论文标题
单纯方程的设置理论解决方案
Set-theoretical solutions of simplex equations
论文作者
论文摘要
A. B. Zamolodchikov引入了$ n $ -simplex方程（$ n $ -SE），作为Yang-baxter方程的概括，即$ 2 $ -Simplex方程。在本文中，我们建议一些通用方法来构建$ n $ -simplex方程的解决方案，描述某些类型的解决方案，引入一种操作，在某些条件下，该操作使我们可以从$（N+M+K）$ - SE构建$（N+K）$ - SE和$（M+K）$的解决方案的解决方案。我们考虑了理性解决方案的热带化，并讨论了一种概括它的方法。我们证明，如果一个组$ g $是组$ k $的集团$ h $的延伸，那么我们可以从$ h $和$ k $上的该方程式的$ g $上找到$ n $ se的解决方案。此外，我们在$ h $上找到了$ h $的参数杨巴克斯特方程的解决方案，其中$ k $。为了研究三个简单方程的解决方案，我们介绍了具有三元操作的代数系统，并提供了这些系统的示例，这些系统提供了3美元的解决方案。我们在免费组上找到了$ 3 $ -SE的所有基本语言解决方案。
The $n$-simplex equation ($n$-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang--Baxter equation, which is the $2$-simplex equation in these terms. In the present paper we suggest some general approaches to constructing solutions of $n$-simplex equations, describe some types of solutions, introduce an operation which under some conditions allows us to construct a solution of $(n+m+k)$-SE from solution of $(n+k)$-SE and $(m+k)$-SE. We consider the tropicalization of rational solutions and discuss a way to generalize it. We prove that if a group $G$ is an extension of a group $H$ by a group $K$, then we can find a solution of the $n$-SE on $G$ from solutions of this equation on $H$ and on $K$. Also, we find solutions of the parametric Yang-Baxter equation on $H$ with parameters in $K$. For studying solutions of the 3-simplex equations we introduce algebraic systems with ternary operations and give examples of these systems which gives solutions of the $3$-SE. We find all elementary verbal solutions of the $3$-SE on free groups.
