Euclidean Jordan algebras, hidden actions, and J-Kepler problems

Abstract

For a simple euclidean Jordan algebra, let $\mathfrak {co}$co be its conformal algebra, $\mathscr P$P be the manifold consisting of its semi-positive rank-one elements, $C^\infty (\mathscr P)$C∞(P) be the space of complex-valued smooth functions on $\mathscr P$P. An explicit action of $\mathfrak {co}$co on $C^\infty (\mathscr P)$C∞(P), referred to as the hidden action of $\mathfrak {co}$co on $\mathscr P$P, is exhibited. This hidden action turns out to be mathematically responsible for the existence of the Kepler problem and its recently discovered vast generalizations, referred to as J-Kepler problems. The J-Kepler problems are then reconstructed and re-examined in terms of the unified language of euclidean Jordan algebras. As a result, for a simple euclidean Jordan algebra, the minimal representation of its conformal group can be realized either as the Hilbert space of bound states for its J-Kepler problem or as $L^2({\mathscr P}, {1\over r}\mathrm{vol})$L2(P,1r vol ), where vol is the volume form on $\mathscr P$P and r is the inner product of $x\in \mathscr P$x∈P with the identity element of the Jordan algebra.