Deformation Quantization of Polynomial Poisson Algebras

Details Deformation Quantization of Polynomial Poisson Algebras Deformation Quantization of Polynomial Poisson Algebras

1998-04-03

arxiv.org/abs/math/9804022v1

Mathematics

Quantum Algebra

33 pages, no figures

Scientific paper

This paper discusses the notion of a deformation quantization for an arbitrary polynomial Poisson algebra A. We examine the Hochschild cohomology group H^3(A) and find that if a deformation of A exists it can be given by bidifferential operators. We then compute an explicit third order deformation quantization of A and show that it comes from a quantized enveloping algebra. We show that the deformation extends to a fourth order deformation if and only if the quantized enveloping algebra gives a fourth order deformation; moreover we give an example where the deformation does not extend. A correction term to the third order quantization given by the enveloping algebra is computed, which precisely cancels the obstruction.

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