Set-theoretical solutions of simplex equations

TL;DR

This paper explores methods to construct solutions for n-simplex equations, generalizes solution techniques, and provides explicit solutions for specific algebraic structures, extending the understanding of these higher-dimensional equations.

Contribution

It introduces new approaches and operations for constructing solutions to n-simplex equations, including group extension methods and algebraic systems for the 3-simplex case.

Findings

Constructed solutions on group extensions.

Generalized tropicalization of rational solutions.

Found all elementary verbal solutions on free groups.

Abstract

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.