Tetrahedron Equation and Quantum $R$ Matrices for modular double of   $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n})$ and $U_q(C^{(1)}_{n})$
  We introduce a homomorphism from the quantum affine algebras $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}), U_q(C^{(1)}_{n})$ to the $n$-fold tensor product of the $q$-oscillator algebra ${\mathcal A}_q$. Their action commute with the solutions of the Yang-Baxter equation obtained by reducing the solutions of the tetrahedron equation associated with the modular and the Fock representations of ${\mathcal A}_q$. In the former case, the commutativity is enhanced to the modular double of these quantum affine algebras. 
Atsuo Kuniba, Masato Okado, Sergey Sergeev
math-ph hep-th math.MP math.QA nlin.SI