Lie groupoids and Lie algebroids in physics and noncommutative geometry. (English) Zbl 1088.58009
This is a survey article on the links between Poisson geometry, noncommutative geometry and quantization, which is accessible for non-specialists of the subject. The author starts from the imprimitivity theorem by Mackey and the notion of Morita equivalence. He introduces Poisson groupoids and associated \(C^*\)-algebras. Then he defines Lie algebroids and focuses on the links between Lie algebroids and Poisson geometry. This allows him to introduce a classical analogue of Mackey’s theorem in Poisson geometry. Finally, he explains the different notions of quantization. The numerous references should help the reader who would want to discover this area.
Reviewer: Jean-François Quint (Villetaneuse)
MSC:
| 58H05 | Pseudogroups and differentiable groupoids |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 53D50 | Geometric quantization |
| 58B34 | Noncommutative geometry (à la Connes) |
| 81R60 | Noncommutative geometry in quantum theory |
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