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

Abstract We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock–Goncharov quivers associated with the longest elements of the Weyl groups of type A and C, and investigate the cluster transformations corresponding to changing a reduced expression into a “most distant” one. By devising a new realization of the quantum y-variables in terms of q-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum y-variables. Explicit formulas of their matrix elements are also derived for some typical representations.