Abstract

Li-Bland’s correspondence between linear Courant algebroids and Lie $2$-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 $2$-representations is a split Lie $2$-algebroid and we explain this result geometrically, as a consequence of the equivalence of VB-Courant algebroids and Lie $2$-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.

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