Deformation Quantization

Abstract

In this chapter we present a second application of Poisson structures: the theory of deformations of commutative associative algebras, also called deformation quantization. By definition, it consists of considering for a given Poisson manifold (M,{⋅,⋅}) all associative products ⋆ on the vector space C ^∞(M)[[ν]], which are of the form

$$ F\star G = FG + {1\over2}\{F,G\}\nu + C(F,G)\nu^2 + \cdots $$

Kontsevich’s formality theorem implies that for every Poisson manifold a deformation quantization exists. Its full proof is beyond the scope of this book, but we provide the material which allows the reader to understand and appreciate Kontsevich’s formality theorem and its farreaching consequences. Thus we treat in detail the relation between deformations and cohomology (both in the context of associative products and Poisson structures), and differential graded Lie algebras, covering partly the case of L _∞-morphisms of differential graded Lie algebras.