DERIVATIONS ET CALCUL DIFFERENTIEL NON COMMUTATIF. II

We study canonical operation of the Lie algebra Der(#7B-A) of derivations of an algebra #7B-A with a unit in the graded differential algebra Ω(#7B-A). We introduce different graded differential algebras associated with this operation. In the case where #7B-A is the algebra C ∞ (V) of smooth functions on a smooth manifold V, one of these graded differential algebras denoted by Ω D (#7B-A) reduces to the algebra of differential forms on V. We define a filtration of Ω(#7B-A) associated with the operation of Der(#7B-A); the corresponding spectral sequence connects the cohomologies of the introduced differential algebras. Finally, by factorizing interior derivations, we define a graded differential sub-algebra Ω Out (#7B-A) of Ω D (#7B-A) which coincides with Ω D (#7B-A) whenever #7B-A is commutative. In the case where #7B-A is the algebra of n×n complex matrices, the cohomology of Ω D (#7B-A) identifies with the Lie algebra cohomology of sl(n, C) On etudie l'operation canonique de l'algebre de Lie Der(#7B-A) des derivations d'une algebre #7B-A possedant une unite dans l'algebre differentielle graduee Ω(#7B-A). On introduit differentes algebres differentielles graduees associees a cette operation. Dans le cas ou #7B-A est l'algebre des fonctions C ∞ sur une variete V, une de ces algebres differentielles graduees notee Ω D (#7B-A) se reduit a l'algebre des formes differentielles sur V. On definit une filtration de Ω(#7B-A) associee a l'operation de Der(#7B-A); la suite spectrale correspondante relie les cohomologies des differentes algebres differentielles introduites. En factorisant les derivations interieures, on definit une sous-algebre differentielle graduee Ω Out (#7B-A) de Ω D (#7B-A) qui coincide avec Ω D (#7B-A) si #7B-A est commutative. Dans le cas ou #7B-A est l'algebre des matrices n×n complexes, la cohomologie de Ω D (#7B-A) s'identifie a la cohomologie de l'algebre de Lie complexe sl(n, C) tandis que la cohomologie de Ω Out (#7B-A) est triviale