Split Lie 2-algebroids and matched pairs of 2-representations

A matched pair of Lie algebroid representations is equivalent to two seemingly different objects; the bicrossproduct Lie algebroid and the double Lie algebroid of the matched pair. In similar manner, a Lie bialgebroid is equivalent to a ‘cotangent double Lie algebroid’, and to a ‘bicrossproduct' Courant algebroid.

We define in this talk matched pairs of 2 term Lie algebroid representations up to homotpy (2-representations for short). We prove that the bicrossproduct of a matched pair of 2-representations is a split Lie 2-algebroid, and we explain the geometric insight -- in the context of double Lie algebroids and VB-Lie bialgebroids -- at the origin of our algebraic construction. This explains in particular how the double of a matched pair of representations can be seen as the geometrisation of its bicrossproduct Lie algebroid.

We sketch more generally Li-Bland's equivalence of Lie 2-algebroids with VB-Courant algebroids, and, if time permits, we explain how this presents as well the cotangent double of a Lie bialgebroid as a geometrisation of its bicrossproduct Courant algebroid.