dorsal/arxiv
View SchemaSubalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for $\complex \mbox{\LARGE P}^n$
| Authors | M. Bordemann, M. Brischle, C. Emmrich, S. Waldmann |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9512019 |
| URL | https://arxiv.org/abs/q-alg/9512019 |
| DOI | 10.1063/1.531779 |
Abstract
Based on a closed formula for a star product of Wick type on $\CP^n$, which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the uniformly dense subspace of representative functions with respect to the canonical action of the unitary group) that consists of {\em converging} power series in the formal parameter, thereby giving an elementary algebraic proof of a convergence result already obtained by Cahen, Gutt, and Rawnsley. In this subalgebra the formal parameter can be substituted by a real number $\alpha$: the resulting associative algebras are infinite-dimensional except for the case $\alpha=1/K$, $K$ a positive integer, where they turn out to be isomorphic to the finite-dimensional algebra of linear operators in the $K$th energy eigenspace of an isotropic harmonic oscillator with $n+1$ degrees of freedom. Other examples like the $2n$-torus and the Poincar\'e disk are discussed.
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"abstract": "Based on a closed formula for a star product of Wick type on $\\CP^n$, which\nhas been discovered in an earlier article of the authors, we explicitly\nconstruct a subalgebra of the formal star-algebra (with coefficients contained\nin the uniformly dense subspace of representative functions with respect to the\ncanonical action of the unitary group) that consists of {\\em converging} power\nseries in the formal parameter, thereby giving an elementary algebraic proof of\na convergence result already obtained by Cahen, Gutt, and Rawnsley. In this\nsubalgebra the formal parameter can be substituted by a real number $\\alpha$:\nthe resulting associative algebras are infinite-dimensional except for the case\n$\\alpha=1/K$, $K$ a positive integer, where they turn out to be isomorphic to\nthe finite-dimensional algebra of linear operators in the $K$th energy\neigenspace of an isotropic harmonic oscillator with $n+1$ degrees of freedom.\nOther examples like the $2n$-torus and the Poincar\\\u0027e disk are discussed.",
"arxiv_id": "q-alg/9512019",
"authors": [
"M. Bordemann",
"M. Brischle",
"C. Emmrich",
"S. Waldmann"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1063/1.531779",
"title": "Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for $\\complex \\mbox{\\LARGE P}^n$",
"url": "https://arxiv.org/abs/q-alg/9512019"
},
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