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Paper IPM / Physic / 7298  


Abstract:  
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 AdS_{2} and S^{2} split into infinite direct sums of infinite and finitedimensional Hilbert subspaces which represent the Lie algebras u(1,1) and u(2) with the infinite and finitefold 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 nonunitary parasupersymmetry algebra of arbitrary order. The representation of parasupersymmetry algebra by the Hilbert subspaces which describe the motion of a free particle on AdS_{2} and S^{2} with the dynamical symmetry groups U(1,1) and U(2) is concluded as well.
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