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Register a new userDeformations of vector bundles in the categories of Lie algebroids and groupoids
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Authors
Pier Paolo La Pastina
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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 infinitesim
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VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.
We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Mosers deformation arguments for groupoids, we obtain several rigidity and normal form results.
A VB-algebroid is a vector bundle object in the category of Lie algebroids. We attach to every VB-algebroid a differential graded Lie algebra and we show that it controls deformations of the VB-algebroid structure. Several examples and applications are discussed. This is the first in a series of papers devoted to deformations of vector bundles and related structures over differentiable stacks.
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation groupoid/algebroid.
We describe the deformation cohomology of a symplectic groupoid, and use it to study deformations via Moser path methods, proving a symplectic groupoid version of the Moser Theorem. Our construction uses the deformation cohomologies of Lie groupoids and of multiplicative forms, and we prove that in the symplectic case, deformation cohomology of both the underlying groupoid and of the symplectic groupoid have de Rham models in terms of differential forms. We use the de Rham model, which is intimately connected to the Bott-Shulman-Stasheff double complex, to compute deformation cohomology in several examples. We compute it for proper symplectic groupoids using vanishing results; alternatively, for groupoids satisfying homological 2-connectedness conditions we compute it using a simple spectral sequence. Finally, without making assumptions on the topology, we constructed a map relating differentiable and deformation cohomology of the underlying Lie groupoid of a symplectic groupoid, and related it to its Lie algebroid counterpart via van Est maps.
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