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arXiv · 1007.3658

VB-groupoids and representation theory of Lie groupoids

Alfonso Gracia-Saz·Rajan Amit Mehta
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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 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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Alfonso Gracia-Saz, Rajan Amit Mehta. 2016-06-05. VB-groupoids and representation theory of Lie groupoids. https://doi.org/10.4310/jsg.2017.v15.n3.a5

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