Deformation Quantization of Poisson Structures Associated to Lie Algebroids

TL;DR

This paper constructs explicit deformation quantizations of Poisson structures on the dual of a Lie algebroid using a geometric Fedosov approach, avoiding Kontsevich's formality theorem, and explores their properties and relations to universal enveloping algebras.

Contribution

It introduces a geometric Fedosov-based method for quantizing Poisson structures on Lie algebroid duals, providing explicit formulas and analyzing their algebraic and trace properties.

Findings

Explicit formulas for star products on E^*

Relation of some products to universal enveloping algebras

Existence of trace functionals for certain star products

Abstract

In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E^*, where E -> 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.