Quantum Cluster Algebras and 3D Integrability: Tetrahedron and 3D Reflection Equations.

The article presents a novel construction of solutions to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra framework developed by Sun and Yagi. Focusing on Fock–Goncharov quivers associated with the longest elements of Weyl groups of types \( A \) and \( C \), the authors realize quantum \( y \)-variable cluster transformations as adjoint actions within a \( q \)-Weyl algebra embedding. They explicitly define operators \(\mathcal{R}_{ijk}\) and \(\mathcal{K}_{ijkl}\), expressed via the quantum dilogarithm, that satisfy the tetrahedron and 3D reflection equations with spectral parameters. The paper further provides infinite-dimensional and modular double representations of these operators, deriving explicit matrix elements and integral kernels involving the noncompact quantum dilogarithm, thereby establishing new integrable structures in three-dimensional quantum systems through cluster algebra techniques.