As Sudeep says: K(X,Y) is the trace of a map (ad_X ad_Y) and the easiest way to define the map is to say what its value is on a given Z (ad_X ad_Y (Z) = [X,[Y,Z]], but Z is just a variable that's there to help defining ad_X ad_Y. It's a bit like saying "max(sin(x)) = 1". Sine is a function and max(sin) =1 makes sense, but often people write sin(x) for the function sin rather than the value of sin at x.
It is a good and useful video. I think it deserves this comment and a like.
why killing form can be viewed as an inner product? Normally, y needs to be transposed tr(xy^T) to make it positive definite.
Thanks!
So K(X,Y) is independent of the choice of Z?
For fixed X and Y, ad_X ad_Y is a (linear)function of Z, but it's trace, K(X, Y), is a unique complex number.
As Sudeep says: K(X,Y) is the trace of a map (ad_X ad_Y) and the easiest way to define the map is to say what its value is on a given Z (ad_X ad_Y (Z) = [X,[Y,Z]], but Z is just a variable that's there to help defining ad_X ad_Y. It's a bit like saying "max(sin(x)) = 1". Sine is a function and max(sin) =1 makes sense, but often people write sin(x) for the function sin rather than the value of sin at x.
guess I don't really know the meaning of the basis of Lie algebra, should probably catch up with that