Boundary quantum Knizhnik-Zamolodchikov equations and fusion

TL;DR

This paper extends solutions of boundary quantum Knizhnik-Zamolodchikov equations to higher-spin representations, developing fusion techniques for K-operators and constructing Jackson integral solutions.

Contribution

It introduces a systematic fusion method for K-operators in higher-spin representations, expanding the class of solutions to boundary qKZ equations.

Findings

Constructed diagonal K-operators for higher-spin representations.

Developed fusion procedures for K-operators in quantum affine $rak{sl}_2$.

Produced Jackson integral solutions for boundary qKZ equations.

Abstract

In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine $\mathfrak{sl}_2$. First we give a systematic exposition of known results on $R$-operators acting in the tensor product of evaluation representations in Verma modules over quantum $\mathfrak{sl}_2$. We develop the corresponding fusion of $K$-operators, which we use to construct diagonal $K$-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure.