Physicist here and I am struggling to understand my lecturers construction of the roots and weights system.
As I understand it the killing form for a semi-simple lie algebra is non degenerate on $\mathfrak{g}$ and $\mathfrak{h}$ where $\mathfrak{h}$ is the Cartan subalgebra. This induces an inner product and an isomorphism between $\mathfrak{h}$ and $\mathfrak{h}*$. Roots are then elements of the dual cartan subalgebra $\mathfrak{h}*$.
I understand that if the Lie Group is compact then we can choose a basis in the lie algebra such that the killing form takes the form $\kappa_{ij}= -\kappa \delta_{ij}$ for $\kappa >0$.
In my lecture notes my lecturer has equipped the cartan subalgebra and its dual with the euclidean inner product, ie the dot product. I do not understand why this is ok.
