Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 082, 32 pages      arXiv:2510.12905      https://doi.org/10.3842/SIGMA.2026.082

Constructing Solutions of Simplex Equations from Polygon Equations

Serban Matei Mihalache a and Tomoro Mochida b
a) Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
b) Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki-aza, Aoba-ku, Sendai, 980-8578, Japan

Received January 27, 2026, in final form August 12, 2026; Published online August 25, 2026

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.

Key words: polygon equations; simplex equations; pentagon equation; Yang-Baxter equation; Hopf algebra.

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