Freidel-Maillet type equations on fused K-matrices over the positive part of U_q(\widehat\mathfraksl₂)

2025/11/30 by Ruan, Chenwei
#81R12 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 17B37 #Quantum Algebra (math.QA) #Secondary: 16T25

paper · doi:10.48550/arxiv.2512.00819

Abstract

The positive part U_q⁺ of the quantized enveloping algebra U_q(\widehat\mathfraksl₂) has a reflection equation presentation of Freidel-Maillet type, due to Baseilhac 2021. This presentation involves a K-matrix of dimension 2 × 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.

Citations

• Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum symmetric pairs
• Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras
• Universal K-matrices for quantum Kac-Moody algebras
• Using Catalan words and a q-shuffle algebra to describe a PBW basis for the positive part of U_q(\widehat \mathfraksl₂)
• Universal K-matrix for quantum symmetric pairs
• Monoidal categorification of cluster algebras
• Quantum Symmetric Pairs and the Reflection Equation
• Dual canonical bases, quantum shuffles and q-characters
• Quantum groups and quantum shuffles
• Quantum symmetric spaces and related q-orthogonal polynomials
• Macdonald's symmetric polynomials as zonal spherical functions on some\n quantum homogeneous spaces

Related

• Fusion Rules and R-Matrices For Representations of SL(2)_q at Roots of Unity
• On quantum Freidel-Maillet algebra for non-ultralocal integrable systems
• A uniform approach to the Damiani, Beck, and alternating PBW bases for the positive part of U_q(\widehat\mathfraksl₂)
• Fusion and exchange matrices for quantized sl(2) and associated q-special functions
• Using Catalan words and a q-shuffle algebra to describe a PBW basis for the positive part of U_q(\widehat \mathfraksl₂)