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.

Let G be a connected and simply connected Lie group with Lie algebra g of finite dimension.

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the * − product formula are weight homogeneous. In the linear Poisson case X = g * for a semi simple Lie algebra g. As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra H 0 (g, m, χ) and the W − algebra (U (g)/U (g)m χ ) m . We also show that in the W − algebra setting,m is polynomial. Finally, we compute generators of H 0 (g, m, χ) as a deformation of (S(g)/S(g)m χ ) m .MSC 2010: 53D55, 20G42, 17B35, 17B20.

Presented by Michel DufloIn this Note we explain how the techniques of deformation quantization in the sense of Kontsevich can be used to describe the algebra of invariant differential operators on Lie groups.

No abstract