# Heun algebras of Lie type

## TL;DR

This paper introduces Heun algebras derived from Lie algebras of dimensions three or four, linking them to orthogonal polynomials and quantum Hamiltonians, and explores their algebraic structures and transformations.

## Contribution

It defines new Heun algebras of Lie type, connects them to classical orthogonal polynomials, and identifies their roles in quantum Hamiltonian models.

## Key findings

- Heun-Krawtchouk algebra linked to $rak{su}(2)$ and quantum gyrostat.
- Heun algebras for $rak{su}(1,1)$ related to Meixner, Pollaczek, and Laguerre polynomials.
- Heun-Charlier algebra associated with the harmonic oscillator algebra.

## Abstract

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For $\mathfrak{su}(1,1)$, one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra $\mathfrak{ho}$ leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the $\mathfrak{su}(1,1)$ cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively.