Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems

TL;DR

This paper explores the hidden actions of conformal algebras of simple Euclidean Jordan algebras on certain manifolds, revealing their role in the structure of generalized Kepler problems and their minimal representations.

Contribution

It explicitly constructs the hidden action of the conformal algebra on smooth functions, linking Euclidean Jordan algebras to generalized Kepler problems and their minimal representations.

Findings

Explicit hidden action of conformal algebra on smooth functions

Reconstruction of J-Kepler problems using Jordan algebra framework

Realization of minimal representations as bound states or L^2 spaces

Abstract

For a {\em simple Euclidean Jordan algebra}, let $\mathfrak{co}$ be its conformal algebra, $\mathscr P$ be the manifold consisting of its semi-positive rank-one elements, $C^\infty(\mathscr P)$ be the space of complex-valued smooth functions on $\mathscr P$. An explicit action of $\mathfrak{co}$ on $C^\infty(\mathscr P)$, referred to as the {\em hidden action} of $\mathfrak{co}$ on $\mathscr P$, 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 $L^2({\mathscr P}, {1\over r}\mathrm{vol})$, where $\mathrm{vol}$ is the volume form on $\mathscr P$ and $r$ is the inner product of $x\in \mathscr P$ with the identity element of the Jordan algebra.