Generalized q-Onsager Algebras and Dynamical K-matrices

TL;DR

This paper introduces a method to construct dynamical K-matrices from generalized q-Onsager algebras, extending existing techniques to models with boundary degrees of freedom, relevant for quantum integrable systems.

Contribution

It extends intertwiner techniques to dynamical solutions of the reflection equation using generalized q-Onsager algebras, linking to boundary integrable models.

Findings

Constructed dynamical K-matrices from generalized q-Onsager algebras.

Demonstrated the method with specific algebra cases $\\cO_{q}(a^{(2)}_{2})$ and $\\cO_{q}(a^{(1)}_{2})$.

Showed relevance to quantum affine Toda field theories with boundary conditions.

Abstract

A procedure to construct $K$-matrices from the generalized $q$-Onsager algebra $\cO_{q}(\hat{g})$ is proposed. This procedure extends the intertwiner techniques used to obtain scalar (c-number) solutions of the reflection equation to dynamical (non-c-number) solutions. It shows the relation between soliton non-preserving reflection equations or twisted reflection equations and the generalized $q$-Onsager algebras. These dynamical $K$-matrices are important to quantum integrable models with extra degrees of freedom located at the boundaries: for instance, in the quantum affine Toda field theories on the half-line they yield the boundary amplitudes. As examples, the cases of $\cO_{q}(a^{(2)}_{2})$ and $\cO_{q}(a^{(1)}_{2})$ are treated in details.