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On Quantizing Nilpotent and Solvable Basic Algebras

Published online by Cambridge University Press:  20 November 2018

Mark J. Gotay  and
Janusz Grabowski
Mark J. Gotay
Affiliation:
Department of Mathematics University of Hawai‘i 2565 The Mall Honolulu, HI 96822 USA, e-mail: gotay@math.hawaii.edu
Janusz Grabowski
Affiliation:
Institute of Mathematics University of Warsaw ul. Banacha 2 02-097 Warsaw Poland, e-mail: jagrab@mimuw.edu.pl
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Abstract

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We prove an algebraic “no-go theorem” to the effect that a nontrivial Poisson algebra cannot be realized as an associative algebra with the commutator bracket. Using it, we show that there is an obstruction to quantizing the Poisson algebra of polynomials generated by a nilpotent basic algebra on a symplectic manifold. This result generalizes Groenewold’s famous theorem on the impossibility of quantizing the Poisson algebra of polynomials on ${{\mathbf{R}}^{2n}}$. Finally, we explicitly construct a polynomial quantization of a symplectic manifold with a solvable basic algebra, thereby showing that the obstruction in the nilpotent case does not extend to the solvable case.

Keywords

81S9958F06
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 2 , 01 June 2001 , pp. 140 - 149
Copyright
Copyright © Canadian Mathematical Society 2001

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