Atiyah classes and dg-Lie algebroids for matched pairs

Abstract

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.

Introduction to graded geometry

“…In recent developments (see [1,15]) formal polynomial functions on graded manifolds have been considered with respect to a Z-graded vector space that has a non-trivial part of degree zero. This means that, in Definition 3.1, we could choose W such that dim W 0 1.…”

“…The graded manifold G T M [1] = (|M|, ) has dimension ( p j ) j∈Z , with p j = n if j = 0, 1 and p j = 0 otherwise. If (x i ) n i=1 are local coordinates on M, then (x i , dx i ) n i=1 is a local coordinate system on G T M [1] whose degrees are respectively 0 and 1.…”

“…The graded manifold G T M [1] = (|M|, ) has dimension ( p j ) j∈Z , with p j = n if j = 0, 1 and p j = 0 otherwise. If (x i ) n i=1 are local coordinates on M, then (x i , dx i ) n i=1 is a local coordinate system on G T M [1] whose degrees are respectively 0 and 1. The exterior differential d is a vector field on G T M [1] , since its restriction on any open set V ⊆ |M| is a derivation of (V ).…”

“…If (x i ) n i=1 are local coordinates on M, then (x i , dx i ) n i=1 is a local coordinate system on G T M [1] whose degrees are respectively 0 and 1. The exterior differential d is a vector field on G T M [1] , since its restriction on any open set V ⊆ |M| is a derivation of (V ). It is given locally by d = i dx i ∂/∂ x i .…”

“…If V 0 is a Lie algebra, one can show that its Chevalley-Eilenberg differential is a vector field of degree 1 which squares to zero. It endows G V 0 [1] with a homological vector field. Conversely, one can construct a Lie bracket, known as the derived bracket, from the homological vector field.…”