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Actions of skew braces and set-theoretic solutions of the reflection equation

Part of: Hopf algebras, quantum groups and related topics Other generalizations of groups

Published online by Cambridge University Press:  25 June 2019

K. De Commer
K. De Commer*
Affiliation:
Vakgroep Wiskunde, Vrije Universiteit Brussel (VUB), B-1050 Brussels, Belgium ([email protected])
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Abstract

A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang–Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.

Keywords

reflection equationYang–Baxter equationbracesbraided groups

MSC classification

Primary: 16T25: Yang-Baxter equations
Secondary: 20N99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1089 - 1113
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1.Balagović, M. and Kolb, S., Universal K-matrix for quantum symmetric pairs, J. Reine Angew. Math. 747 (2019), 299–353.Google Scholar
2.Baxter, R. J., Partition function of the eight-vertex lattice model, Ann. Phys. 70 (1972), 193–228.Google Scholar
3.Ben-Zvi, D., Brochier, A. and Jordan, D., Quantum character varieties and braided module categories, Selecta Math. (N.S.) 24(5) (2018), 4711–4748.Google Scholar
4.Brochier, A., Cyclotomic associators and finite type invariants for tangles in the solid torus, Algebr. Geom. Topol. 13(6) (2013), 3365–3409.Google Scholar
5.Caudrelier, V. and Zhang, Q. C., Vector nonlinear Schrödinger equation on the half-line, J. Phys. A 45(10) (2012), 105201.Google Scholar
6.Caudrelier, V., Crampé, N. and Zhang, Q. C., Set-theoretical reflection equation: classification of reflection maps, J. Phys. A 46 (2013), 095203.Google Scholar
7.Cedó, F., Jespers, E. and del Río, Á., Involutive Yang–Baxter groups, Trans. Amer. Math. Soc. 362(5) (2010), 2541–2558.Google Scholar
8.Cedó, F., Jespers, E. and Okniński, J., Retractability of set theoretic solutions of the Yang–Baxter equation, Adv. Math. 224(6) (2010), 2472–2484.Google Scholar
9.Cedó, F., Jespers, E. and Okniński, J., Braces and the Yang–Baxter equation, Comm. Math. Phys. 327(1) (2014), 101–116.Google Scholar
10.Cherednik, I., Factorizing particles on a half line, and root systems, Teoret. Mat. Fiz. 61(1) (1984), 35–44.Google Scholar
11.Dehornoy, P., Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs, Adv. Math. 282 (2015), 93–127.Google Scholar
12.tom Dieck, T., Categories of rooted cylinder ribbons and their representations, J. Reine Angew. Math. 494 (1998), 35–63.Google Scholar
13.tom Dieck, T. and Häring-Oldenburg, R., Quantum groups and cylinder braiding, Forum Math. 10(5) (1998), 619–639.Google Scholar
14.Donin, J., Kulish, P. P. and Mudrov, A. I., On a universal solution to the reflection equation, Lett. Math. Phys. 63 (2003), 179–194.Google Scholar
15.Drinfeld, V. G., Quantum groups, Proceedings of the International Congress of Mathematicians, (Berkeley, CA, 1986), Volumes 1 and 2, pp. 798–820 (American Mathematical Society, Providence, RI, 1987).Google Scholar
16.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, Berlin, 1992).Google Scholar
17.Etingof, P., Schedler, T. and Soloviev, A., Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100(2) (1999), 169–209.Google Scholar
18.Gateva-Ivanova, T., A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation, J. Math. Phys. 45(10) (2004), 3828–3858.Google Scholar
19.Gateva-Ivanova, T., Quadratic algebras, Yang–Baxter equation, and Artin–Schelter regularity, Adv. Math. 230(4–6) (2012), 2152–2175.Google Scholar
20.Gateva-Ivanova, T. and Cameron, P., Multipermutation solutions of the Yang–Baxter equation, Comm. Math. Phys. 309(3) (2012), 583–621.Google Scholar
21.Gateva-Ivanova, T. and Majid, S., Set-theoretic solutions of the Yang–Baxter equation, graphs and computations, J. Symbolic Comput. 42(11–12) (2007), 1079–1112.Google Scholar
22.Gateva-Ivanova, T. and Majid, S., Matched pairs approach to set theoretic solutions of the Yang–Baxter equation, J. Algebra 319(4) (2008), 1462–1529.Google Scholar
23.Gateva-Ivanova, T. and Van den Bergh, M., Semigroups of I-type, J. Algebra 206(1) (1998), 97–112.Google Scholar
24.Guarnieri, L. and Vendramin, L., Skew braces and the Yang–Baxter equation, Math. Comp. 86(307) (2017), 2519–2534.Google Scholar
25.Jespers, E. and Van Antwerpen, A., Left semi-braces and solutions to the Yang–Baxter equation, Forum Math. 31(1) (2018), 241–263.Google Scholar
26.Kolb, S., Quantum symmetric pairs and the reflection equation, Algebr. Represent. Theory 11(6) (2008), 519–544.Google Scholar
27.Kolb, S., Braided module categories via quantum symmetric pairs, preprint (arXiv:1705.04238, 2017).Google Scholar
28.Kolb, S. and Stokman, J., Reflection equation algebras, coideal subalgebras, and their centres, Selecta Math. (N.S.) 15(4) (2009), 621–664.Google Scholar
29.Lu, J.-H., Yan, M. and Zhu, Y.-C., On the set-theoretical Yang–Baxter equation, Duke Math. J. 104(1) (2000), 1–18.Google Scholar
30.Majid, S., Foundations of quantum group theory (Cambridge University Press, 1995).Google Scholar
31.Regelskis, V. and Vlaar, B., Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type, preprint (arXiv:1602.08471, 2017).Google Scholar
32.Rump, W., Modules over braces, Algebra Discrete Math. (2) (2006), 127–137.Google Scholar
33.Rump, W., Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307(1) (2007), 153–170.Google Scholar
34.Rump, W., The brace of a classical group, Note Mat. 34(1) (2014), 115–144.Google Scholar
35.Smoktunowicz, A. and Vendramin, L., On skew braces (with an Appendix by N. Byott and L. Vendramin), J. Comb. Algebra 2(1) (2018), 47–86.Google Scholar
36.Takeuchi, M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981), 841–882.Google Scholar
37.Takeuchi, M., Survey on matched pairs of groups—an elementary approach to the ESS-LYZ theory, Banach Center Publ. 61(1) (2003), 305–331.Google Scholar
38.Yang, C. N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312–1315.Google Scholar