Affiliation:
1. Belarusian State Pedagogical University Named After Maxim Tank
2. Octonion Technology Ltd.
Abstract
In the light of the Howe duality, two different, but isomorphic representations of one algebra as Higgs algebra and Hahn algebra are considered in this article. The first algebra corresponds to the symmetry algebra of a harmonic oscillator on a 2-sphere and a polynomially deformed algebra SU(2), and the second algebra encodes the bispectral properties of corresponding homogeneous orthogonal polynomials and acts as a symmetry algebra for the Hartmann and certain ring-shaped potentials as well as the singular oscillator in two dimensions. The realization of this algebra is shown in explicit form, on the one hand, as the commutant O(4) ⊕ O(4) of subalgebra U(8) in the oscillator representation of universal algebra U (u(8)) and, on the other hand, as the embedding of the discrete version of the Hahn algebra in the double tensor product SU(1,1) ⊗ SU(1,1). These two realizations reflect the fact that SU(1,1) and U(8) form a dual pair in the state space of the harmonic oscillator in eight dimensions. The N-dimensional, N-fold tensor product SU(1,1)⊗N аnd q-generalizations are briefly discussed.
Publisher
Publishing House Belorusskaya Nauka
Subject
Computational Theory and Mathematics,General Physics and Astronomy,General Mathematics
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