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

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 (Baseilhac 2022). Its defining K-matrix has size $2\times 2$ and can be expressed, using Rosso's embedding of $U_q^{+}$ into a $q$-shuffle algebra, as generating functions whose coefficients are Terwilliger's alternating PBW basis elements. This and older PBW bases of $U_q^{+}$ due to Damiani and Beck are unified by linear combinations of Catalan words (Ruan 2025). In this paper, we use this unification to define K-matrices of any dimension $\ge 2$ whose entries are explicit generating functions over $U_q^{+}$. Our main result is that any pair of such K-matrices, possibly of different dimensions, satisfy a Freidel-Maillet type equation. This yields a family of algebraic relations over $U_q^{+}$ that generalize Baseilhac's equation and can be used to study integrable systems or higher-spin representations.

22 pages; abstract rewritten; minor changes