Tetrahedron Equation and Quantum $R$ Matrices for modular double of $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n})$ and $U_q(C^{(1)}_{n})$

TL;DR

This paper constructs homomorphisms from certain quantum affine algebras to tensor products of q-oscillator algebras, linking solutions of the tetrahedron and Yang-Baxter equations, and extends this to the modular double case.

Contribution

It introduces a new homomorphism connecting quantum affine algebras with q-oscillator algebras and relates tetrahedron solutions to Yang-Baxter solutions, including the modular double extension.

Findings

Homomorphisms from quantum affine algebras to q-oscillator tensor products

Solutions of the tetrahedron equation reduce to Yang-Baxter solutions

Modular double extension of the quantum affine algebra actions

Abstract

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.