Reflection equation for the N=3 Cremmer-Gervais R-matrix

TL;DR

This paper analyzes the reflection equation for the N=3 Cremmer-Gervais R-matrix, revealing a parameter-independent set of 38 equations and characterizing the solution space using algebraic varieties.

Contribution

It explicitly solves the reflection equation for the N=3 Cremmer-Gervais R-matrix and describes the structure of the solution space in algebraic geometric terms.

Findings

Solution space consists of two algebraic varieties.

Reflection equation reduces to 38 parameter-independent equations.

Solution parametrized by products of projective spaces.

Abstract

We consider the reflection equation of the N=3 Cremmer-Gervais R-matrix. The reflection equation is shown to be equivalent to 38 equations which do not depend on the parameter of the R-matrix, q. Solving those 38 equations. the solution space is found to be the union of two types of spaces, each of which is parametrized by the algebraic variety $\mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^2(\mathbb{C})$ and $ \mathbb{C} \times \mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^2(\mathbb{C})$.