Let Ω*(M) be the sheaf of exterior algebras of differential forms on a smooth manifold M. This is a graded algebra in which forms are graded by degree:
A graded derivation of degree ℓ is a mapping
which is linear with respect to constants and satisfies
Thus, in particular, the interior product with a vector defines a graded derivation of degree ℓ = −1, whereas the exterior derivative is a graded derivation of degree ℓ = 1.
The vector space of all derivations of degree ℓ is denoted by DerℓΩ*(M). The direct sum of these spaces is a graded vector space whose homogeneous components consist of all graded derivations of a given degree; it is denoted
This forms a graded Lie superalgebra under the anticommutator of derivations defined on homogeneous derivations D1 and D2 of degrees d1 and d2, respectively, by
Any vector-valued differential form K in Ωk(M, TM) with values in the tangent bundle of M defines a graded derivation of degree k − 1, denoted by iK, and called the insertion operator. For ω ∈ Ωℓ(M),
The Nijenhuis–Lie derivative along K ∈ Ωk(M, TM) is defined by
where d is the exterior derivative and iK is the insertion operator.
The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form
such that
Hence,
If k = 0, so that K ∈ Ω0(M, TM)
is a vector field, the usual homotopy formula for the Lie derivative is recovered
If k=ℓ=1, so that K,L ∈ Ω1(M, TM),
one has for any vector fields X and Y
If k=0 and ℓ=1, so that K=Z∈ Ω0(M, TM) is a vector field and L ∈ Ω1(M, TM), one has for any vector field X
An explicit formula for the Frölicher–Nijenhuis bracket of and (for forms φ and ψ and vector fields X and Y) is given by