Connections Adapted to Non-Negatively Graded Structures
arXiv:1810.04479 · doi:10.1142/S021988781950021X
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 -connection on a graded bundle. In a natural sense weighted -connections are adapted to the basic geometric structure of a graded bundle in the same way as linear -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 -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.
arXiv comment19 pages. Accepted for publication in the International Journal of Geometric Methods in Modern Physics
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