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Symplectic groupoids, geometric quantization, and irrational rotation algebras. (English) Zbl 0731.58031

Symplectic geometry, groupoids, and integrable systems, Sémin. Sud- Rhodan. Geom. VI, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 20, 281-290 (1991).
[For the entire collection see Zbl 0722.00026.]
The paper is devoted to the groupoid approach to quantization of Poisson manifolds. Under fairly general circumstances, with a Poisson manifold P there can be associated a symplectic manifold \(\Gamma_ p\) called its symplectic groupoid. If one can carry out the process of geometric quantization on \(\Gamma_ p\) in a way consistent with the groupoid structure, the groupoid operations on \(\Gamma_ p\) induce on the vector space H quantizing \(\Gamma_ p\) a bilinear multiplication making H to a *-algebra with identity. The case of P being a torus with a translation- invariant Poisson structure is considered in detail.

MSC:

53D50 Geometric quantization
55R91 Equivariant fiber spaces and bundles in algebraic topology
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0722.00026