killing form and the dot product

Question

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot product equivalent? Humphreys just went straight from the killing form to the dot product. Because the killing of two roots such that one is not a negative of the other is 0, but the dot product is less than zero, if they are simple roots.

Thanks

Answer 1

The killing form restricted to the Cartan subalgebra is an inner product, thus there is an isometry between the Cartan subalgebra with the killing form and $\mathbb{R}^k$ with the dot product given by choosing an orthonormal (w.r.t. the killing form) basis. This is the equivilance I think you're talking about.

What you're thinking of, is $ < e_\alpha, e_\beta >_K = 0 $ if $\alpha$ and $\beta$ are not scalar multiples. These $e$s are not the roots but act on the Cartan subalgebra via the roots.