Differential calculus over double Lie algebroids

M. Van den Bergh [20] defined the notion of a double Lie algebroid and showed that a double quasi-Poisson algebra gives rise to a double Lie algebroid. We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative Karoubi de Rham complex [7, 9] and the double Poisson Lichnerowicz cohomology [16] as particular cases of our construction.