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

Part of: Hopf algebras, quantum groups and related topics Other nonassociative rings and algebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  06 May 2020

Wolfgang Rump
Wolfgang Rump*
Affiliation:
Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550Stuttgart, Germany ([email protected])
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Abstract

An unexpected relationship between indecomposable involutive set-theoretic solutions to the Yang–Baxter equation and one-generator braces has recently been discovered by Agata and Alicja Smoktunowicz. We extend these results and answer three open questions which arose in this context.

Keywords

Yang–Baxter equationindecomposable solutionbracecycle set

MSC classification

Primary: 17D99: None of the above, but in this section
Secondary: 16T25: Yang-Baxter equations 81R50: Quantum groups and related algebraic methods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 676 - 696
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Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Footnotes

Dedicated to B. V. M.

References

1.Andruskiewitsch, N. and Graña, M., From racks to pointed Hopf algebras, Adv. Math. 178(2) (2003), 177–243.CrossRefGoogle Scholar
2.Angiono, I., Galindo, C. and Vendramin, L., Hopf braces and Yang–Baxter operators, Proc. Amer. Math. Soc. 145(5) (2017), 1981–1995.CrossRefGoogle Scholar
3.Brieskorn, E. and Saito, K., Artin–Gruppen und Coxeter–Gruppen, Invent. Math. 17 (1972), 245–271.10.1007/BF01406235CrossRefGoogle Scholar
4.Carter, J. S., Elhamdadi, M. and Saito, M., Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles, Fund. Math. 184 (2004), 31–54.CrossRefGoogle Scholar
5.Castelli, M., Catino, F. and Pinto, G., Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation, J. Pure Appl. Algebra 223 (2019), 4477–4493.CrossRefGoogle Scholar
6.Cedó, F., Jespers, E. and Okniński, J., Retractability of set theoretic solutions of the Yang–Baxter equation, Adv. Math. 224(6) (2010), 2472–2484.CrossRefGoogle Scholar
7.Cedó, F., Jespers, E. and del Río, Á., Involutive Yang–Baxter groups, Trans. Amer. Math. Soc. 362(5) (2010), 2541–2558.CrossRefGoogle Scholar
8.Cedó, F., Smoktunowicz, A. and Vendramin, L., Skew left braces of nilpotent type, Proc. Lond. Math. Soc. 118(6) (2019), 1367–1392.10.1112/plms.12209CrossRefGoogle Scholar
9.Childs, L. N., Fixed-point free endomorphisms and Hopf Galois structures, Proc. Amer. Math. Soc. 141(4) (2013), 1255–1265.CrossRefGoogle Scholar
10.Chouraqui, F., Garside groups and Yang–Baxter equation, Commun. Algebra 38(12) (2010), 4441–4460.CrossRefGoogle Scholar
11.Chouraqui, F. and Godelle, E., Finite quotients of groups of I-type, Adv. Math. 258 (2014), 46–68.CrossRefGoogle Scholar
12.Dehornoy, P., Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs, Adv. Math. 282 (2015), 93–127.CrossRefGoogle Scholar
13.Dehornoy, P. and Paris, L., Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. Lond. Math. Soc. 79(3) (1999), 569–604.CrossRefGoogle Scholar
14.Deligne, P., Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302.CrossRefGoogle Scholar
15.Drinfeld, V. G., On some unsolved problems in quantum group theory, in Quantum groups (Leningrad, 1990), Lecture Notes in Mathematics, Volume 1510, pp. 1–8 (Springer-Verlag, Berlin, 1992).Google Scholar
16.Eisermann, M., Quandle coverings and their Galois correspondence, Fund. Math. 225(1) (2014), 103–168.CrossRefGoogle Scholar
17.Etingof, P. and Graña, M., On rack cohomology, J. Pure Appl. Algebra 177(1) (2003), 49–59.CrossRefGoogle Scholar
18.Etingof, P., Schedler, T. and Soloviev, A., Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999), 169–209.CrossRefGoogle Scholar
