In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940).
A graded derivation of degree ℓ is a mapping
which is linear with respect to constants and satisfies
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 algebra under the anticommutator of derivations defined on homogeneous derivations D1 and D2 of degrees d1 and d2, respectively, by
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
If k = 0, so that K ∈ Ω0(M, TM) is a vector field, the usual homotopy formula for the Lie derivative is recovered
An explicit formula for the Frölicher–Nijenhuis bracket of and (for forms φ and ψ and vector fields X and Y) is given by
Derivations of the ring of forms
Every derivation of Ω*(M) can be written as
for unique elements K and L of Ω*(M, TM). The Lie bracket of these derivations is given as follows.
- The derivations of the form form the Lie superalgebra of all derivations commuting with d. The bracket is given by
- where the bracket on the right is the Frölicher–Nijenhuis bracket. In particular the Frölicher–Nijenhuis bracket defines a graded Lie algebra structure on , which extends the Lie bracket of vector fields.
- The derivations of the form form the Lie superalgebra of all derivations vanishing on functions Ω0(M). The bracket is given by
- where the bracket on the right is the Nijenhuis–Richardson bracket.
- The bracket of derivations of different types is given by
- for K in Ωk(M, TM), L in Ωl+1(M, TM).
The Nijenhuis tensor of an almost complex structure J, is the Frölicher–Nijenhuis bracket of J with itself. An almost complex structure is a complex structure if and only if the Nijenhuis tensor is zero.
With the Frölicher–Nijenhuis bracket it is possible to define the curvature and cocurvature of a vector-valued 1-form which is a projection. This generalizes the concept of the curvature of a connection.
There is a common generalization of the Schouten–Nijenhuis bracket and the Frölicher–Nijenhuis bracket; for details see the article on the Schouten–Nijenhuis bracket.
- Frölicher, A.; Nijenhuis, A. (1956), "Theory of vector valued differential forms. Part I.", Indagationes Mathematicae 18: 338–360.
- Frölicher, A.; Nijenhuis, A. (1960), "Invariance of vector form operations under mappings", Communicationes Mathematicae Helveticae 34: 227–248.
- P. W. Michor (2001), "Frölicher–Nijenhuis bracket", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Schouten, J. A. (1940), "Über Differentialkonkomitanten zweier kontravarianten Grössen", Indagationes Mathematicae 2: 449–452.