Double Lie algebroids and representations up to homotopy
  We show that double Lie algebroids, together with a chosen linear splitting, are equivalent to pairs of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the tangent of a Lie algebroid. 
Alfonso Gracia-Saz, Madeleine Jotz Lean, Kirill C. H. Mackenzie and   Rajan A. Mehta
math.DG