19.Etingof, P., Soloviev, A. and Guralnick, R., 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.CrossRefGoogle Scholar
20.Farinati, M. A. and García Galofre, J., A differential bialgebra associated to a set theoretical solution of the Yang–Baxter equation, J. Pure Appl. Algebra 220(10) (2016), 3454–3475.CrossRefGoogle Scholar
21.Featherstonhaugh, S. C., Caranti, A. and Childs, L. N., Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364(7) (2012), 3675–3684.CrossRefGoogle Scholar
22.Fenn, R. and Rourke, C., Racks and links in codimension two, J. Knot Theor. Ramif. 1(4) (1992), 343–406.CrossRefGoogle Scholar
23.Gateva-Ivanova, T., Noetherian properties of skew-polynomial rings with binomial relations, Trans. Amer. Math. Soc. 343 (1994), 203–219.CrossRefGoogle Scholar
24.Gateva-Ivanova, T., Quadratic algebras, Yang–Baxter equation, and Artin–Schelter regularity, Adv. Math. 230(4-6) (2012), 2152–2175.10.1016/j.aim.2012.04.016CrossRefGoogle Scholar
25.Gateva-Ivanova, T. and Van den Bergh, M., Semigroups of I-type, J. Algebra 206 (1998), 97–112.CrossRefGoogle Scholar
26.Guarnieri, L. and Vendramin, L., Skew braces and the Yang–Baxter equation, Math. Comp. 86 (2017), 2519–2534.CrossRefGoogle Scholar
27.Hilton, P. J. and Stammbach, U., A course in homological algebra, Graduate Texts in Mathematics, Volume 4 (Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
28.Jacobson, N., Structure of rings, American Mathematical Society Colloquium Publication, Volume 37 (American Mathematical Society, Providence, RI, 1974).Google Scholar
29.Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23(1) (1982), 37–65.CrossRefGoogle Scholar
30.Kassel, C., Quantum groups, Graduate Texts in Mathematics, Volume 155 (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
31.Lu, J.-H., Yan, M. and Zhu, Y.-C., On the set-theoretical Yang–Baxter equation, Duke Math. J. 104 (2000), 1–18.Google Scholar
32.Rump, W., A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005), 40–55.CrossRefGoogle Scholar
33.Rump, W., Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307 (2007), 153–170.CrossRefGoogle Scholar
34.Rump, W., Right l-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation, J. Algebra 439 (2015), 470–510.CrossRefGoogle Scholar
35.Rump, W., A covering theory for non-involutive set-theoretic solutions to the Yang–Baxter equation, J. Algebra 520 (2019), 136–170.CrossRefGoogle Scholar
36.Rump, W., Construction of finite braces, Ann. Comb. 23 (2019), 391–416.CrossRefGoogle Scholar
37.Rump, W., Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation, Forum Math. DOI:10.1515/forum-2019-0274.Google Scholar
38.Smoktunowicz, A. and Smoktunowicz, A., Set-theoretic solutions of the Yang–Baxter equation and new classes of R-matrices, Linear Algebra Appl. 546 (2018), 86–114.CrossRefGoogle Scholar
39.Takeuchi, M., Survey on matched pairs of groups – an elementary approach to the ESS-LYZ theory, in Noncommutative geometry and quantum groups, Banach Center Publications, Volume 61, pp. 305–331 (Polish Academy of Sciences, Warsaw, 2003).CrossRefGoogle Scholar
40.Tate, J. and Van den Bergh, M., Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), 619–647.CrossRefGoogle Scholar
41.Vendramin, L., Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra 220(5) (2016), 2064–2076.CrossRefGoogle Scholar
42.Veselov, A. O., Yang–Baxter maps and integrable dynamics, Phys. Lett. A 314(3) (2003), 214–221.CrossRefGoogle Scholar
43.Weinstein, A. and Xu, P., Classical solutions of the quantum Yang–Baxter equation, Commun. Math. Phys. 148(2) (1992), 309–343.10.1007/BF02100863CrossRefGoogle Scholar