The Lie Derivative on Forms
Well now we know how to translate -forms by pulling back, and thus we can define another Lie derivative:
What happens if is a -form — a function ? We check
That is, the Lie derivative by acts on exactly the same as the vector field does itself.
I also say that the Lie derivative by is a degree-zero derivation of the algebra . That is, it’s a real-linear transformation, and it satisfies the Leibniz rule:
for any -form and -form . Linearity is straightforward, and given linearity the Leibniz rule follows if we can show
for -forms . Indeed, we can write and as linear combinations of such - and -fold wedges, and then the Leibniz rule is obvious.
So, let us calculate:
So we see how we can peel off one of the -forms. A simple induction gives us the general case.