In a general setting, a deformation of an algebra A over a commutative ring is its Gerstenhaber extension to an algebra over the ring of formal power series in a real variable h such that ? A [167]. By deformation quantization is meant a deformation of a Poisson algebra of functions on a Poisson manifold where h is treated as a Plank constant [30]. One also considers a generalized deformation where a deformation parameter no longer commutes with elements of the original algebra [330; 332; 358].

7.1 Gerstenhaber's Deformation of Algebras

Subsections: A. Formal deformation, 433; B. Deformation of associative algebras, 436; C. Relative deformation, 439; D. Commutative deformation; 441; E. Deformation of Lie algebras. 442.

This Section summarizes the relevant material on deformations of algebraic structures, especially, associative algebras and Lie algebras [152; 167; 168]. A deformation parameter h throughout is real.

A. Formal Deformation

Let A be a (not necessarily associative) algebra. Let us consider the set of formal power series

in a real variable h whose coefficients are elements of A. This set is customarily denoted by A[[ h]]. It is naturally an algebra, called the power series algebra, with respect to the formal sum and product of power series.

For instance, let be a commutative ring. Then is also a commutative ring. Without a loss of generality, we further assume that any algebra A