Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 090, 25 pages      arXiv:1502.06092      https://doi.org/10.3842/SIGMA.2015.090
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

Graded Bundles in the Category of Lie Groupoids

Andrew James Bruce a, Katarzyna Grabowska b and Janusz Grabowski a
a) Institute of Mathematics, Polish Academy of Sciences, Poland
b) Faculty of Physics, University of Warsaw, Poland

Received February 25, 2015, in final form November 05, 2015; Published online November 11, 2015

Abstract
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.

Key words: graded manifolds; homogeneity structures; Lie groupoids; Lie algebroids.

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