Constructing solutions of simplex equations from polygon equations

We study polygon equations and their connections to simplex equations, which generalize the pentagon and Yang–Baxter equations, respectively. First, we show that certain “commutative” pairs of solutions of (dual) polygon equations give rise to solutions of higher-order polygon equations. Next, we define an explicit compatibility condition between solutions of the $n$-gon and dual $n$-gon equations and use it to construct solutions of the $(n-2)$- and $(n-1)$-simplex equations. This extends earlier work by Kashaev–Sergeev and Dimakis–Müller-Hoissen.