216
H. De Bie et al.
This embedding gives us a good idea of what should be the abstract embedding
of R n in U(sl n−1 ). Achieving the abstract construct remains, however, nontrivial.
The differential embedding simplifies the problem since, e.g. all central elements
C i
become scalars and it is not immediate therefore how to lift the differential case to
the abstract one. An alternative construction might be better suited for that purpose
and we believe that the route taken in [2] for the case of the Heisenberg algebra is
promising in this respect.
Acknowledgments The work of HDB is supported by the Research Foundation Flanders (FWO)
under Grant EOS 30889451. HDB and WVDV are grateful for the hospitality extended to them by
the Centre de Recherches Mathématiques in Montréal, where part of this research was carried out.
The research of LV is funded in part by a discovery grant of the Natural Sciences and Engineering
Council (NSERC) of Canada.
References
1. N. Crampé, L. Poulain d’Andecy, L. Vinet, Temperley-Lieb, Brauer and Racah algebras and
other centralizers of su(2). Trans. Amer. Math. Soc. 373(7), 4907–4932 (2020)
2. N. Crampé, W. van de Vijver, L. Vinet, Racah problems for the oscillator algebra, the Lie
algebra sl n and multivariate Krawtchouk polynomials (2019). arXiv:1909.12643, 28 pages
3. H. De Bie, W. van de Vijver, A discrete realization of the higher rank Racah algebra. Constr.
Approx. 52, 1–29 (2020). https://doi.org/10.1007/s00365-019-09475-0
4. H. De Bie, V.X. Genest, W. van de Vijver, L. Vinet, A higher rank Racah algebra and the (Z 2 ) n
Laplace-Dunkl operator. J. Phys. A Math. Theor. 51, 025203 (20 pp.) (2018)
5. H. De Bie, P. Iliev, L. Vinet, Bargmann and Barut-Giradello models for the Racah algebra.
J. Math. Phys. 60, 011701 (2019)
6. J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The generalized Racah algebra as a commutant.
J. Phys. Conf. Ser. 1194(1), 012034 (2019)
7. V.X. Genest, L. Vinet, A. Zhedanov, The equitable Racah algebra from three su(1, 1) algebras.
J. Phys. A Math. Theor. 47(2), 025203, 12 pp. (2014)
8. V.X. Genest, L. Vinet, A. Zhedanov, The Racah algebra and superintegrable models. J. Phys.
Conf. Ser. 512, 012011 (2014)
9. P. Iliev, The generic quantum superintegrable system on the sphere and Racah operators. Lett.
Math. Phys. 107(11), 2029–2045 (2017)
10. P. Iliev, Symmetry algebra for the generic superintegrable system on the sphere. J. High Energy
Phys. 2018(2), 44 (2018), front matter+22 pp.
11. R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their
q-Analogues (Springer, New York, 2010)
12. M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete
families. J. Math. Phys. 32, 2337–2342 (1991)
13. A.S. Zhedanov, “Hidden symmetry” of the Askey-Wilson polynomials. Theor. Math. Phys.
89, 1146–1157 (1991)
H. De Bie et al.
This embedding gives us a good idea of what should be the abstract embedding
of R n in U(sl n−1 ). Achieving the abstract construct remains, however, nontrivial.
The differential embedding simplifies the problem since, e.g. all central elements
C i
become scalars and it is not immediate therefore how to lift the differential case to
the abstract one. An alternative construction might be better suited for that purpose
and we believe that the route taken in [2] for the case of the Heisenberg algebra is
promising in this respect.
Acknowledgments The work of HDB is supported by the Research Foundation Flanders (FWO)
under Grant EOS 30889451. HDB and WVDV are grateful for the hospitality extended to them by
the Centre de Recherches Mathématiques in Montréal, where part of this research was carried out.
The research of LV is funded in part by a discovery grant of the Natural Sciences and Engineering
Council (NSERC) of Canada.
References
1. N. Crampé, L. Poulain d’Andecy, L. Vinet, Temperley-Lieb, Brauer and Racah algebras and
other centralizers of su(2). Trans. Amer. Math. Soc. 373(7), 4907–4932 (2020)
2. N. Crampé, W. van de Vijver, L. Vinet, Racah problems for the oscillator algebra, the Lie
algebra sl n and multivariate Krawtchouk polynomials (2019). arXiv:1909.12643, 28 pages
3. H. De Bie, W. van de Vijver, A discrete realization of the higher rank Racah algebra. Constr.
Approx. 52, 1–29 (2020). https://doi.org/10.1007/s00365-019-09475-0
4. H. De Bie, V.X. Genest, W. van de Vijver, L. Vinet, A higher rank Racah algebra and the (Z 2 ) n
Laplace-Dunkl operator. J. Phys. A Math. Theor. 51, 025203 (20 pp.) (2018)
5. H. De Bie, P. Iliev, L. Vinet, Bargmann and Barut-Giradello models for the Racah algebra.
J. Math. Phys. 60, 011701 (2019)
6. J. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov, The generalized Racah algebra as a commutant.
J. Phys. Conf. Ser. 1194(1), 012034 (2019)
7. V.X. Genest, L. Vinet, A. Zhedanov, The equitable Racah algebra from three su(1, 1) algebras.
J. Phys. A Math. Theor. 47(2), 025203, 12 pp. (2014)
8. V.X. Genest, L. Vinet, A. Zhedanov, The Racah algebra and superintegrable models. J. Phys.
Conf. Ser. 512, 012011 (2014)
9. P. Iliev, The generic quantum superintegrable system on the sphere and Racah operators. Lett.
Math. Phys. 107(11), 2029–2045 (2017)
10. P. Iliev, Symmetry algebra for the generic superintegrable system on the sphere. J. High Energy
Phys. 2018(2), 44 (2018), front matter+22 pp.
11. R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their
q-Analogues (Springer, New York, 2010)
12. M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete
families. J. Math. Phys. 32, 2337–2342 (1991)
13. A.S. Zhedanov, “Hidden symmetry” of the Askey-Wilson polynomials. Theor. Math. Phys.
89, 1146–1157 (1991)