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It is shown that the set of all quantum states corresponding to
the motion of a free particle on the group manifold $AdS_{3}$ as
the bases with two different labels, constitute a Hilbert space.
The second label is bounded by the first one however, the first
label is semibounded. The Casimir operator corresponding to the
simultaneous and agreeable shifting generators of both labels
along with the Cartan subalgebra generator describe the
Hamiltonian of a free particle on $AdS_{3}$ with dynamical
symmetry group U(1,1) and infinite-fold degeneracy for the energy
spectrum. The Hilbert space for the Lie algebra of the dynamical
symmetry group is a reducible representation space. But the
Hilbert subspaces constructed by all the bases which have a given
constant value for the difference of two their labels, constitute
an irreducible representation for it. It is also shown that the
irreducible representation subspaces of the Lie algebras u(1,1)
and u(2) are separately spanned by the bases which have the same
value for the second and first labels, respectively. These two
bunches of Hilbert subspaces present two different types of
quantum splittings on the Hilbert space.
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