Contractions of Lie algebras and separation of variables. Two-dimensional hyperboloid

The Inonu-Wigner contraction from the Lorentz group O(2,1) to the Euclidean group E(2) is used to relate the separation of variables in the Laplace-Beltrami operators on the two corresponding homogeneous spaces. We consider the contractions on four levels: the Lie algebra, the commuting sets of second order operators in the enveloping algebra of o(2,1), the coordinate systems and some eigenfunctions of the Laplace-Beltrami operators.