On the Quantization of Poisson Brackets

✍ Scribed by Joseph Donin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 355 KB
• Volume: 127
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1626

✦ Synopsis

In this paper we introduce two classes of Poisson brackets on algebras (or on sheaves of algebras). We call them locally free and nonsingular Poisson brackets. Using the Fedosov's method we prove that any locally free nonsingular Poisson bracket can be quantized. In particular, it follows from this that all Poisson brackets on an arbitrary field of characteristic zero can be quantized. The well-known theorem about the quantization of nondegenerate Poisson brackets on smooth manifolds follows from the main result of this paper as well.

1997 Academic Press a skew-symmetric Hochschild cocycle, which gives a deformation of order one. Then, a skew-symmetric bilinear form A_A Ä A is a Hochschild cocycle if and only if it defines a biderivation with respect to the original multiplication, i.e. satisfies the Leibniz rule. This form must also satisfy the Jacobi identity if there exists an extension of the deformation up to order article no.