Graded Bundles in the Category of Lie Groupoids

Symmetry Integrability and Geometry-Methods and Applications

article

Andrew James Bruce¹  Katarzyna Grabowska²  Janusz Grabowski¹ 

[1] Institute of Mathematics, Polish Academy of Sciences;Faculty of Physics, University of Warsaw

关键词: graded manifolds;    homogeneity structures;    Lie groupoids;    Lie algebroids;   

DOI  :  10.3842/SIGMA.2015.090

来源: National Academy of Science of Ukraine

【 摘 要 】

We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids . One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids , objects defined in a previous publication of the authors, which are graded manifolds in the category of Lie algebroids, showing that they are naturally related via differentiation and integration. In this work we also make an initial study of weighted Poisson-Lie groupoids and weighted Lie bi-algebroids , as well as weighted Courant algebroids .

【 授权许可】

Unknown