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A VB-algebroid is essentially defined as a Lie algebroid object in thecategory of vector bundles. There is a one-to-one correspondence betweenVB-algebroids and certain flat Lie algebroid superconnections, up to a naturalnotion of equivalence. In this setting, we are able to construct characteristicclasses, which in special cases reproduce characteristic classes constructed byCrainic and Fernandes. We give a complete classification of regularVB-algebroids, and in the process we obtain another characteristic class of Liealgebroids that does not appear in the ordinary representation theory of Liealgebroids.

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids