Abstract
The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the d-sphere. Appropriately normalized, the symmetry operators preserve the space of polynomials. Under mild conditions on the free parameters, maximal abelian subalgebras of the symmetry algebra, generated by Jucys-Murphy elements, have unique common eigenfunctions consisting of families of Jacobi polynomials in d variables. We describe the action of the symmetries on the basis of Jacobi polynomials in terms of multivariable Racah operators, and combine this with different embeddings of symmetry algebras of lower dimensions to prove that the representations restricted on the space of polynomials of a fixed total degree are irreducible.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Appell and J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques - polynomes d’Hermite, Gauthier-Villars et Cie, Paris France (1926).
H. De Bie et al., A higher rank Racah algebra and the ℤ n2 Laplace-Dunkl operator, J. Phys. A 51 (2018) 025203 [arXiv:1610.02638].
C. F. Dunkl, Orthogonal polynomials with symmetry of order three, Canad. J. Math. 36 (1984) 685.
C.F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, 2nd edition, Encyclopedia of Mathematics and its Applications volume 155, Cambridge University Press, Cambridge U.K. (2014).
V. X. Genest and L. Vinet, The generic superintegrable system on the 3-sphere and the 9j symbols of \( \mathfrak{s}\mathfrak{u}\left(1,1\right) \), SIGMA 10 (2014) 108.
V. X. Genest, L. Vinet and A Zhedanov, Superintegrability in two dimensions and the Racah-Wilson algebra, Lett. Math. Phys. 104 (2014) 931.
J. Geronimo and P. Iliev, Bispectrality of multivariable Racah-Wilson polynomials, Constr. Approx. 31 (2010) 417.
P. Iliev, The generic quantum superintegrable system on the sphere and Racah operators, Lett. Math. Phys. 107 (2017) 2029.
P. Iliev and Y. Xu, Connection coefficients for classical orthogonal polynomials of several variables, Adv. Math. 310 (2017) 290 [arXiv:1506.04682].
E.G. Kalnins, J.M. Kress and W. Miller Jr., Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory, J. Math. Phys. 46 (2005) 103507.
E.G. Kalnins, W. Miller Jr. and S. Post, Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A 40 (2007) 11525.
E.G. Kalnins, W. Miller Jr. and S. Post, Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere, SIGMA 7 (2011) 051 [arXiv:1010.3032].
E.G. Kalnins, W. Miller Jr. and M.V. Tratnik, Families of orthogonal and biorthogonal polynomials on the N -sphere, SIAM J. Math. Anal. 22 (1991) 272.
T. Kohno, Conformal field theory and topology, Translations of Mathematical Monographs 210, American Mathematical Society Providence, U.S.A. (2002).
W. Miller Jr., S. Post and P. Winternitz, Classical and quantum superintegrability with applications, J. Phys. A 46 (2013) 423001.
W. Miller Jr. and A. V. Turbiner, (Quasi)-exact-solvability on the sphere S n, J. Math. Phys. 56 (2015) 023501.
G. Munschy and P. Pluvinage, Résolution de l’équation de Schrödinger des atomes à deux électrons. II. Méthode rigoureuse. États s symétriques, J. Phys. Radium 18 (1957) 157.
S. Post, Racah polynomials and recoupling schemes of \( \mathfrak{s}\mathfrak{u}\left(1,1\right) \), SIGMA 11 (2015) 057 [arXiv:1504.03705].
J. Proriol, Sur une famille de polynomes à deux variables orthogonaux dans un triangle, C. R. Acad. Sci. Paris 245 (1957) 2459.
M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys. 32 (1991) 2337.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1712.06422
The author is partially supported by Simons Foundation Grant #280940. (Plamen Iliev)
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Iliev, P. Symmetry algebra for the generic superintegrable system on the sphere. J. High Energ. Phys. 2018, 44 (2018). https://doi.org/10.1007/JHEP02(2018)044
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2018)044