  Dynamics of small-amplitude perturbations in the global anti-de Sitter (AdS)
spacetime is restricted by selection rules that forbid effective energy
transfer between certain sets of normal modes. The selection rules arise
algebraically because some integrals of products of AdS mode functions vanish.
Here, we reveal the relation of these selection rules to AdS isometries. The
formulation we discover through this systematic approach is both simpler and
stronger than what has been reported previously. In addition to the selection
rule considerations, we develop a number of useful representations for the
global AdS mode functions, with connections to algebraic structures of the
Higgs oscillator, a superintegrable system describing a particle on a sphere in
an inverse cosine-squared potential, where the AdS isometries play the role of
a spectrum-generating algebra.
