Deformations of vector bundles in the categories of Lie algebroids and groupoids

TL;DR

This thesis develops a cohomological framework for understanding deformations of VB-algebroids and VB-groupoids, linking their properties to differentiable stacks and establishing invariance results.

Contribution

It introduces a deformation complex with DGLA structure for VB-algebroids and VB-groupoids, and proves a linear van Est theorem and Morita invariance for their deformation cohomology.

Findings

Deformation complexes relate to total and base space complexes.

Linear van Est theorem links VB-groupoid and VB-algebroid cohomologies.

Morita invariance shows cohomology as an algebraic invariant.

Abstract

This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they are geometric models for some kind of representations of Lie algebroids (groupoids), namely 2-term representations up to homotopy. Finally, it is well known that Lie groupoids are "concrete" incarnations of differentiable stacks, hence VB-groupoids can be considered as representatives of vector bundles over differentiable stacks, and VB-algebroids their infinitesimal versions. In this work, we attach to every VB-algebroid and VB-groupoid a cochain complex controlling its deformations, their linear deformation complex. Moreover, the deformation complex of a VB-algebroid is equipped with a DGLA structure. The basic properties of these complexes are discussed: their relationship with the deformation complexes of the total spaces and the base spaces, particular cases and generalizations. The main theoretical results are a linear van Est theorem, that gives conditions for the linear deformation cohomology of a VB-groupoid to be isomorphic to that of the corresponding VB-algebroid, and a Morita invariance theorem, that implies that the linear deformation cohomology of a VB-groupoid is really an algebraic invariant of the associated vector bundle of differentiable stacks. Finally, several examples are discussed, showing how the linear deformation cohomologies are related to other well-known cohomologies.