Heun algebras of Lie type

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 Zhukovsky–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 to 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.