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A natural basis of states for the noncommutative sphere and its Moyal bracket

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<mark>Journal publication date</mark>08/1997
<mark>Journal</mark>Journal of Mathematical Physics
Issue number8
Number of pages18
Pages (from-to)4283-4300
<mark>Original language</mark>English


An infinite-dimensional algebra which is a nondecomposable reducible representation of su(2) is given. This algebra is defined with respect to two real parameters. If one of these parameters is zero, the algebra is the commutative algebra of functions on the sphere, otherwise it is a noncommutative analog. This is an extension of the algebra normally referred to as the (Berezin) quantum sphere or ''fuzzy'' sphere. A natural indefinite ''inner'' product and a basis of the algebra orthogonal with respect to it are given. The basis elements are homogeneous polynomials, eigenvectors of a Laplacian, and related to the Hahn polynomials. It is shown that these elements tend to the spherical harmonics far the sphere. A Moyal bracket is constructed and shown to be the standard Moyal bracket for the sphere. (C) 1997 American Institute of Physics.