Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 074, 29 pages      arXiv:0708.0516      https://doi.org/10.3842/SIGMA.2009.074
Contribution to the Special Issue on Deformation Quantization

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

Nikolai Neumaier and Stefan Waldmann
Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann Herder Straße 3, D-79104 Freiburg, Germany

Received September 26, 2008, in final form May 25, 2009; Published online July 16, 2009

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.

Key words: deformation quantization; Fedosov construction; duals of Lie algebroids; trace functionals.

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