Subgroups of Lie groups and separation of variables

Separable systems of coordinates for the Helmholtz equation Δ_dΨ = EΨ pseudo-Riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator Δ. They have also been characterized geometrically by the form that the metric ds² = g _ik (x)dxⁱdx^k can take. We complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of G, the symmetry group of Δ_d. For d = 3 we study all separable coordinates that can be characterized in terms of the Lie algebra L of G and show that they are of eight types, seven of which are related to the subgroup structure of G. Our method clearly generalizes to the case d > 3. Although each separable system corresponds to a pair of commuting symmetry operators, there do exist pairs of commuting symmetries S₁,S₂ that are not associated with separable coordinates. For subgroup related operators we show in detail just which symmetries S₁,S₂ fail to define separation and why this failure occurs.