论文标题
向量捆绑包的变形属于谎言代数和群体类别的类别
Deformations of vector bundles in the categories of Lie algebroids and groupoids
论文作者
论文摘要
该论文涉及VB - 甲状腺和VB组的变形。它们可以被视为谎言代数和群体类别中的矢量捆绑包，并包括几个经典对象，包括lie代数和谎言组表示，2矢量空间以及切线和切线和cotangengent elgebroid（groupoid）到lie elgebroid（groupBroid（groupoid））。此外，它们是用于谎言代数（群体）的某种表示形式的几何模型，即直到均匀的2项表示。最后，众所周知，li lie classoids是可微分堆栈的“混凝土”化身，因此，VB类别可以将VB类别视为矢量捆绑包的代表，而VB - 地球化学的无限版本。在这项工作中，我们附着在每个VB-Algebroid和Vb-Groupoid上，一个控制其变形的Cochain络合物，即线性变形复合物。此外，VB-Algebroid的变形复合物配备了DGLA结构。讨论了这些复合物的基本特性：它们与总空间和基本空间，特定情况和概括的变形复合物的关系。主要的理论结果是线性定理，它为VB组的线性变形共同体学提供了与相应的VB-Algebroid的同构以及Morita不变性定理的同构的条件，这意味着VB-GROUPTER的线性不适合vb-groups a aLgebraiant a al aLairaic the aLaim a aL varraic的线性不利共同学。最后，讨论了几个示例，展示了线性变形的共同体如何与其他众所周知的共同体相关。
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.
