Set-theoretical solutions of the Yang–Baxter and pentagon equations on semigroups

Abstract

The Yang–Baxter and pentagon equations are two well-known equations of Mathematical Physic. If S is a set, a map $s:S\times S\rightarrow S\times S$ is said to be a set-theoretical solution of the quantum Yang–Baxter equation if

$$\begin{aligned} s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23}, \end{aligned}$$

where $s_{12}=s\times {{\,\mathrm{id}\,}}_S$, $s_{23}={{\,\mathrm{id}\,}}_S\times s$, and $s_{13}=({{\,\mathrm{id}\,}}_S\times \tau )\,s_{12}\,({{\,\mathrm{id}\,}}_S\times \tau )$ and $\tau$ is the flip map, i.e., the map on $S\times S$ given by $\tau (x,y)=(y,x)$. Instead, s is called a set-theoretical solution of the pentagon equation if

$$\begin{aligned} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}. \end{aligned}$$

The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang–Baxter equation. Specifically, we present a new construction of solutions of the Yang–Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.