# Truncation of the reflection algebra and the Hahn algebra

## TL;DR

This paper establishes an isomorphism between the truncated reflection algebra of the Yangian of sl(2) and the Hahn algebra, linking quantum integrable systems with orthogonal polynomial algebras.

## Contribution

It introduces a novel connection between reflection algebras and the Hahn algebra, and develops a general framework using Euler polynomials for studying algebra truncations.

## Key findings

- Proves the isomorphism between truncated reflection algebra and Hahn algebra.
- Provides a new method using Euler polynomials for algebra truncation analysis.
- Enhances understanding of algebraic structures in quantum integrable systems.

## Abstract

In the context of connections between algebras coming from quantum integrable systems and algebras associated to the orthogonal polynomials of the Askey scheme, we prove that the truncated reflection algebra attached to the Yangian of sl(2) is isomorphic to the Hahn algebra. As a by-product, we provide a general set-up based on Euler polynomials to study truncations of reflection algebras.