Contracted and expanded integrable structures

TL;DR

This paper introduces a unified framework for deriving contracted and centrally extended algebras from quadratic algebras related to boundary integrable models, clarifying misconceptions and expanding algebraic structures.

Contribution

It presents a novel method to generate contracted and extended algebras using quadratic algebras like reflection algebras and twisted Yangians, resolving old misconceptions.

Findings

Resolved misconception about E_2 to sl_2 expansion

Derived centrally extended algebras from rational and trigonometric R-matrices

Provided a unified framework for algebraic contractions in integrable models

Abstract

We propose a generic framework to obtain certain types of contracted and centrally extended algebras. This is based on the existence of quadratic algebras (reflection algebras and twisted Yangians), naturally arising in the context of boundary integrable models. A quite old misconception regarding the "expansion" of the E_2 algebra into sl_2 is resolved using the representation theory of the aforementioned quadratic algebras. We also obtain centrally extended algebras associated to rational and trigonometric (q-deformed) R-matrices that are solutions of the Yang--Baxter equation.