Involutive latin solutions of the Yang-Baxter equation

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10436289" target="_blank" >RIV/00216208:11320/21:10436289 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=EM70T2VOhh" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=EM70T2VOhh</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jalgebra.2020.09.001" target="_blank" >10.1016/j.jalgebra.2020.09.001</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Involutive latin solutions of the Yang-Baxter equation

Popis výsledku v původním jazyce

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations L, are bijections, the squaring map is a bijection, and the identity (xy) (xz) = (yx) (yz) holds. We call these algebras rumples in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order n exists if and only if n = p(1)(p1k1) ... p(m)(pmkm) for some distinct primes p(i) and positive integers k(i). A large class of affine solutions is obtained from nonsingular near-circulant matrices A, B satisfying [A, B] = A(2). We characterize affine latin rumples as those latin rumples for which the displacement group generated by LxLy-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 LxLy-1 of their displacement group have order dividing four. (C) 2020 Elsevier Inc. All rights reserved.

Název v anglickém jazyce

Involutive latin solutions of the Yang-Baxter equation

Popis výsledku anglicky

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations L, are bijections, the squaring map is a bijection, and the identity (xy) (xz) = (yx) (yz) holds. We call these algebras rumples in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order n exists if and only if n = p(1)(p1k1) ... p(m)(pmkm) for some distinct primes p(i) and positive integers k(i). A large class of affine solutions is obtained from nonsingular near-circulant matrices A, B satisfying [A, B] = A(2). We characterize affine latin rumples as those latin rumples for which the displacement group generated by LxLy-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 LxLy-1 of their displacement group have order dividing four. (C) 2020 Elsevier Inc. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA18-20123S" target="_blank" >GA18-20123S: Rozšíření záběru univerzální algebry</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Algebra

ISSN

0021-8693

e-ISSN

—

Svazek periodika

2021

Číslo periodika v rámci svazku

565

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

32

Strana od-do

128-159

Kód UT WoS článku

000581500500006

EID výsledku v databázi Scopus

2-s2.0-85090565492