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 $\geq 2$ 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.
From: Chenwei Ruan [view email]
[v1]
Sun, 30 Nov 2025 09:59:53 UTC (20 KB)
[v2]
Wed, 24 Dec 2025 09:05:44 UTC (20 KB)
[v3]
Sat, 3 Jan 2026 07:26:20 UTC (20 KB)
[v4]
Sun, 2 Aug 2026 11:16:44 UTC (20 KB)
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