Freidel-Maillet type equations on fused K-matrices over the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
Chenwei Ruan
TL;DR
The paper addresses constructing and proving Freidel-Maillet-type reflection equations for fused K-matrices associated with the positive part $U_q^+(\widehat{\mathfrak{sl}}_2)$. It leverages Rosso's embedding into a $q$-shuffle algebra and a uniform fusion framework to produce closed-form fused $R$-, $\widehat{R}$-, and $K$-matrices, with $K$ expressed via Catalan-word generating functions tied to PBW bases. The main result is a generalized Freidel-Maillet type equation for all fused spins, accompanied by a fusion recurrence for $K^{(j)}$ and corollaries that yield KRKR-type relations, along with connections to the quasi R-matrix. This work advances boundary integrability and provides explicit higher-dimensional boundary operators that may impact quantum integrable systems and representation theory.
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.