“School of Physic”
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| Paper IPM / Physic / 7298 |
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Using two different types of the ladder equations realized simultaneously by the associated Gegenbauer functions, we show that all quantum states corresponding to the motion of a free particle on AdS2 and S2 split into infinite direct sums of infinite- and finite-dimensional Hilbert subspaces which represent the Lie algebras u(1,1) and u(2) with the infinite- and finite-fold degeneracies, respectively. In addition, it is shown that the representation bases of Lie algebras with rank one, i.e. gl(2,C), realize the representation of non-unitary parasupersymmetry algebra of arbitrary order. The representation of parasupersymmetry algebra by the Hilbert subspaces which describe the motion of a free particle on AdS2 and S2 with the dynamical symmetry groups U(1,1) and U(2) is concluded as well.
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