# Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$

## TL;DR

This paper explicitly constructs reflection $K$ matrices for certain quantum affine algebras, providing the first proof of the reflection equation for non-type A cases through intertwiners of Onsager coideal subalgebras.

## Contribution

It introduces a novel approach using intertwiners of Onsager coideal subalgebras to derive reflection $K$ matrices for non-type A quantum affine algebras.

## Key findings

- Derived reflection $K$ matrices for $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$, $U_p(D^{(2)}_{n+1})$
- Provided the first proof of the reflection equation for non-type A cases
- Connected the intertwiners to three-dimensional integrability and matrix product constructions.

## Abstract

We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They reproduce the reflection $K$ matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type $A$ cases.