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# On Quantizing Nilpotent and Solvable Basic Algebras

Published:2001-06-01
Printed: Jun 2001
• Mark J. Gotay
• Janusz Grabowski
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## Abstract

We prove an algebraic no-go theorem'' to the effect that a nontrivial \pa\ cannot be realized as an associative algebra with the commu\-ta\-tor bracket. Using it, we show that there is an obstruction to quantizing the \pa\ of polynomials generated by a nilpotent \ba\ on a \sm. This result generalizes \gr 's famous theorem on the impossibility of quantizing the Poisson algebra of polynomials on $\r^{2n}$. Finally, we explicitly construct a polynomial quantization of a \sm\ with a solvable \ba, thereby showing that the obstruction in the nilpotent case does not extend to the solvable case.
 MSC Classifications: 81S99 - None of the above, but in this section 58F06 - unknown classification 58F06