New solutions to the tetrahedron equation associated with quantized six-vertex models

TL;DR

This paper introduces new solutions to the tetrahedron equation using quantized six-vertex models linked to $q$-oscillator and $q$-Weyl algebras, revealing both known and novel intertwiners with explicit series representations.

Contribution

It provides a family of solutions to the tetrahedron equation connecting quantized six-vertex models with $q$-algebras, including new $R$-matrices with explicit series forms.

Findings

When using $q$-oscillator algebra, $R$ matches known intertwiners of $A_q(sl_3)$.

$q$-Weyl algebra leads to new $R$ matrices with factorized or series expressions.

The solutions extend the understanding of integrable models related to quantum groups.

Abstract

We present a family of new solutions to the tetrahedron equation of the form $RLLL=LLLR$, where $L$ operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the $q$-oscillator or $q$-Weyl algebras. When the three $L$'s are associated with the $q$-oscillator algebra, $R$ coincides with the known intertwiner of the quantized coordinate ring $A_q(sl_3)$. On the other hand, $L$'s based on the $q$-Weyl algebra lead to new $R$'s whose elements are either factorized or expressed as a terminating $q$-hypergeometric type series.