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The universal Kepler problem. (English) Zbl 1353.70041

Summary: For each simple Euclidean Jordan algebra \(V\), we introduce the analogue of Hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. Being referred to as the universal Hamiltonian, universal angular momentum and universal Laplace-Runge-Lenz vector respectively, they are elements in (essentially) the TKK (Tits-Kantor-Koecher) algebra of \(V\) and satisfy commutation relations similar to the ones for the Hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. We also give some examples of Poisson realization of the TKK algebra, along with the resulting classical generalized Kepler problems. For the simplest simple Euclidean Jordan algebra (i.e., \(R\)), we give examples of operator realization for the TKK algebra, along with the resulting quantum generalized Kepler problems.

MSC:

70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
17C50 Jordan structures associated with other structures
70F10 \(n\)-body problems
17B63 Poisson algebras
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