Advances in Mathematics
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.
Characteristic classes, Double category, Lie algebroid, Representation, Superconnection
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Gracia-Saz, Alfonso and Mehta, Rajan Amit, "Lie Algebroid Structures on Double Vector Bundles and Representation Theory of Lie Algebroids" (2010). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.