On Quantizing Nilpotent and Solvable Basic Algebras

Details On Quantizing Nilpotent and Solvable Basic Algebras On Quantizing Nilpotent and Solvable Basic Algebras

1999-02-08

arxiv.org/abs/math-ph/9902012v2

Canadian Math. Bull. 44 (2001), 140-149.

Physics

Mathematical Physics

12 pages, Latex2e. Main result (Theorem 1) substantially strengthened; some rewriting

Scientific paper

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 this, we show that there is an obstruction to quantizing the Poisson algebra of polynomials generated by a nilpotent basic algebra on a symplectic manifold. 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.

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