$\mathcal{VB}$-groupoids and representation theory of Lie groupoids

Abstract

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint representation" of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not.We define a cochain complex that is canonically associated to any VBgroupoid. The cohomology of this complex is isomorphic to the groupoid cohomology with values in the corresponding representations up to homotopy. When applied to the tangent bundle of a Lie groupoid, this construction produces a canonical complex that computes the cohomology with values in the adjoint representation.Finally, we give a classification of regular 2-term representations up to homotopy. By considering the adjoint representation, we find a new cohomological invariant associated to regular Lie groupoids.

Differential forms with values in VB-groupoids and its Morita invariance

“…and Ω g1,g2 : E s(g2) → C t(g1) the curvature term associated to the representation (see [1,22]). The semidirect product of G with the ruth is the vector bundle V = t * C ⊕ s * E → G endowed with the VB-groupoid structure V ⇒ E given by:…”

“…Definitions and the fat representations. Following [22], we define the fat category of V as the category whose space of objects is F (V), the space of pointwise splittings of the core exact sequence (2.2). More precisely, an element of…”

“…Recall that VB-groupoids are, roughly, vector bundles in the category of Lie groupoids. They naturally extend the notion of Lie-groupoid representations by enconding the information of representations up to homotopy [22] (see also [1]), with the tangent bundle playing the role of the adjoint representation. The infinitesimal counterparts of VB-groupoids are known as VB-algebroids.…”

“…, g p ) is the 0th face map. Note that C •,0 (V) = C • VB (V), the VB-groupoid complex introduced in [22] to calculate the cohomology of G with values in 2-term representation up to homotopy. Also, the differential δ : C •,q (V) → C •+1,q (V) (see (4.2) for the explicit formula) is a natural extension of the differential on C • VB (V).…”

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

“…Contrary to the situation for Lie groups, general Lie groupoids do not admit an "adjoint representation" (see however 3.13 and Example 3.14). As a consequence, one was led to the generalised notion of a "representations up to homotopy", see [GSM17]. In the present article, we will only consider the classical concept, which appears naturally in applications (e.g.…”