Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids

Details Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids

2006-04-03

arxiv.org/abs/math/0604027v4

Mathematics

Symplectic Geometry

40 pages, corrected version 11-01-2006

Scientific paper

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of K\"{a}hler quantization suitable for this setting. We proceed by defining a Marsden-Weinstein quotient for our setting and prove a ``quantization commutes with reduction'' theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of families of Lie groups, foliations, as well as some general constructions from differential geometry.

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