## Heun algebras of Lie type

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- by Nicolas Crampé, Luc Vinet and Alexei Zhedanov PDF
- Proc. Amer. Math. Soc.
**148**(2020), 1079-1094 Request permission

## 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 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.## References

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## Additional Information

**Nicolas Crampé**- Affiliation: Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d’Orléans, Parc de Grammont, 37200 Tours, France; Centre de recherches, mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7, Canada
- Email: crampe1977@gmail.com
**Luc Vinet**- Affiliation: Centre de recherches, mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7, Canada
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Email: vinet@CRM.UMontreal.ca
**Alexei Zhedanov**- Affiliation: School of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
- MR Author ID: 234560
- Email: zhedanov@ruc.edu.cn
- Received by editor(s): April 24, 2019
- Received by editor(s) in revised form: April 25, 2019, and July 29, 2019
- Published electronically: October 28, 2019
- Additional Notes: The first author is gratefully holding a CRM–Simons professorship.

The research of the second author was supported in part by a Natural Science and Engineering Council (NSERC) of Canada discovery grant.

The research of the third author was supported by the National Science Foundation of China (Grant No. 11711015). - Communicated by: Mourad Ismail
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 1079-1094 - MSC (2010): Primary 17B60, 33C45, 33C80
- DOI: https://doi.org/10.1090/proc/14788
- MathSciNet review: 4055936