It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940).
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.
With the Frölicher–Nijenhuis bracket it is possible to define the curvature and cocurvature of a vector-valued 1-form which is a projection.