Constructing solutions of simplex equations from polygon equations

TL;DR

The paper develops a graphical, tensor-based framework to connect polygon equations and simplex equations, showing that commutative combinations of polygon solutions lift to higher polygons and that a uniform mixed (compatibility) condition between solutions of the $n$-gon and its dual yields explicit constructions of $(n-1)$- and $(n-2)$-simplex solutions. It provides precise mixed-relations, explicit formulas for constructing simplex solutions from polygon data, and both even/odd-case proofs, including non-constant versions, thereby generalizing earlier work by Kashaev–Sergeev and Dimakis–Müller-Hoissen. The results are supported by graphical calculus, and concrete Hopf-algebra-based examples (notably via the O(H) construction) illustrate the method and yield explicit low-dimensional simplex solutions, with potential topological implications through Pachner moves. Overall, the work systematizes how polygon equations underpin simplex equations and offers a versatile toolkit for producing higher-dimensional integrability structures with applications in topology and mathematical physics.

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 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.

Figures (16)

• Figure 1: Coordinates of the input/output vector spaces.
• Figure 2: Right trace version of $R^{(n-2)}$, from odd-gon (left), and even-gon (right)
• Figure 6: Mixed relations for 4-gon (left) and 5-gon (right). The blue boxes represent a map $T$ and the red boxes represent a map $S$ (dual). The free outputs/inputs are labeled with inner simplices, and the inner edges are labeled with the boundary simplices in the Pachner move (compare with Figure \ref{['fig:graphical45goneq']}).
• Figure 7: Left: the solution $R(p)$ of the 3-simplex equation from the (dual) 4-gon equations, where $p$ is any 2-simplex of $\Delta^3$. Right: the solution $R(p)$ of the 4-simplex equation from the (dual) 5-gon equations, where $p$ is any 3-simplex of $\Delta^4$.
• Figure 8: A construction of $Q(q)$ from the 5-gon equation, in which we take the $2$-simplex $q=[023]$ as an example. The first part connects the edges of $R(0134)$ with the label $134$, and the second part arranges the diagram and reindexes the edge labels.

Theorems & Definitions (39)

• Definition 3.1: Odd-gon equation
• Definition 3.2: Dual odd-gon equation
• Definition 3.3: Even-gon equation
• Definition 3.4: Dual even-gon equation
• Example 3.5
• Remark 3.1
• Proposition 3.6
• proof
• Proposition 4.1
• proof