Abstract

This paper is a revisitation of the work by A.Froelicher and A.Nijenhuis on the algebra of vector valued forms.

The algebra of derivations of scalar forms is studied and shown to be naturally splitted into the direct sum of two subalgebras: the "contraction type" and the "Lie type" derivations. Both these spaces are canonically dtffeomorphic to the space of vector valued forms and induce on it two different algebra structures, one of which is the Froelicher-Nijenhuis algebra.

Introduction

Some results of the work of Froelicher and Nijenhuis on vector valued forms have reached a stable place in current differential geometry, while others have been used only occasionally. However, we believe that the whole of that work will turn out to be fundamental, through its extension to fibred manifolds, in the developments of the general theory of connections, not subordinated to the theory of G-structures. We expect that, in this way, important results concerning the unification of different concepts of differential geometry, and new ideas for gauge theories, can be found; some work in this direction has already been done. Having in mind a systematic setting of this subject, we give here a review of the foundations. The originality of this work consists in a more organic, tidy and deductive exposition.