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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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 , 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
, 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
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
cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn, and Confluent Heun operators, respectively.
- [1] I. Basak, Explicit solution of the Zhukovski-Volterra gyrostat, Regul. Chaotic Dyn. 14 (2009), no. 2, 223–236. MR 2505426, https://doi.org/10.1134/S1560354709020038
- [2] P. Baseilhac and R. Pimenta, Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz, to appear.
- [3] P. Baseilhac, L. Vinet, and A. Zhedanov, The q-Heun operator of big q-Jacobi type and the q-Heun algebra, arXiv:1808.06695.
- [4] P. Baseilhac, S. Tsujimoto, L. Vinet, and A. Zhedanov, The Heun-Askey-Wilson algebra and the Heun operator of Askey-Wilson type, arXiv:1811.11407.
- [5] N. Crampe, R. I. Nepomechie, and L. Vinet, Free-Fermion entanglement and orthogonal polynomials, arXiv:1907.00044.
- [6] N. Crampe, E. Ragoucy, L. Vinet, and A. S. Zhedanov, Truncation of the reflection algebra and the Hahn algebra, arXiv:1903.05674.
- [7] Roberto Floreanini, Jean LeTourneux, and Luc Vinet, Quantum mechanics and polynomials of a discrete variable, Ann. Physics 226 (1993), no. 2, 331–349. MR 1240360, https://doi.org/10.1006/aphy.1993.1072
- [8] Luc Frappat, Julien Gaboriaud, Luc Vinet, Stéphane Vinet, and Alexei Zhedanov, The Higgs and Hahn algebras from a Howe duality perspective, Phys. Lett. A 383 (2019), no. 14, 1531–1535. MR 3944835, https://doi.org/10.1016/j.physleta.2019.02.024
- [9] V. X. Genest, L. Vinet, and A. S. Zhedanov, The Racah algebra and superintegrable models, J. Phys.: Conf. Ser. 512 (2014) 012011 and arXiv:1309.3540.
- [10] Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov, Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), no. 1, 1–20. MR 1173277, https://doi.org/10.1016/0003-4916(92)90336-K
- [11] Ya. A. Granovskiĭ and A. S. Zhedanov, Nature of the symmetry group of the 6𝑗-symbol, Zh. Èksper. Teoret. Fiz. 94 (1988), no. 10, 49–54 (Russian); English transl., Soviet Phys. JETP 67 (1988), no. 10, 1982–1985 (1989). MR 997934
- [12] Ya. I. Granovskiĭ and A. S. Zhedanov, Orthogonal polynomials on Lie algebras, Izv. Vyssh. Uchebn. Zaved. Fiz. 29 (1986), no. 5, 60–66 (Russian). MR 853633
- [13] F. Alberto Grünbaum, Luc Vinet, and Alexei Zhedanov, Tridiagonalization and the Heun equation, J. Math. Phys. 58 (2017), no. 3, 031703, 12. MR 3623072, https://doi.org/10.1063/1.4977828
- [14] F. Alberto Grünbaum, Luc Vinet, and Alexei Zhedanov, Algebraic Heun operator and band-time limiting, Comm. Math. Phys. 364 (2018), no. 3, 1041–1068. MR 3875822, https://doi.org/10.1007/s00220-018-3190-0
- [15] Gerhard Kristensson, Second order differential equations, Springer, New York, 2010. Special functions and their classification. MR 2682403
- [16] H. T. Koelink and J. Van Der Jeugt, Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. Math. Anal. 29 (1998), no. 3, 794–822. MR 1617724, https://doi.org/10.1137/S003614109630673X
- [17] Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their 𝑞-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096
- [18] A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, Painlevé VI, rigid tops and reflection equation, Comm. Math. Phys. 268 (2006), no. 1, 67–103. MR 2249796, https://doi.org/10.1007/s00220-006-0089-y
- [19] J. Patera and P. Winternitz, A new basis for the representations of the rotation group. Lamé and Heun polynomials, J. Math. Phys. 14 (1973), 1130-1139.
- [20] D. R. Masson and J. Repka, Spectral theory of Jacobi matrices in 𝑙²(𝑍) and the 𝑠𝑢(1,1) Lie algebra, SIAM J. Math. Anal. 22 (1991), no. 4, 1131–1146. MR 1112070, https://doi.org/10.1137/0522073
- [21] A. Ronveaux (Ed.), Heun's Differential Equations, Oxford University Press, Oxford, 1995.
- [22] R. F. Streater, The representations of the oscillator group, Comm. Math. Phys. 4 (1967), 217–236. MR 207908
- [23] Kouichi Takemura, On 𝑞-deformations of the Heun equation, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 061, 16. MR 3815312, https://doi.org/10.3842/SIGMA.2018.061
- [24] A. V. Turbiner, The Heun operator as a Hamiltonian, J. Phys. A 49 (2016), no. 26, 26LT01, 8. MR 3512091, https://doi.org/10.1088/1751-8113/49/26/26LT01
- [25] L. Vinet, and A. Zhedanov, The Heun operator of Hahn type, arXiv:1808.00153.
- [26] L. Vinet and A. Zhedanov, Solvability in classical and quantum mechanics and algebraic Heun observables, to appear.
- [27] O. B. Zaslavskiĭ and V. V. Ul′yanov, Periodic effective potentials for spin systems and new exact solutions of the one-dimensional Schrödinger equation for energy zones, Teoret. Mat. Fiz. 71 (1987), no. 2, 260–271 (Russian, with English summary). MR 911671
- [28] A. S. Zhedanov, “Hidden symmetry” of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991), no. 2, 190–204 (Russian, with English summary); English transl., Theoret. and Math. Phys. 89 (1991), no. 2, 1146–1157 (1992). MR 1151381, https://doi.org/10.1007/BF01015906
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 17B60, 33C45, 33C80
Retrieve articles in all journals with MSC (2010): 17B60, 33C45, 33C80
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