Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures

Abstract

Introduction. Let L be a finite dimensional Lie algebra over the real numbers, R, and let L* be its dual vector space. It is well-known [24] that the Lie algebra structure on L defines a natural Poisson structure on L*-in fact this was already known to Lie [24]-and these Poisson structures are exactly what are now called the linear Poisson structures. Given a manifold M equipped with a Poisson structure, { , one can seek deformation quantizations the direction of { , }, as first studied in [3]. These are, loosely speaking, one-parameter families, {*h}lER, of deformations of the pointwise multiplication on C<(M) (or an appropriate subalgebra), such that