The deformation quantizations of the hyperbolic plane
The paper gives an explicit description of all invariant deformation quantizations of the hyperbolic plane. The formal deformation quantization program, inaugurated by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer, proposes a generalization of the non formal Weyl product to an arbitrary Poisson manifold. In this work, the authors concentrate on the hyperbolic plane. Observing that the hyperbolic plane admits a unique contraction into a generic co-adjoint orbit of the Poincaré group, they give a detailed description of the space of all invariant, both formal and non formal, deformation quantizations of the hyperbolic plane. The method used gives in the 2-dimensional context an exact solution to Weinstein's WKB quantization program within geometric terms.
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