Contractions of Lie algebras and separation of variables. The n -dimensional sphere

Inönü–Wigner contractions from the rotation group O(n+1) to the Euclidean group E(n) are used to relate the separation of variables in Laplace–Beltrami operators on n -dimensional spheres and Euclidean spaces, respectively. In this article we consider all subgroup type coordinates corresponding to d...

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Bibliographic Details
Published in:Journal of mathematical physics 1999-03, Vol.40 (3), p.1549-1573
Main Authors: Izmest’ev, A. A., Pogosyan, G. S., Sissakian, A. N., Winternitz, P.
Format: Article
Language:English
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Summary:Inönü–Wigner contractions from the rotation group O(n+1) to the Euclidean group E(n) are used to relate the separation of variables in Laplace–Beltrami operators on n -dimensional spheres and Euclidean spaces, respectively. In this article we consider all subgroup type coordinates corresponding to different chains of subgroups of O(n+1) and E(n), respectively. In particular, the contractions relate the graphical formalism of “trees” on spheres to the “clusters” on Euclidean spaces (introduced in this article). The contractions are considered analytically on several levels: the vector fields realizing the Lie algebras, the complete sets of commuting operators characterizing separable coordinate systems, the coordinate systems themselves and the separated eigenfunctions.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.532820