Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T13:09:37.678Z Has data issue: false hasContentIssue false

Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups

Published online by Cambridge University Press:  20 April 2021

Wolfgang Rump*
Affiliation:
Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550Stuttgart, Germany

Abstract

Set-theoretic solutions to the Yang–Baxter equation can be classified by their universal coverings and their fundamental groupoids. Extending previous results, universal coverings of irreducible involutive solutions are classified in the degenerate case. These solutions are described in terms of a group with a distinguished self-map. The classification in the nondegenerate case is simplified and compared with the description in the degenerate case.

Type
Article
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Angiono, I., Galindo, C., and Vendramin, L., Hopf braces and Yang–Baxter operators. Proc. Amer. Math. Soc. 145(2017), no. 5, 19811995.CrossRefGoogle Scholar
Catino, F., Colazzo, I., and Stefanelli, P., On regular subgroups of the affine group. Bull. Aust. Math. Soc. 91(2015), no. 1, 7685 10.1017/S000497271400077XCrossRefGoogle Scholar
Catino, F., Colazzo, I., and Stefanelli, P., Regular subgroups of the affine group and asymmetric product of radical braces. J. Algebra 455(2016), 164182.10.1016/j.jalgebra.2016.01.038CrossRefGoogle Scholar
Catino, F. and Rizzo, R., Regular subgroups of the affine group and radical circle algebras. Bull. Aust. Math. Soc. 79(2009), no. 1, 103107.10.1017/S0004972708001068CrossRefGoogle Scholar
Cedó, F., Jespers, E., and Okniński, J., Retractability of set theoretic solutions of the Yang–Baxter equation. Adv. Math. 224(2010), no. 6, 24722484.10.1016/j.aim.2010.02.001CrossRefGoogle Scholar
Childs, L. N., Fixed-point free endomorphisms and Hopf–Galois structures. Proc. Amer. Math. Soc. 141(2013), no. 4, 12551265.10.1090/S0002-9939-2012-11418-2CrossRefGoogle Scholar
Chouraqui, F. and Godelle, E., Finite quotients of groups of I-type. Adv. Math. 258(2014), 4668.10.1016/j.aim.2014.02.009CrossRefGoogle Scholar
Dehornoy, P., Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs. Adv. Math. 282(2015), 93127.10.1016/j.aim.2015.05.008CrossRefGoogle Scholar
Drinfeld, V. G., On some unsolved problems in quantum group theory. In: Quantum groups (Leningrad, 1990), Lecture Notes in Mathematics, 1510, Springer, Berlin, Germany, 1992, pp. 18.Google Scholar
Etingof, P., Schedler, T., and Soloviev, A., Set-theoretical solutions to the quantum Yang–Baxter equation. Duke Math. J. 100(1999), 169209.CrossRefGoogle Scholar
Featherstonhaugh, S. C., Caranti, A., and Childs, L. N., Abelian Hopf–Galois structures on prime-power Galois field extensions. Trans. Amer. Math. Soc. 364(2012), no. 7, 36753684.CrossRefGoogle Scholar
Gateva-Ivanova, T., Noetherian properties of skew-polynomial rings with binomial relations. Trans. Amer. Math. Soc. 343(1994), 203219.CrossRefGoogle Scholar
Gateva-Ivanova, T., Skew polynomial rings with binomial relations. J. Algebra 185(1996), 710753.10.1006/jabr.1996.0348CrossRefGoogle Scholar
Gateva-Ivanova, T., Quadratic algebras, Yang–Baxter equation, and Artin–Schelter regularity. Adv. Math. 230(2012), nos. 4–6, 21522175.CrossRefGoogle Scholar
Gateva-Ivanova, T. and Van den Bergh, M., Semigroups of I-type. J. Algebra 206(1998), 97112.10.1006/jabr.1997.7399CrossRefGoogle Scholar
Guarnieri, L. and Vendramin, L., Skew braces and the Yang–Baxter equation. Math. Comp. 86(2017), 25192534.CrossRefGoogle Scholar
Jacobson, N., Structure of rings. Vol. 37. American Mathematical Society Colloquium Publications, Providence, RI, 1964.Google Scholar
Lebed, V. and Vendramin, L., Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation. Adv. Math. 304(2017), 12191261.10.1016/j.aim.2016.09.024CrossRefGoogle Scholar
Lu, J. H., Yan, M., and Zhu, Y. C., On the set-theoretical Yang–Baxter equation. Duke Math. J. 104(2000), 118.CrossRefGoogle Scholar
Mac Lane, S., Homology. Springer, Berlin, Göttingen, and Heidelberg, Germany, 1963.10.1007/978-3-642-62029-4CrossRefGoogle Scholar
Rump, W., A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation. Adv. Math. 193(2005), 4055.10.1016/j.aim.2004.03.019CrossRefGoogle Scholar
Rump, W., Braces, radical rings, and the quantum Yang–Baxter equation. J. Algebra 307(2007), 153170.CrossRefGoogle Scholar
Rump, W., Right $l$ groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation. J. Algebra 439(2015), 470510.10.1016/j.jalgebra.2015.04.045CrossRefGoogle Scholar
Rump, W., Quasi-linear cycle sets and the retraction problem for set-theoretic solutions of the quantum Yang–Baxter equation. Algebra Colloq. 23(2016), no. 1, 149166.CrossRefGoogle Scholar
Rump, W., Classification of cyclic braces, II. Trans. Amer. Math. Soc. 372(2019), no. 1, 305328.10.1090/tran/7569CrossRefGoogle Scholar
Rump, W., A covering theory for non-involutive set-theoretic solutions to the Yang–Baxter equation. J. Algebra 520(2019), 136170.10.1016/j.jalgebra.2018.11.007CrossRefGoogle Scholar
Rump, W., Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation. Forum Math. 32(2020), no. 4, 891903.CrossRefGoogle Scholar
Rump, W., One-generator braces and indecomposable set-theoretic solutions to the Yang–Baxter equation. Proc. Edinb. Math. Soc. 63(2020), 676696.10.1017/S0013091520000073CrossRefGoogle Scholar
Tate, J. and Van den Bergh, M., Homological properties of Sklyanin algebras. Invent. Math. 124(1996), 619647.10.1007/s002220050065CrossRefGoogle Scholar
Weinstein, A. and Xu, P., Classical solutions of the quantum Yang–Baxter equation. Comm. Math. Phys. 148(1992), no. 2, 309343.10.1007/BF02100863CrossRefGoogle Scholar