Connections adapted to non-negatively graded structures

Abstract

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we define the notion of a weighted[Formula: see text]-connection on a graded bundle. In a natural sense weighted [Formula: see text]-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear [Formula: see text]-connections are adapted to the structure of a vector bundle. This notion generalizes directly to multi-graded bundles and in particular we present the notion of a bi-weighted[Formula: see text]-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.