Freidel-Maillet type equations on fused K-matrices over the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

TL;DR

This paper constructs fused K-matrices of arbitrary size for the positive part of $U_q( ext{sl}_2)$, demonstrating they satisfy Freidel-Maillet equations, using a uniform approach based on PBW bases and fusion techniques.

Contribution

It introduces a method to explicitly construct fused K-matrices of any size within $U_q^+$ and proves they satisfy Freidel-Maillet type equations, extending previous results.

Findings

Fused K-matrices can be expressed in closed form.

Any pair of fused K-matrices satisfy Freidel-Maillet equations.

The approach unifies previous PBW basis methods for constructing K-matrices.

Abstract

The positive part $U_q^+$ of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ has a reflection equation presentation of Freidel-Maillet type, due to Baseilhac 2021. This presentation involves a K-matrix of dimension $2 \times 2$. Under an embedding of $U_q^+$ into a $q$-shuffle algebra due to Rosso 1995, this K-matrix can be written in closed form using a PBW basis for $U_q^+$ due to Terwilliger 2019. This PBW basis, together with two PBW bases due to Damiani 1993 and Beck 1994, can be obtain from a uniform approach by Ruan 2025. Following a natural fusion technique, we will construct fused K-matrices of arbitary meaningful dimension in closed form using the uniform approach. We will also show that any pair of these fused K-matrices satisfy Freidel-Maillet type equations.