Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

TL;DR

This paper constructs a new solution to the 3D reflection equation using quantum cluster algebra techniques related to the symmetric butterfly quiver, extending previous work on quantum dilogarithm operators.

Contribution

It introduces a novel $K$-operator involving quantum dilogarithms, derived from quantum cluster algebra associated with the symmetric butterfly quiver, providing new solutions to the 3D reflection equation.

Findings

New $K$-operator involving ten quantum dilogarithms

Solution extends previous 3D reflection equation solutions

Utilizes quantum cluster algebra from symmetric butterfly quiver

Abstract

We construct a new solution $(R,K)$ to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The $R$-operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum dilogarithms with arguments in the $q$-Weyl algebra. The new $K$-operator similarly involves ten such quantum dilogarithms. Our approach is based on the quantum cluster algebra associated with the symmetric butterfly quiver on the wiring diagram of type C.