Graded Bundles in the Category of Lie Groupoids

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 $\mathcal{V}\mathcal{B}$-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.

References in corpus (6)

• AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
• Lectures on Integrability of Lie Brackets
• Supergroupoids, double structures, and equivariant cohomology
• $Q$-manifolds and Higher Analogs of Lie Algebroids
• Remarks on Contact and Jacobi Geometry
• Higher order mechanics on graded bundles

Cited by in corpus (8)

• Remarks on Contact and Jacobi Geometry
• Pre-Courant Algebroids
• Polarisation of Graded Bundles
• Introduction to graded bundles
• Connections Adapted to Non-Negatively Graded Structures
• Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms
• The category of ${\mathbb{Z}}_2^n$-supermanifolds
• Lie theory of vector bundles, Poisson geometry and double structures