ar X iv : 0 80 1 . 32 33 v 1 [ m at h . A G ] 2 1 Ja n 20 08 Twisted Deformation Quantization of Algebraic Varieties Lecture

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold (the phase space), and C = C ∞ (X), the ring of differentiable R-valued functions on X. Example 1.2. Lie theory: let g be a finite dimensional Lie algebra. Then C := Sym g can be identified with O(g *), the ring of algebraic functions on the dual vector space g *. There is an intrinsically defined Poisson bracket called the Kostant-Kirillov bracket.