Set-Theoretical Solutions of the $$n$$ -Simplex Equation

Siberian Advances in Mathematics Pub Date : 2024-03-11 DOI:10.1134/s1055134424010012

V. G. Bardakov, B. B. Chuzhinov, I. A. Emelyanenkov, M. E. Ivanov, T. A. Kozlovskaya, V. E. Leshkov

{"title":"Set-Theoretical Solutions of the $$n$$ -Simplex Equation","authors":"V. G. Bardakov, B. B. Chuzhinov, I. A. Emelyanenkov, M. E. Ivanov, T. A. Kozlovskaya, V. E. Leshkov","doi":"10.1134/s1055134424010012","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3> The \$n \$-simplex equation was introduced by Zamolodchikov\nas a generalization of the Yang–Baxter equation which becomes the \$2 \$-simplex equation in this terms. In the present\narticle, we suggest general approaches to construction of solutions of the \$n \$-simplex equation, describe certain types of\nsolutions, and introduce an operation that allows us to construct, under certain conditions,\na solution of the \$(n + m + k)\$-simplex equation from solutions of the\n\$(n + k) \$-simplex equation and \$(m + k) \$-simplex equation. We consider the tropicalization\nof rational solutions and discuss its generalizations. We prove that a solution of the\n\$n \$-simplex equation on \$G \$ can be constructed from solutions of this equation\non \$H \$ and \$K \$ if \$G \$ is an extension of a group \$H \$ by a group \$K \$. We also find solutions of the parametric\nYang–Baxter equation on \$H\$ with parameters in\n\$K \$. We introduce ternary algebras for studying\nthe 3-simplex equation and present examples of such algebras that provide us with solutions of\nthe 3-simplex equation. We find all elementary verbal solutions of the 3-simplex equation on a free\ngroup. \$|| \$\n","PeriodicalId":39997,"journal":{"name":"Siberian Advances in Mathematics","volume":"56 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siberian Advances in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1055134424010012","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

The $n $-simplex equation was introduced by Zamolodchikov as a generalization of the Yang–Baxter equation which becomes the $2 $-simplex equation in this terms. In the present article, we suggest general approaches to construction of solutions of the $n $-simplex equation, describe certain types of solutions, and introduce an operation that allows us to construct, under certain conditions, a solution of the $(n + m + k)$-simplex equation from solutions of the $(n + k) $-simplex equation and $(m + k) $-simplex equation. We consider the tropicalization of rational solutions and discuss its generalizations. We prove that a solution of the $n $-simplex equation on $G $ can be constructed from solutions of this equation on $H $ and $K $ if $G $ is an extension of a group $H $ by a group $K $. We also find solutions of the parametric Yang–Baxter equation on $H$ with parameters in $K $. We introduce ternary algebras for studying the 3-simplex equation and present examples of such algebras that provide us with solutions of the 3-simplex equation. We find all elementary verbal solutions of the 3-simplex equation on a free group. $|| $

Abstract Image

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

$$n$$ -Simplex 方程的集合论解法

摘要 $n $-simplex 方程是由 Zamolodchikov 作为 Yang-Baxter 方程的广义化引入的，在此条件下成为 $2 $-simplex 方程。在本文中，我们提出了构建（n）-二元一次方程解的一般方法，描述了某些类型的解，并介绍了一种运算，这种运算允许我们在特定条件下，从（（n + k））-二元一次方程和（（m + k））-二元一次方程的解中构建（（n + m + k））-二元一次方程的解。我们考虑了有理解的热带化，并讨论了它的一般化。我们证明，如果$G$是一个群$H$由一个群$K$的延伸，那么$G$上的$n$-二元方程的解可以从这个方程在$H$和$K$上的解构造出来。我们还找到了参数Yang-Baxter方程在(H\)上的解（参数在(K\)中）。我们引入了用于研究三元复数方程的三元代数，并举例说明了这些代数为我们提供了三元复数方程的解。我们找到了 3-复数方程在自由组合上的所有基本言解。$|| $

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Siberian Advances in Mathematics Mathematics-Mathematics (all)

CiteScore

0.70

自引率

0.00%

发文量

期刊介绍： Siberian Advances in Mathematics is a journal that publishes articles on fundamental and applied mathematics. It covers a broad spectrum of subjects: algebra and logic, real and complex analysis, functional analysis, differential equations, mathematical physics, geometry and topology, probability and mathematical statistics, mathematical cybernetics, mathematical economics, mathematical problems of geophysics and tomography, numerical methods, and optimization theory.