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

TL;DR

This paper constructs multiple solutions to the Yang-Baxter equation related to quantum affine algebras, acting on Fock spaces with mixed particles and holes, using novel reductions of the tetrahedron equation and $q$-oscillator algebra embeddings.

Contribution

It introduces new solutions to the Yang-Baxter equation for various quantum affine algebras via reductions of the tetrahedron equation and a novel embedding into $q$-oscillator algebras.

Findings

Constructed $2^n+1$ solutions to the Yang-Baxter equation.

Solutions act on Fock spaces with mixed particles and holes.

Developed new reductions of the tetrahedron equation.

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.