nLab Nijenhuis–Richardson bracket

Definition

Suppose $M$ is a smooth manifold. Recall that any differential $(k+1)$-form $K\in\Omega^{k+1}(M,T M)$ valued in the tangent bundle of $M$ gives rise to a graded derivation $\iota_K$ of degree $k$ on the algebra of differential forms on $M$: on 1-forms we have $\iota_K \omega=\omega\circ K$ and on higher forms we extend using the Leibniz identity.

Concretely,

$$\iota_K \omega(X_1,\ldots,X_{k+l})=1/((k+1)!(l-1)!)\sum_\sigma (-1)^\sigma \omega(K(Y_1,\ldots,Y_{k+1}),Y_{k+2},\ldots),$$

where $Y_i=X_{\sigma(i)}$.

The map $\iota$ defined an injective homomorphism of graded vector spaces from $\Omega^{\bullet+1}(M,T M)$ to graded derivations of $\Omega(M)$. Its image comprises precisely those derivations that vanish on 0-forms and is closed under the commutator operation. Transferring the bracket to its source yields the Nijenhuis–Richardson bracket:

$$[K,L]^\wedge = \iota_K L-(-1)^{k l}\iota_L K,$$

where $\iota_K(\omega\otimes X)=\iota_K \omega\otimes X$.

Classification of graded derivations of differential forms

The Nijenhuis–Richardson bracket is an important ingredient in the classification of graded derivations of differential forms.

See the article Frölicher–Nijenhuis bracket for more information.

Related concepts

• Frölicher–Nijenhuis bracket

• Schouten–Nijenhuis bracket

References

The formula for the NR bracket is made explicit in and thus derives its name from:

• Albert Nijenhuis, Roger W. Richardson, Section 2 of: Cohomology and deformations of algebraic structures, Bulletin of the American Mathematical Society 70:3 (1964), 406–412. doi.

• Albert Nijenhuis, Roger W. Richardson, Section 5 of: Deformations of Lie Algebra Structures, Journal of Mathematics and Mechanics 17 1 (1967) 89-105 [jstor:24902154]

but the existence of the bracket is already implicit in:

• Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359. doi:10.1016/s1385-7258(56)50046-7 and doi:10.1016/s1385-7258(56)50047-9.

and refined to almost complex structures in

• Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part II. Almost-complex structures, Indagationes Mathematicae (Proceedings) 61 (1958), 414–421 and 422-429. doi:10.1016/s1385-7258(58)50057-2 and doi:10.1016/s1385-7258(58)50058-4.

See also:

• Wikipedia, Nijenhuis–Richardson bracket

A textbook account: Chapter 16 of

• Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics 93 (2008). PDF.