Tetrahedron Equation and Quantum $R$ Matrices for $q$-Oscillator Representations Mixing Particles and Holes

Abstract: We construct $2^n+1$ solutions to the Yang-Baxter equation associated with the quantum affine algebras $U_q\big(A^{(1)}_{n-1}\big)$, $U_q\big(A^{(2)}_{2n}\big)$, $U_q\big(C^{(1)}_n\big)$ and $U_q\big(D^{(2)}_{n+1}\big)$. They act on the Fock spaces of arbitrary mixture of particles and holes in general. Our method is based on new reductions of the tetrahedron equation and an embedding of the quantum affine algebras into $n$ copies of the $q$-oscillator algebra which admits an automorphism interchanging particles and holes.