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Heun algebras of Lie type


Authors: Nicolas Crampé, Luc Vinet and Alexei Zhedanov
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 17B60, 33C45, 33C80
DOI: https://doi.org/10.1090/proc/14788
Published electronically: October 28, 2019
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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
Email: vinet@CRM.UMontreal.ca

Alexei Zhedanov
Affiliation: School of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
Email: zhedanov@ruc.edu.cn

DOI: https://doi.org/10.1090/proc/14788
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