Infinitesimal Automorphisms of VB-groupoids and algebroids

TL;DR

This paper investigates the infinitesimal automorphisms of VB-groupoids and algebroids, revealing their structure and relation to multiplicative sections, with implications for Poisson geometry and singular spaces.

Contribution

It characterizes infinitesimal automorphisms of VB-groupoids and algebroids, linking them to multiplicative sections in the case of representation-derived structures.

Findings

Infinitesimal automorphisms preserve both linear and groupoid/algebroid structures.

Automorphisms correspond to multiplicative sections in certain derivation groupoids/algebroids.

Results have applications in Poisson geometry and models for vector bundles over singular spaces.

Abstract

VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation groupoid/algebroid.