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)

Text
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## 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 |
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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 |

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