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Subgroups of Lie groups and separation of variables

J. Math. Phys. 22, 251 (1981); doi:10.1063/1.524896

Issue Date: February 1981

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W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

J. Patera and P. Winternitz
Centre de recherches mathématiques, Université de Montréal, Montréal, Québec, Canada
Separable systems of coordinates for the Helmholtz equation DeltadPsi =EPsi 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 Deltad. They have also been characterized geometrically by the form that the metric ds2=gik(x)dxidxk 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 Deltad. 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. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
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KEYWORDS and PACS

Keywords
PACS
  • 02.20.Qs
    Mathematical methods in physics Group theory General properties, structure, and representation of Lie groups
  • YEAR: 1981

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (39)
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