Subgroups of Lie groups and separation of variables

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Abstract
Separable systems of coordinates for the Helmholtz equation Δ d Ψ =EΨ in 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 d . They have also been characterized geometrically by the form that the metric d s 2=g i k (x)d x i d x 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 algebraL 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 S 1,S 2 that are not associated with separable coordinates. For subgroup related operators we show in detail just which symmetries S 1,S 2 fail to define separation and why this failure occurs.