Questions on Killing form: its definition and a root space decomposition.

Question

I have a question on Killing form.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Consider the adjoint representation $(\mathrm{ad},\mathfrak{g})$ of $\mathfrak g$, i.e. $$ \mathrm{ad}: \mathfrak{g}\to \mathrm{Der}(\mathfrak{g}).$$ It is injective and surjective, so for $X\in\mathfrak{g}$, $\mathrm{ad}X$ is a derivation on $\mathfrak g$, and all derivations are of the form (inner forms).

Therefor, to study $\mathfrak{g}$, it is natural to study the derivation $\mathrm{ad} X$ on $\mathfrak{g}$.

The killing form is defined by $$B(\,,\,):\mathfrak{g}\times\mathfrak{g}\rightarrow \mathbb{C},\quad B(X,Y)=\mathrm{Tr}(\mathrm{ad}X\mathrm{ad}Y).$$

I know the properties of Killing norm. But why is it defined in this form? Is there any means of $\mathrm{Tr(PQ)}$ for $P$ and $Q$ derivations?

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Let $G$ be a compact Lie group with Lie algebra $\mathfrak g_0$ and the complexification $\mathfrak g=\mathfrak g_0^\mathbb{C}$. The Cartan involution on $\mathfrak{g}$ is $$\theta:\mathfrak{g_0}\oplus i\mathfrak{g_0}\rightarrow \mathfrak{g_0}\oplus i\mathfrak{g_0},\quad X+iY\mapsto X-iY.$$ Let $T$ be a maximal torus with Lie algebra $\mathfrak t_0$.

For $(\mathrm{Ad},\mathfrak g)$ of $G$, we have root space decomposition $$\mathfrak g=\mathfrak t\oplus\bigoplus_{\alpha\in\Phi(G,T)} \mathfrak g_\alpha.$$

My second question is on the $\mathfrak{su}(2)$ sub Lie algebra. I found there are two different arguments.

1. In D. Bump's book, Lie groups. For $X_\alpha\in \mathfrak{g}_\alpha$, let $X_{-\alpha}:=-\theta X_\alpha\in\mathfrak {g}_{-\alpha}$ and $H_\alpha=[X_\alpha,X_{-\alpha}]\in i\mathfrak t_0$.

2. In A.W. Knapp's book, Lie groups beyond an introduction. For $X_\alpha\in\mathfrak{g}_\alpha$, choose $X_{-\alpha}\in \mathfrak{g}_{-\alpha}$ such that $B(X_\alpha,X_{-\alpha})=1$, then $[X_\alpha,X_{-\alpha}]=H_\alpha$.

I am confused about the two arguments. However, I notice that in the first argument, $$B(X_\alpha,X_{-\alpha})=B(X_\alpha,-\theta X_\alpha)=B_\theta(X_\alpha,X_\alpha)$$

Answer 1

For the first question:

If you want to introduce some invariant form on the operators then the trace of product is natural in the following sense: the product is the only natural map $V \otimes V \to V$ you have. $id$ is the only natural element in $End V=V \otimes V^*$ and the trace corresponds to this element in the isomorphism $End(V)^*=(V^* \otimes V)^*=V \otimes V^*=V^* \otimes V$. So the Killing-type forms are "natural" (and they appear not only in the Lie algebras context).

For the second question:

These are not two different arguments, just different notations. I am used to the Knapp notation, but if you want to study real forms then inserting Cartan involution somewhere is useful. The first type notations are usually used over algebraically closed fields.