Frölicher-Nijenhuis bracket

Let be a smooth manifold (cf. also Differentiable manifold) and let . One calls

the space of all vector-valued differential forms. The Frölicher–Nijenhuis bracket is a -graded Lie bracket:

It extends the Lie bracket of smooth vector fields, since . The identity on generates the one-dimensional centre. It is called the Frölicher–Nijenhuis bracket since it appeared with its full properties for the first time in [a1], after some indication in [a8]. One formula for it is:

where and are vector fields, is a -form, and is an -form. It is a bilinear differential operator of bi-degree .

The Frölicher–Nijenhuis bracket is natural in the same way as the Lie bracket for vector fields: if is smooth and are -related to , then is also -related to , .

Details.

A convenient source is [a3], Sect. 8. The basic formulas of the calculus of differential forms extend naturally to include the Frölicher–Nijenhuis bracket: Let

be the algebra of differential forms. One denotes by the space of all (graded) derivations of degree , i.e. all bounded linear mappings with and for . The space is a -graded Lie algebra with the graded commutator as bracket.

A derivation with satisfies for , thus is of tensorial character and induces a derivation for each . It is uniquely determined by its restriction to -forms , which can be viewed as an element depending smoothly on ; this is expressed by writing , where , and one has

for and (or ).

By putting , one obtains a bracket on which defines a graded Lie algebra structure with the grading as indicated, and for , one has

where . The bracket is called the the Nijenhuis–Richardson bracket, see [a6] and [a7]. If viewed on a vector space , it recognizes Lie algebra structures on : A mapping is a Lie bracket if and only if . This can be used to study deformations of Lie algebra structures: is again a Lie bracket on if and only if ; this can be written in the form of a Maurer–Cartan equation (cf. also Maurer–Cartan form) as , since is the coboundary operator for the Chevalley cohomology (cf. also Cohomology) of the Lie algebra with values in the adjoint representation . See [a4] for a multi-graded elaboration of this.

The exterior derivative is an element of . In view of the formula for vector fields , one defines for the Lie derivation by . The mapping is injective. One has .

For any graded derivation there are unique and such that

One has if and only if . Moreover, if and only if .

Let and . Then, obviously, , so

for a uniquely defined . This vector-valued form is the Frölicher–Nijenhuis bracket of and .

For and one has

The space is a graded module over the graded algebra with the action , because is graded commutative. Let the degree of be , of be , and of be . Let the other degrees be as indicated. Then:

For and , the Lie derivative of along is given by:

For and , the Frölicher–Nijenhuis bracket is given by:

The Frölicher–Nijenhuis bracket expresses obstructions to integrability in many different situations: If is an almost-complex structure, then is complex structure if and only if the Nijenhuis tensor vanishes (the Newlander–Nirenberg theorem, [a5]). If is a fibre-wise projection on the tangent spaces of a fibre bundle , then is a version of the curvature (see [a3], Sects. 9; 10). If is fibre-wise diagonalizable with all eigenvalues real and of constant multiplicity, then the eigenspaces of are integrable if and only if .

References

[a1]	A. Frölicher, A. Nijenhuis, "Theory of vector valued differential forms. Part I." Indag. Math. , 18 (1956) pp. 338–359
[a2]	A. Frölicher, A. Nijenhuis, "Invariance of vector form operations under mappings" Comment. Math. Helvetici , 34 (1960) pp. 227–248
[a3]	I. Kolář, Peter W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993)
[a4]	Pierre Lecomte, Peter W. Michor, Hubert Schicketanz, "The multigraded Nijenhuis–Richardson Algebra, its universal property and application" J. Pure Appl. Algebra , 77 (1992) pp. 87–102
[a5]	A. Newlander, L. Nirenberg, "Complex analytic coordinates in almost complex manifolds" Ann. of Math. , 65 (1957) pp. 391–404
[a6]	A. Nijenhuis, R. Richardson, "Cohomology and deformations in graded Lie algebras" Bull. Amer. Math. Soc. , 72 (1966) pp. 1–29
[a7]	A. Nijenhuis, R. Richardson, "Deformation of Lie algebra structures" J. Math. Mech. , 17 (1967) pp. 89–105
[a8]	J.A. Schouten, "Über Differentialkonkomitanten zweier kontravarianten Grössen" Indag. Math. , 2 (1940) pp. 449–452