A Cohomology for Vector Valued Differential Forms

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Frölicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of ”traceless” vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed. 1. Notation 1.1 The Frölicher-Nijenhuis bracket. Let M be a smooth manifold of dimension m throughout the paper. We consider the space Ω(M ;TM) = ⊕m k=0 Ω (M ;TM) of all tangent bundle valued differential forms on M . Below K and L will be elements of Ω(M ;TM) of degree k and `, respectively. It is well known that Ω(M ;TM) is a graded Lie algebra with the so called Frölicher-Nijenhuis bracket [ , ] : Ω(M ;TM)× Ω(M ;TM)→ Ω(M ;TM). For its definition, properties, and notation we refer to [Mi, 1987]. 1.2. In the investigation of the Lie algebra cohomology of the graded Lie algebra (Ω(M ;TM), [ , ]) in [Sch,1988] the following exterior graded derivation of degree 1 appeared: δ : Ω(M ;TM)→ Ω(M ;TM) Before its definition we need another operator. Let the contraction or trace c : Ω(M ;TM)→ Ωk−1(M) be given by c(φ⊗X) = iXφ, linearly extended. We also put c̄ | Ω(M ;TM) := (−1) k−1 m−k+1 c 1991 Mathematics Subject Classification. 17B70, 58A12.