In this note I shall introduce the Frölicher-Nijenhuis bracket – a powerful tool, especially when dealing with connections on fibre bundles.
Let be a fibre bundle. In the previous note we have seen that the connection form is a one-form on with values in (in fact, in values in ) This leads us to a more general subject of natural operations on tangent valued-forms.
Let therefore, for now, be any differentiable manifold, and let us consider the space of tangent valued forms on Let be the graded algebra of ordinary differential forms on Then the (graded) space of tangent-valued forms can be written as
In the following we will use the capital indices to refer to local charts of
The elements of are simply vector fields on The elements of are tangent-valued one-forms. If is such a form, then assigns to each tangent vector another tangent vector In coordinates it is represented as In general, an element of is represented by antisymmetric in indices or
In the following it will be convenient to introduce the following notation:
We will write
Operations on differential forms
Frölicher-Nijenhuis (FN) bracket will assign to any two tangent valued forms another form
We will see (in the following posts) that the curvature of a connection can be simply expressed in terms of the FN bracket of the connection form with itself. Also Bianchi identities for the curvature will follow immediately from this definition and from the properties of the FN bracket. But first let us recall the three fundamental operations on differential forms These are: exterior derivative, Lie derivative, insertion. Exterior derivative maps to according to the formula
Given a vector field the insertion operator maps to according to the formula
Lie derivative maps into itself, the definition being
These three operations on number-valued differential forms have the following important properties (note: the commutators are graded commutators):
- If is a function on then
Frölicher-Nijenhuis bracket associates to each pair of tangent-valued forms on the third tangent-valued form. Because of its bilinearity it is enough to define it on simple tensors. For and for it is defined by the following formula:
In coordinates the Frölicher-Nijenhuis bracket is given by the following explicit expression:
If we want to write the sam formula in a more explicit form, then antisymmetrizations and combinatorial factors enter:
As an exercise let us compute where is the identity tangent-valued form (the natural soldering form), with coordinates . Thus we set The first term gives
The second and fourth terms vanishs because are constant.
The third is
It follows that – therefore commutes with every tangent valued form.
The most important properties of the FN bracket are the following ones
- – graded antisymmetry
- – Jacobi identity
Suppose now is a vector field. It can be also considered as a tangent-valued 0-form: We then find that
Therefore for vector fields the FN bracket reduces to an ordinary Lie derivative.
Another insight into the nature of the FN bracket comes from considering tangent-valued forms as operators on diferential forms. First of all, for any we can define a graded derivation by the formula:
Now I have a problem. In references  and  we have the following local formulas:
In our notation this would be:
But I am not able to reproduce this result. It seems to me that the factor is missing in front of the expression on the RHS. I will return to this enigma at the end of this note.
Remark. Notice that in these formulas we assume and not in
Then we define the Lie derivative of a differential form along a tangent-valued form by the formula
Again it is instructive to calculate the action of and of this extended Lie derivative for Using the local formula from  or 
we obtain that for , that is for we have
But the identity map is not a graded derivative (of degree 0 in this case). On the other hand, from the formula involving vector fields I am getting
This looks good, because then
as it should be.
At this point I have to pause until this point is clarified. Perhaps I am missing something evident? I don’t know.
And here are the conventions concerning differential forms used in  and :
 Peter W. Michor, Topics in Differential Geometry (see hxxp://www.mat.uniroma1.it/people/manetti/GeoDiff0809/dgbook.pdf )
 Ivan Kolar, Peter W. Michor, Jan Slovak, Natural Operations in Differentila Geometry (see hxxp://126.96.36.199/pub/muni.cz/EMIS/monographs/KSM/kmsbookh.ps.gz )
 Andreas Kriegl, Peter W. Michor, The Convenient Setting for Global Analysis, Ch. 33.18