Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Created:
Suppose $M$ is a smooth manifold. Recall that any differential $(k+1)$-form $K\in\Omega^{k+1}(M,TM)$ 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 $\Omega^{\bullet+1}(M,TM)$ 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 $\iota_K(\omega\otimes X)=\iota_K \omega\otimes X$.
Added references:
Original definition:
Refinements for almost complex structures:
Or maybe you made a copy-and-paste error: The original references seem to be these here, instead:
Albert Nijenhuis, R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966) 1-29 [doi:10.1090/S0002-9904-1966-11401-5]
Albert Nijenhuis, R. W. Richardson, Deformations of Lie Algebra Structures, Journal of Mathematics and Mechanics 17 1 (1967) 89-105 [jstor:24902154]
No, the original references were correct. Frölicher and Nijenhuis computed the whole graded Lie algebra of graded derivations, and as explained in the article Frölicher–Nijenhuis bracket, both brackets participate in the description.
The first paper you added appears not to mention the Nijenhuis–Richardson bracket at all. I am not sure why Richard Borcherds added it as a reference to the Wikipedia article.
The second paper does explore it briefly in Section 5 and gives explicit formulas for it, unlike the Frölicher–Nijenhuis paper, which simply transfers the bracket along the injective homomorphism.
However, it came out much later than the Frölicher–Nijenhuis paper.
Added an earlier reference:
So then it looks like on Wikipedia here they confused the two articles by Nijenhuis & Richardson which both start with “Cohomology and deformations…”
Added:
A textbook account: Chapter 16 of
1 to 9 of 9