Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules

Abstract

For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an $\mr{Spin}(2, 2n+2)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n+1)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\mr{Spin}(2, 2n+2)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\mr{Spin}(2, 2n+2)$-module.

27 pages, Refs. updated

References in corpus (4)

• The MICZ-Kepler Problems in All Dimensions
• Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules -- II
• Dirac and Yang Monopoles Revisited
• Generalized Dirac monopoles in non-Abelian Kaluza-Klein theories

Cited by in corpus (8)

• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems
• Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules -- II
• The Sp(1)-Kepler Problems
• The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems
• The classical dynamic symmetry for the $\mathrm{U}(1)$-Kepler problems
• A characterization of the unitary highest weight modules by Euclidean Jordan algebras
• Families of Symmetries and the Hydrogen Atom
• Algebraic structure underlying spherical, parabolic and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problem