Solutions to quantum tetrahedron equation with two colors and nonnegative matrix entries

TL;DR

This paper constructs solutions to the quantum tetrahedron equation with two-color variables and nonnegative matrix entries, introducing two methods involving $\\ ext{\cal R}$-operators and a modified tetrahedron approach.

Contribution

It presents two novel constructions of quantum tetrahedron solutions with two colors and nonnegative matrices, expanding the set of known solutions.

Findings

Two $\\text{\cal R}$-operator based solutions with 12 or 14 nonzero entries

A modified tetrahedron in a direct sum producing additional solutions

All matrix entries are nonnegative with proper parameter choices

Abstract

In this short note, we construct solutions to quantum tetrahedron equation of the kind "with variables on the edges". Each of these variables takes just two values, called sometimes "colors". We propose two different constructions. The first of them involves, in particular, two $\mathcal R$-operators each depending on one parameter, while these parameters are independent from each other; the number of nonvanishing matrix entries in these two $\mathcal R$-operators is either 12 or 14. In the second construction, we use what may be called "modified tetrahedron in a direct sum" to produce, again, quantum tetrahedron solutions. All matrix entries of our $\mathcal R$-operators are nonnegative, if the relevant parameters are chosen properly.