On Quantizing Nilpotent and Solvable Basic Algebras

http://dx.doi.org/10.4153/CMB-2001-018-x
Canad. Math. Bull. 44(2001), 140-149

  Published:2001-06-01
 Printed: Jun 2001

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.