dorsal/arxiv
View SchemaA Natural Basis of States for the Noncommutative Sphere and its Moyal bracket
| Authors | J. Gratus |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9703038 |
| URL | https://arxiv.org/abs/q-alg/9703038 |
| DOI | 10.1063/1.532003 |
Abstract
An infinite dimensional algebra which is a non-decomposable 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 analogue. This is an extension of the algebra normally refered 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 homogenious polynomials, eigenvectors of a Laplacian, and related to the Hahn polynomials. It is shown that these elements tend to the spherical harmonics for the sphere. A Moyal bracket is constructed and shown to be the standard Moyal bracket for the sphere.
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"abstract": "An infinite dimensional algebra which is a non-decomposable reducible\nrepresentation of $su(2)$ is given. This algebra is defined with respect to two\nreal parameters. If one of these parameters is zero the algebra is the\ncommutative algebra of functions on the sphere, otherwise it is a\nnoncommutative analogue. This is an extension of the algebra normally refered\nto as the (Berezin) quantum sphere or ``fuzzy\u0027\u0027 sphere. A natural indefinite\n``inner\u0027\u0027 product and a basis of the algebra orthogonal with respect to it are\ngiven. The basis elements are homogenious polynomials, eigenvectors of a\nLaplacian, and related to the Hahn polynomials. It is shown that these elements\ntend to the spherical harmonics for the sphere. A Moyal bracket is constructed\nand shown to be the standard Moyal bracket for the sphere.",
"arxiv_id": "q-alg/9703038",
"authors": [
"J. Gratus"
],
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"q-alg",
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"doi": "10.1063/1.532003",
"title": "A Natural Basis of States for the Noncommutative Sphere and its Moyal bracket",
"url": "https://arxiv.org/abs/q-alg/9703038"
},
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