Li-Bland’s correspondence between linear Courant algebroids and Lie
-algebroids
is explained at the level of linear and core sections versus graded functions, and
shown to be an equivalence of categories. More precisely, decomposed VB-Courant
algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as
decomposed VB-algebroids are equivalent to 2-term representations up to homotopy
(Gracia-Saz and Mehta). Several special cases are discussed, yielding new examples of
split Lie 2-algebroids.
We prove that the bicrossproduct of a matched pair of
-representations is a
split Lie
-algebroid
and we explain this result geometrically, as a consequence of the equivalence of VB-Courant algebroids
and Lie
-algebroids.
This explains in particular how the two notions of the “double” of a matched
pair of representations are geometrically related. In the same manner, we
explain the geometric link between the two notions of the double of a Lie
bialgebroid.
Keywords
Lie 2-algebroids, VB-Courant algebroids, Dorfman
connections, representations up to homotopy, matched pairs,
Lie bialgebroids