Quantization of Poisson algebras associated to Lie algebroids

Abstract

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded as a result of a quantization procedure. The C*-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra g) is the C*-algebra of a continuous field of C*-algebras over R with fibers A_0=C*(g)=C_0(g*) and A_h=C*(G) for nonzero h. The same is true for the corresponding reduced C*-algebras. Our results have applications to, e.g., transformation group C*-algebras, K-theory, and index theory.

arXiv comment34 pages, minor corrections

Cited by in corpus (7)

• Strict Deformation Quantization for a Particle in a Magnetic Field
• Some remarks about the weak containment property for groupoids and semigroups
• Nambu-Poisson manifolds and associated n-ary Lie algebroids
• A Continuous Field of C*-algebras and the Tangent Groupoid for Manifolds with Boundary
• Fell bundles associated to groupoid morphisms
• Approximation of Quantum Tori by Finite Quantum Tori for the Quantum Gromov-Hausdorff Distance
• Quantum Systems as Lie Algebroids