Abstract. For every Lie pair (L, A) of algebroids we construct a dg-manifold structure on theThe vertical tangent bundle T p M then inherits a structure of dg-Lie algebroid over M. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion ι induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras.

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.

Given a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of L-invariant fibrewise affine connections on the homogeneous space L/A. For Lie groupoid pairs with vanishing Atiyah class, we show that the left A-action on the quotient space L/A can be linearized.In addition to giving an alternative proof of a result of Calaque about the Poincaré-Birkhoff-Witt map for Lie algebroid pairs with vanishing Atiyah class, this result specializes to a necessary and sufficient condition for the linearization of dressing actions, and gives a clear interpretation of the Molino class as an obstruction to simultaneous linearization of all the monodromies.In the course of the paper, a general theory of connections on Lie groupoid equivariant principal bundles is developed. Contents2010 Mathematics Subject Classification. 53C05,53C30,53C12.

We introduce symplectic structures on "Lie pairs" of (real or complex) Lie algebroids as studied by Chen, Stiénon, and the second author in [4], encompassing homogeneous symplectic spaces, symplectic manifolds with a g-action, and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.