Abstract
In this paper, we will prove that the reproducing kernels \(Z_k({\underline{x}},{\underline{u}})\) for the spaces \({\mathcal {H}}_k({\mathbb {R}}^m,{\mathbb {C}})\) of k-homogeneous harmonics can be seen as elements of an infinite-dimensional ladder operator representation for a cubic polynomial angular momentum algebra which is known as the Higgs algebra. This algebra will be shown to be one of two direct summands in a transvector algebra which is related to the harmonic Fischer decomposition in two vector variables.
Similar content being viewed by others
References
Bonatsos, D., Daskaloyannis, C., Kokkotas, K.: Deformed oscillator algebras for two-dimensional quantum superintegrable systems. Phys. Rev. A 50, 3700 (1994)
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics 76. Pitman, London (1982)
Constales, D., Sommen, F., Van Lancker, P.: Models for irreducible Spin\((m)\)-modules. Adv. Appl. Cliff. Algebra 11, 271–289 (2001)
Coulembier, K., Kieburg, M.: Pizzetti formulae for Stiefel manifolds and applications. Lett. Math. Phys. 105(10), 1333–1376 (2015)
De Bie, H., Eelbode, D., Roels, M.: The harmonic transvector algebra in two vector variables. J. Algebra 473, 247–282 (2017)
De Bie, H., Genest, V., van de Vijver, W., Vinet, L.: A higher rank Racah algebra and the \({\mathbb{Z} }_{2}^{n}\) Laplace-Dunkl operator. J. Phys. A (2018). https://doi.org/10.1088/1751-8121/aa9756
Delanghe, R., Sommen, F., Souček, V.: Clifford Analysis and Spinor Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)
Eastwood, M.: Higher symmetries of the Laplacian. Ann. Math. 161(3), 1645–1665 (2005)
Eelbode, D., Homma, Y.: Pizzetti formula on the Grassmannian of 2-planes. Ann. Glob. Anal. Geom. 25(3), 1–26 (2020)
Gilbert, J., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser-Verlag, Basel (1990)
Higgs, P.W.: Dynamical symmetries in a spherical geometry I. J. Phys. A Math. Gen. 12, 309 (1979)
Zhedanov, A.S.: The Higgs algebra as a quantum deformation of \(SU(2)\). Mod. Phys. Lett. A 7, 507 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Daniele Struppa.
Dedicated to John Ryan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of Topical Collection in Honor of Prof. John Ryan’s Retirement
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Eelbode, D. Higgs Algebras in Classical Harmonic Analysis. Complex Anal. Oper. Theory 18, 42 (2024). https://doi.org/10.1007/s11785-023-01458-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-023-01458-1