Howe Duality and AW Algebras
233
5 Conclusion
This paper has offered a summary of how the quadratic algebras of Racah, Hahn,
Bannai–Ito, Askey–Wilson, and q-Hahn types can be given dual descriptions as
commutant of Lie algebras, superalgebras, and quantum algebras. The connection
between these dual pictures is rooted in Howe dualities whose various expressions
have been stressed. The attentive reader will have noticed that the Clebsch–Gordan
problem for osp(1|2) has not been mentioned; this is because it has not been
analyzed yet. We plan on adding this missing piece to complete the picture.
Acknowledgments 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.
References
1. D.J. Rowe, M.J. Carvalho, J. Repka, Dual pairing of symmetry groups and dynamical groups
in physics. Rev. Mod. Phys. 84, 711 (2012)
2. V.X. Genest, L. Vinet, A. Zhedanov, The Racah algebra and superintegrable models. J. Phys.
Conf. Ser. 512, 012011 (2014)
3. 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)
4. 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)
5. J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The generalized Racah algebra as a commutant.
J. Phys. Conf. Ser. 1194, 012034 (2019)
6. 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)
7. 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)
8. J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The dual pair P in(2n) ⊗ osp(1|2), the Dirac
equation and the Bannai–Ito algebra. Nucl. Phys. B 937, 226–239 (2018)
9. H. De Bie, V.X. Genest, L. Vinet, The Z n
2 Dirac-Dunkl operator and a higher rank Bannai-Ito
algebra. Adv. Math. 5, 390–414 (2016)
10. L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet, The dual pair
U q (su(1, 1)), o q 1/2 (2n)
, qoscillators and the higher rank Askey–Wilson algebra AW (n). J. Math. Phys. 61, 041701
(2020). https://doi.org/10.1063/1.5124251
11. 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)
12. L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet, The q-Higgs and Askey–Wilson algebras.
Nucl. Phys. B 944, 114632 (2019)
233
5 Conclusion
This paper has offered a summary of how the quadratic algebras of Racah, Hahn,
Bannai–Ito, Askey–Wilson, and q-Hahn types can be given dual descriptions as
commutant of Lie algebras, superalgebras, and quantum algebras. The connection
between these dual pictures is rooted in Howe dualities whose various expressions
have been stressed. The attentive reader will have noticed that the Clebsch–Gordan
problem for osp(1|2) has not been mentioned; this is because it has not been
analyzed yet. We plan on adding this missing piece to complete the picture.
Acknowledgments 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.
References
1. D.J. Rowe, M.J. Carvalho, J. Repka, Dual pairing of symmetry groups and dynamical groups
in physics. Rev. Mod. Phys. 84, 711 (2012)
2. V.X. Genest, L. Vinet, A. Zhedanov, The Racah algebra and superintegrable models. J. Phys.
Conf. Ser. 512, 012011 (2014)
3. 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)
4. 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)
5. J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The generalized Racah algebra as a commutant.
J. Phys. Conf. Ser. 1194, 012034 (2019)
6. 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)
7. 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)
8. J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The dual pair P in(2n) ⊗ osp(1|2), the Dirac
equation and the Bannai–Ito algebra. Nucl. Phys. B 937, 226–239 (2018)
9. H. De Bie, V.X. Genest, L. Vinet, The Z n
2 Dirac-Dunkl operator and a higher rank Bannai-Ito
algebra. Adv. Math. 5, 390–414 (2016)
10. L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet, The dual pair
U q (su(1, 1)), o q 1/2 (2n)
, qoscillators and the higher rank Askey–Wilson algebra AW (n). J. Math. Phys. 61, 041701
(2020). https://doi.org/10.1063/1.5124251
11. 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)
12. L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet, The q-Higgs and Askey–Wilson algebras.
Nucl. Phys. B 944, 114632 (2019)