Deformation quantization of Poisson structures associated to Lie algebroids. (English) Zbl 1189.53086

Neumaier, Nikolai; Waldmann, Stefan

SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 074, 29 p. (2009).

Summary: In the present paper we explicitly construct deformation quantizations of certain Poisson structures on \(E^{*}\), where \(E \rightarrow M\) is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich’s formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of \(E\) we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and self-equivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on \(E^{*}\) the integration with respect to a density with vanishing modular vector field defines a trace functional.

MSC:

53D55	Deformation quantization, star products
53D17	Poisson manifolds; Poisson groupoids and algebroids

Keywords:

deformation quantization; Fedosov construction; duals of Lie algebroids; trace functionals