Deformation Quantization of Poisson Structures Associated to Lie Algebroids

Abstract

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 selfequivalences 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.

The Strong Homotopy Structure of BRST Reduction

“…The anchor is given by ρ(p, ξ) = −ξ C p . In particular, one can check that π KKS is the negative of the linear Poisson structure on its dual C × g * in the convention of [30].…”

On Almost Complex Lie Algebroids

“…We notice that by the relation It is well known, see [33,41], that there is an one-to-one correspondence between Lie algebroid structures on the vector bundle p : E → M and specific Poisson structures on the total space of the corresponding dual bundle E * → M , called the dual Poisson structures, defined by the following brackets of basic and fiber-linear functions (with respect to foliation by fibres of E * ):…”

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

“…To name just a few, work on the various generalizations continues, for example on Jacobi-Nijenhuis algebroids [18], while the implications of the properties of the modular classes of Poisson-Nijenhuis structures on manifolds and on Lie algebroids for the theory of integrable systems are being studied [4]. Closely related works are those of Launois and Richard on Poisson algebras [83], that of Dolgushev [26] on the exponentiation of the modular class into an automorphism of an associative, non-commutative algebra, quantizing a Poisson algebra, and that of Neumaier and Waldmann on the relationship between unimodularity and the existence of a trace in a quantized algebra [103]. The relationship between unimodularity and the existence of a trace on non-commutative algebras was already stressed by Weinstein in [124] and should be the subject of further work.…”

“…This vertical lift, in turn, can be identified with a vector field on F , tangent to the fibers and invariant by translations along each fiber. In this way, the modular class of a Lie algebroid E can be identified [124,103] with the modular class of the Poisson manifold E * , when E * is equipped with the linear Poisson structure (see, e.g., [94] [34]). …”