Local description of generalized forms on transitive Lie algebroids and applications

In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a \v{C}ech-de Rham bicomplex with a Leray-Serre spectral sequence. We apply the general theory to Atiyah Lie algebroids and to derivations on a vector bundle.