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 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.

v5: Introduction is completely rewritten, many other improvements in the exposition. v6: Implements numerous corrections and changes suggested by referees, most notably a significant simplification of the calculations in Appendix A.2. Final version, to appear in J. Symp. Geom

References in corpus (3)

• Vector bundles over Lie groupoids and algebroids
• Obstructions to the integrability of VB-algebroids
• Duality and triple structures

Cited by in corpus (28)

• Vector bundles over Lie groupoids and algebroids
• Integration of Exact Courant Algebroids
• Riemannian metrics on differentiable stacks
• Deformations of Lie groupoids
• Morita equivalences of vector bundles
• Holomorphic Jacobi Manifolds and Holomorphic Contact Groupoids
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Differential forms with values in VB-groupoids
• Graded Bundles in the Category of Lie Groupoids
• VB-structures and generalizations
• Invariant connections and PBW theorem for Lie groupoid pairs
• Modular Classes of Lie Groupoid Representations up to Homotopy
• Shifted Poisson structures on differentiable stacks
• Poisson double structures
• Double Groupoids, Orbifolds, and the Symplectic Category
• Multiplicative Connections and Their Lie Theory
• Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
• The general linear 2-groupoid
• Linking Lie groupoid representations and representations of infinite-dimensional Lie groups
• Deformations of symplectic groupoids
• The Classification of Dirac Homogeneous Spaces
• On the reduced space of multiplicative multivectors
• On the Lie 2-algebra of sections of an LA-groupoid
• Higher Vector Bundles
• Interior Kasparov product for $\varrho$-classes on Riemannian foliated bundles
• Categorification of VB-Lie algebroids and VB-Courant algebroids
• Weak representations, representations up to homotopy, and VB-groupoids
• Lie theory of vector bundles, Poisson geometry and double structures