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

A $\mathcal{VB}$-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between $\mathcal{VB}$-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 $\mathcal{VB}$-groupoid. 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.

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Received 27 January 2015

Published 8 September 2017