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Heun algebras of Lie type


Authors: Nicolas Crampé, Luc Vinet and Alexei Zhedanov
Journal: Proc. Amer. Math. Soc. 148 (2020), 1079-1094
MSC (2010): Primary 17B60, 33C45, 33C80
DOI: https://doi.org/10.1090/proc/14788
Published electronically: October 28, 2019
MathSciNet review: 4055936
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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.


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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
Article copyright: © Copyright 2019 American Mathematical Society