Deformations of Lie Groupoids

Abstract

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser's deformation arguments for groupoids, we obtain several rigidity and normal form results.

On the Lie 2-algebra of sections of an LA-groupoid

“…As defined in [14], a horizontal lift of V is a right splitting h : s * E −→ V of (5) which at every unit x ∈ M coincides with the canonical splitting (8). As shown in [14], horizontal lifts of a VB-groupoid always exist.…”

“…In [8] the authors define and study the deformation complex C • def (G) of a Lie groupoid G. Propositions 3.9 and 4.3 in loc. cit.…”

Deformation cohomology of Lie algebroids and Morita equivalence

“…Proof. The present proof is inspired by the Crainic, Mestre, and Struchiner proof of the Morita invariance of the deformation cohomology of Lie groupoids [7]. However, notice that, in our statement, the Lie algebroid A needs not to be integrable.…”

“…In this note, we prove the analogous result for the deformation cohomology of Lie algebroids. Notice that, for Lie groupoids, the Morita invariance of Lie groupoid cohomology has been proved by Crainic himself in [4], while the deformation cohomology has been introduced, and its Morita invariance has been proved, only very recently, by Crainic, Mestre, and Struchiner in [7].…”

The general linear 2-groupoid

“…They can be easily defined as differentials on certain bigraded algebra of sections, or alternatively, they can be regarded as a sequence of tensors: the first one is a differential ∂ on V , the second one consists of chain maps ρ g : V x → V y between the fibers, the third one γ h,g provide chain homotopies relating ρ hg and ρ h ρ g , etc. Representation up to homotopy has proven to be a useful concept, for instance, when dealing with cohomology theory [1], deformations [6] and Morita equivalences [8].…”