Involutive Latin solutions of the Yang-Baxter equation. (English) Zbl 1460.16036

Bonatto, Marco; Kinyon, Michael; Stanovský, David; Vojtěchovský, Petr

J. Algebra 565, 128-159 (2021).

Authors’ abstract: We prove that an affine latin rumple of order \(n\) exists if and only if \(n = p_1^{p_1k_1}\cdots p_m^{p_mk_m}\) for some distinct primes \(p_i\) and positive integers \(k_i\). We characterize affine latin rumples as those latin rumples for which the displacement group generated by \(L_xL_y^{-1}\) is abelian and normal in the group generated by all translations.We develop the extension theory of rumples sufficiently to obtain examples of latin rumples that are not affine, not even isotopic to a group. Finally, we investigate latin rumples in which the dual identity \((zx)(yx) = (zy)(xy)\) holds as well, and we show, among other results, that the generators \(L_xL_y^{-1}\) of their displacement group have order dividing four.

Reviewer: J. N. Alonso Alvarez (Vigo)

MSC:

16T25	Yang-Baxter equations
20N05	Loops, quasigroups

Keywords:

quantum Yang-Baxter equation; nondegenerate involutive solution; involutive Latin solution; cycle set; affine quasigroup

Software:

OEIS; GAP

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