Involutive latin solutions of the Yang-Baxter equation
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Involutive latin solutions of the Yang-Baxter equation
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA18-20123S" target="_blank" >GA18-20123S: Expanding the Scope of Universal Algebra</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Algebra
ISSN
0021-8693
e-ISSN
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Volume of the periodical
2021
Issue of the periodical within the volume
565
Country of publishing house
US - UNITED STATES
Number of pages
32
Pages from-to
128-159
UT code for WoS article
000581500500006
EID of the result in the Scopus database
2-s2.0-85090565492