Dérivations et calcul différentiel non commutatif. (Derivations and non-commutative differential calculus). (French) Zbl 0661.17012

Dubois-Violette, Michel

C. R. Acad. Sci., Paris, Sér. I 307, No. 8, 403-408 (1988).

The canonical operation of the Lie algebra Der(\({\mathcal A})\) of derivations of an algebra \({\mathcal A}\) with a unit in the graded differential algebra \(\Omega\) (\({\mathcal A})\) is studied. Different graded differential algebras associated with this operation are introduced. A filtration of \(\Omega\) (\({\mathcal A})\) associated with the operation of Der(\({\mathcal A})\) is defined and investigated. Some natural examples in which \({\mathcal A}=C^{\infty}(V)\) (V-paracompact \(C^{\infty}\)-manifold) and \({\mathcal A}=M_ n({\mathbb{C}})\) (the Lie algebra of endomorphisms of \({\mathbb{C}}^ n)\) are considered.

Reviewer: J.Kubarski

MSC:

17B56	Cohomology of Lie (super)algebras
58A12	de Rham theory in global analysis
53C99	Global differential geometry

Keywords:

non-commutative differential geometry; spectral sequence; algebra of smooth functions; Lie algebra of derivations; canonical operation; graded differential algebra; filtration