Connections Adapted to Non-Negatively Graded Structures

TL;DR

This paper introduces weighted A-connections on graded bundles, generalizing linear connections, and demonstrates their existence and applications to Lie algebroid actions on complex geometric structures.

Contribution

It defines weighted A-connections for graded bundles, extending the concept of linear connections, and develops their theory for multi-graded bundles and double vector bundles.

Findings

Existence of weighted A-connections on graded bundles

Generalization to multi-graded and double vector bundles

Application to Lie algebroid actions on graded structures

Abstract

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$-connection on a graded bundle. In a natural sense weighted $A$-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear $A$-connections are adapted to the structure of a vector bundle. This notion generalises directly to multi-graded bundles and in particular we present the notion of a bi-weighted $A$-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles.