Short Abstracts

Abstract

A cone is a manifold with a free action of the multiplicative group R ₊. A star product on a cone is an associative algebra law (f, g) → f * g on the space of formal series Σ _{k ≤ k 0} f _k of homogeneous functions of degree k, an integer → −∞, for which 1 is unit: f * (g * h) = (f * g) * h, 1 * f = f * 1 = f, the law being given by a formal bilinear differential operator f *g = Σ_k≥0 L _k (f,g) where for each k, L _k is a bilinear differential operator homogeneous of degree −k. Then L ₀(f,g) = fg is the usual product law, and {f, g} = L ₁ (f, g) − L ₁ (g, f) is a Poisson bracket on X, homogeneous of degree −1. Using Fedosov’s method we show that any homogeneous Poisson bracket of constant rank is associated to some star-product.