Tetrahedron equation and quantum $R$ matrices for $q$-oscillator representations

TL;DR

This paper explores the connection between the tetrahedron equation and quantum R matrices for q-oscillator representations, providing new formulas and proofs related to these mathematical structures.

Contribution

It introduces a new formula for the 3d R and a quantum R matrix for n=1, and proves the irreducibility of tensor products of q-oscillator representations.

Findings

New formula for the 3d R matrix

Quantum R matrix for n=1 derived

Irreducibility of tensor products proven

Abstract

We review and supplement the recent result by the authors on the reduction of the three dimensional $R$ (3d $R$) satisfying the tetrahedron equation to the quantum $R$ matrices for the $q$-oscillator representations of $U_q(D^{(2)}_{n+1})$, $U_q(A^{(2)}_{2n})$ and $U_q(C^{(1)}_{n})$. A new formula for the 3d $R$ and a quantum $R$ matrix for $n=1$ are presented and a proof of the irreducibility of the tensor product of the $q$-oscillator representations is detailed.