Representation up to homotopy of double algebroids and their transgression classes

Abstract In this work, we present definition of the representation of double Lie algebroids and find a one-to-one correspondence to introduce representation up to homotopy of double Lie algebroids and gauge equivalence of them. Also dual, tensor product and direct sum representations up to homotopy are made as a double Lie algebroid modules. We conclude this paper by generalization of some Lie algebroid oo-representation properties to oo-representation of double Lie algebroids and introduce transgression classes for double Lie algebroids.