Heun algebras of Lie type

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.

arXiv comment15 pages

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• Painleve VI, Rigid Tops and Reflection Equation
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• The Racah algebra and superintegrable models
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Cited by in corpus (10)

• Entanglement in Fermionic Chains and Bispectrality
• Heun operator of Lie type and the modified algebraic Bethe ansatz
• The Heun-Racah and Heun-Bannai-Ito algebras
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• The rational Sklyanin algebra and the Wilson and para-Racah polynomials
• Bethe ansatz diagonalization of the Heun-Racah operator
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• The missing label of ${\mathfrak{su}}_3$ and its symmetry
• Classical elliptic ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries
• Generalized quadratic commutator algebras of PBW-type