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Abstract. We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Dirac-isospectral noncommutative deformations of the spectral triples of locally anti de Sitter black holes. The study is motivated by physical and cosmological considerations.

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In this paper, we study the relevant structure of the algebra so(2, n) which makes the BTZ black hole possible in the anti de Sitter space AdS = SO(2, n)/SO (1, n). We pay a particular attention on the reductive Lie algebra structures of so(2, n) and we study how this structure evolves when one increases the dimension.As in [1] and [2], we define the singularity as the closed orbits of the Iwasawa subgroup of the isometry group of anti de Sitter, but here, we insist on an alternative (closely related to the original conception of the BTZ black hole) way to describe the singularity as the loci where the norm of fundamental vector vanishes. We provide a manageable Lie algebra oriented formula which describes the singularity and we use it in order to derive the existence of a black hole and to give a geometric description of the horizon. We also define a coherent structure of black hole on AdS2.This paper contains a "short" and a "long" version of the text. In the short version, only the main results are exposed and the proofs are reduced to the most important steps in order to be easier to follow the developments. The long version contains all the intermediate steps and computations for the sake of completeness.