), Springer-Verlag New York · Zbl 0980.16001
[28] Lebed, V.; Vendramin, L., Cohomology and extensions of braces, Pacific J. Math., 284, 191-212, (2016) · Zbl 1357.20009
[29] Lebed, V.; Vendramin, L., Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. in Math., 304, 1219-1261, (2017) · Zbl 1356.16027
[30] Lu, J.; Yan, M.; Zhu, Y., On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1-18, (2000) · Zbl 0960.16043
[31] Majid, S., Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math., 141, 311-332, (1990) · Zbl 0735.17017
[32] Majid, S., Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrosspoduct construction, J. Algebra, 130, 17-64, (1990) · Zbl 0694.16008
[33] Majid, S., More examples of bicrossproduct and double cross product Hopf algebras, Israel J. Math., 72, 133-148, (1990) · Zbl 0725.17015
[34] Majid, S., Foundations of quantum group theory, (1995), Cambridge Univ. Press · Zbl 0857.17009
[35] Matsumoto, D. K., Dynamical braces and dynamical Yang-Baxter maps, J. Pure Appl. Algebra, 217, 195-206, (2013) · Zbl 1266.81112
[36] Matsumoto, D. K.; Shibukawa, Y., Quantum Yang-Baxter equation, braided semigroups, and dynamical Yang-Baxter maps, Tokyo J. Math., 38, 227-237, (2015) · Zbl 1332.16028
[37] Rump, W., Modules over braces, Algebra Discrete Math., 2, 127-137, (2006) · Zbl 1164.81328
[38] Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 153-170, (2007) · Zbl 1115.16022
[39] Rump, W., The brace of a classical group, Note Mat., 34, 115-144, (2014) · Zbl 1344.14029
[40] Smoktunowicz, A., A note on set-theoretic solutions of the Yang-Baxter equation, J. Algebra, 500, 3-18, (2018) · Zbl 1442.16037
[41] Smoktunowicz, A., On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation, Trans. Amer. Math. Soc., 370, 6535-6564, (2018) · Zbl 1440.16040
[42] Takeuchi, M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra, 9, 841-882, (1981) · Zbl 0456.16011
[43] Takeuchi, M., Survey on matched pairs of groups. an elementary approach to the ESS-LYZ theory, Banach Center Publ., 61, 305-331, (2003) · Zbl 1066.16044
[44] Vendramin, L., Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of gateva-ivanova, J. Pure Appl. Algebra, 220, 2064-2076, (2016) · Zbl 1337.16028
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