Abstract
The Askey–Wilson algebra and its relatives such as the Racah and Bannai–Ito algebras were initially introduced in connection with the eponym orthogonal polynomials. They have since proved ubiquitous. In particular they admit presentations as commutants that are related through Howe duality. This paper surveys these results.
Keywords
- Howe duality
- Racah
- Bannai–Ito and Askey–Wilson algebras
- Commutants
- Reductive dual pairs
This is a preview of subscription content, access via your institution.
Buying options
References
D.J. Rowe, M.J. Carvalho, J. Repka, Dual pairing of symmetry groups and dynamical groups in physics. Rev. Mod. Phys. 84, 711 (2012)
V.X. Genest, L. Vinet, A. Zhedanov, The Racah algebra and superintegrable models. J. Phys. Conf. Ser. 512, 012011 (2014)
J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The Racah algebra as a commutant and Howe duality. J. Phys. A Math. Theor. 51, 50LT01 (2018)
H. De Bie, V.X. Genest, W. van de Vijver, L. Vinet, A higher rank Racah algebra and the Laplace-Dunkl operator. J. Phys. A Math. Theor. 51, 025203 (2017)
J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The generalized Racah algebra as a commutant. J. Phys. Conf. Ser. 1194, 012034 (2019)
L. Frappat, J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The Higgs and Hahn algebras from a Howe duality perspective. Phys. Lett. A 383, 1531–1535 (2019)
H. De Bie, V.X. Genest, S. Tsujimoto, L. Vinet, A. Zhedanov, The Bannai–Ito algebra and some applications. J. Phys. Conf. Ser. 597, 012001 (2015)
J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The dual pair
, the Dirac equation and the Bannai–Ito algebra. Nucl. Phys. B 937, 226–239 (2018)H. De Bie, V.X. Genest, L. Vinet, The
Dirac-Dunkl operator and a higher rank Bannai-Ito algebra. Adv. Math. 5, 390–414 (2016)L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet, The dual pair
, q-oscillators and the higher rank Askey–Wilson algebra AW(n). J. Math. Phys. 61, 041701 (2020). https://doi.org/10.1063/1.5124251
H. De Bie, H. De Clerq, W. van de Vijver, The higher rank q-deformed Bannai–Ito and Askey–Wilson algebra. Commun. Math. Phys. 374(1), 277 (2020)
L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet, The q-Higgs and Askey–Wilson algebras. Nucl. Phys. B 944, 114632 (2019)
, the Dirac equation and the Bannai–Ito algebra. Nucl. Phys. B 937, 226–239 (2018)
Dirac-Dunkl operator and a higher rank Bannai-Ito algebra. Adv. Math. 5, 390–414 (2016)
, q-oscillators and the higher rank Askey–Wilson algebra AW(n). J. Math. Phys. 61, 041701 (2020). Acknowledgements
The authors thank Luc Frappat, Eric Ragoucy, and Alexei Zhedanov for collaborations that led to the results reviewed here. JG holds an Alexander-Graham-Bell Scholarship from the Natural Science and Engineering Research Council (NSERC) of Canada. LV gratefully acknowledges his support from NSERC through a Discovery Grant.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Gaboriaud, J., Vinet, L., Vinet, S. (2021). Howe Duality and Algebras of the Askey–Wilson Type: An Overview. In: Paranjape, M.B., MacKenzie, R., Thomova, Z., Winternitz, P., Witczak-Krempa, W. (eds) Quantum Theory and Symmetries. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-55777-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-55777-5_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-55776-8
Online ISBN: 978-3-030-55777-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)