Algebra of Dunkl Laplace–Runge–Lenz vector

Feigin, M. and Hakobyan, T. (2022) Algebra of Dunkl Laplace–Runge–Lenz vector. Letters in Mathematical Physics, 112, 59. (doi: 10.1007/s11005-022-01551-0)

Abstract

We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in RN and with a multiplicity function g. This vector generalizes the usual Laplace–Runge–Lenz vector and its components commute with the Dunkl–Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential γ/r. We study the resulting symmetry algebra Rg, γ(W) and show that it has the Poincaré–Birkhoff–Witt property. In the absence of a Coulomb potential, this symmetry algebra Rg, 0(W) is a subalgebra of the rational Cherednik algebra Hg(W). We show that a central quotient of the algebra Rg, γ(W) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra Hgso(N+1)(W). This gives an interpretation of the algebra Hgso(N+1)(W) as the hidden symmetry algebra of the Dunkl–Coulomb problem in RN. By specialising Rg, γ(W) to g=0, we recover a quotient of the universal enveloping algebra U(so(N+1)) as the hidden symmetry algebra of the Coulomb problem in RN. We also apply the Dunkl Laplace–Runge–Lenz vector to establish the maximal superintegrability of the generalised Calogero–Moser systems.

• Item Type: Articles
• Additional Information: The work of M.F. (Sects. 4, 5) was supported by the Russian Science Foundation Grant No. 20-11-20214. The work of T.H. was supported by the Armenian Science Committee Grants No. 20TTWS-1C035, No. 20TTAT-QTa009, and No. 21AG-1C047.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Feigin, Professor Misha
• Authors: Feigin, M., and Hakobyan, T.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Letters in Mathematical Physics
• Publisher: Springer
• ISSN: 0377-9017
• ISSN (Online): 1573-0530
• Copyright Holders: Copyright © The Author(s) 2022
• First Published: First published in Letters in Mathematical Physics 112:59
• Publisher Policy: Reproduced under a Creative Commons license