Graded Bundles in the Category of Lie Groupoids

TL;DR

This paper introduces weighted Lie groupoids, a new class of graded structures in Lie groupoids, exploring their properties, examples, and related Lie theory, including connections to Lie algebroids and Poisson structures.

Contribution

It defines weighted Lie groupoids and studies their Lie theory, linking them to graded Lie algebroids and extending the framework to Poisson and Courant structures.

Findings

Weighted Lie groupoids generalize VB-groupoids as higher-degree graded structures.

The Lie theory for weighted groupoids and algebroids is established, showing natural differentiation and integration relations.

Initial exploration of weighted Poisson-Lie groupoids and Courant algebroids is provided.

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.