Covariance Properties of Reflection Equation Algebras

TL;DR

This paper explores the algebraic properties and structures of reflection equation algebras, extending the understanding of their relation to quantum groups and their solutions, with implications for integrable models.

Contribution

It provides a detailed analysis of the properties of reflection equation algebras, including their quantum group comodule structure and solution generation methods.

Findings

Reflection equation algebras possess quantum group comodule properties.

New solutions can be generated by composing known solutions.

Extended REA and central elements exhibit notable algebraic features.

Abstract

The reflection equations (RE) are a consistent extension of the Yang-Baxter equations (YBE) with an addition of one element, the so-called reflection matrix or $K$-matrix. For example, they describe the conditions for factorizable scattering on a half line just like the YBE give the conditions for factorizable scattering on an entire line. The YBE were generalized to define quadratic algebras, \lq Yang-Baxter algebras\rq\ (YBA), which were used intensively for the discussion of quantum groups. Similarly, the RE define quadratic algebras, \lq the reflection equation algebras\rq\ (REA), which enjoy various remarkable properties both new and inherited from the YBA. Here we focus on the various properties of the REA, in particular, the quantum group comodule properties, generation of a series of new solutions by composing known solutions, the extended REA and the central elements, etc.