Constructing Solutions of Simplex Equations from Polygon Equations

Abstract

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.