Frölicher-Nijenhuis bracket

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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).

It is related to but not the same as the Nijenhuis-Richardson bracket and the Schouten-Nijenhuis bracket.

1 Definition
2 Derivations of the ring of forms
3 Applications
4 References

[edit] Definition

The Frölicher-Nijenhuis bracket is defined to be the unique vector-valued differential form

$[\cdot, \cdot] : \Omega^k(M,\mathrm{T}M) \times \Omega^l(M,\mathrm{T}M) \to \Omega^{k+l}(M,\mathrm{T}M) : (K, L) \mapsto [K, L]$

such that

$\mathcal{L}_{[K, L]} = [\mathcal{L}_K, \mathcal{L}_L]$

where $\mathcal{L}$ is the Nijenhuis-Lie derivative of $Ω(M)$ , given by

$\mathcal{L}_K = [d,i_K] =di_K+i_Kd$

(the same formula as for the usual Lie derivative, except that K is an element of Ω^*(M, TM) instead of Ω⁰(M, TM)).

An explicit formula for the Frölicher-Nijenhuis bracket of $\phi\otimes X$ and $\psi\otimes Y$ (for forms φ and ψ and vector fields X and Y) is given by

$\left.\right.[\phi \otimes X,\psi \otimes Y] = \phi\wedge\psi\otimes [X,Y] + \phi\wedge\mathcal{L}_X \psi\otimes Y - \mathcal{L}_Y \phi\wedge\psi \otimes X +(-1)^{\deg(\phi)}(d\phi \wedge i_X(\psi)\otimes Y +i_Y(\phi) \wedge d\psi \otimes X)$

[edit] Derivations of the ring of forms

Every derivation of Ω^*(M) can be written as

$i_L + \mathcal{L}_K$

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 $\mathcal{L}_K$ form the Lie superalgebra of all derivations commuting with d. The bracket is given by

$[\mathcal{L}_{K_1},\mathcal{L}_{K_2}]= \mathcal{L}_{[K_1,K_2]}$

where the bracket on the right is the Frölicher-Nijenhuis bracket. In particular the Frölicher-Nijenhuis bracket defines a graded Lie algebra structure on

Ω(M,T M)

, which extends the Lie bracket of vector fields.

The derivations of the form $i L$ form the Lie superalgebra of all derivations vanishing on functions Ω⁰(M). The bracket is given by

$[i_{L_1},i_{L_2}]= i_{[L_1,L_2]^\and}$

where the bracket on the right is the Nijenhuis-Richardson bracket.

The bracket of derivations of different types is given by

$[\mathcal{L}_{K}, i_L]= i_{[K,L]} - (-1)^{kl}\mathcal{L}_{i_LK}$

for K in Ω^k(M, TM), L in Ω^l+1(M, TM).

[edit] Applications

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.

[edit] References

Frölicher, A. and Nijenhuis, A., Theory of vector valued differential forms. Part I., Indagationes Math 18 (1956) 338-360.
Frölicher, A. and Nijenhuis, A., Invariance of vector form operations under mapings, Comm. Math. Helv. 34 (1960), 227-248.
P. W. Michor (2001), “Frölicher–Nijenhuis bracket”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
J.A. Schouten, Über Differentialkonkomitanten zweier kontravarianten Grössen Indag. Math. , 2 (1940) pp. 449–452