Show that $\mathfrak n_- \simeq \mathfrak n_+^{\ast}$ and $\mathfrak h \simeq \mathfrak {h}^{\ast}.$

Question

Let $\mathfrak g$ be a simple Lie algebra with the triangular decomposition $\mathfrak g = \mathfrak n_+ \oplus \mathfrak n_- \oplus \mathfrak h,$ where $\mathfrak h$ is the Cartan subalgebra of $\mathfrak g$ and $\mathfrak n_+$ and $\mathfrak n_-$ are the direct sum of root subspaces of $\mathfrak g$ corresponding to the positive and negative roots respectively. Let $(\cdot, \cdot)$ be a non-degenerate invariant bilinear form on $\mathfrak g$ (e.g. the killing form). Then show that it induces isomorphisms $\mathfrak {n}_- \simeq {\mathfrak {n}_+}^{\ast}$ and $\mathfrak h \simeq \mathfrak h^{\ast}.$

Could anyone please help me in this regard? Thanks for your time.

Answer 1

Let $\Phi$ be the root system and for $\alpha \in \Phi \cup \{0\}$ let $\mathfrak g_\alpha$ be the subspace of $\mathfrak g$ where all $h \in \mathfrak h$ act via $\alpha(h)$ (note $\mathfrak g_0 = \mathfrak h$).

Let $\alpha, \beta \in \Phi \cup \{0\}$, then for all $x \in \mathfrak g_\alpha, y \in \mathfrak g_\beta$, by invariance of the form we have

$$-\alpha(h) \cdot (x,y) = ([x,h],y) = (x,[h,y]) = \beta(h) \cdot (x,y)$$

for all $h \in \mathfrak h$ which entails

$$ \alpha \neq -\beta \implies \mathfrak g_\alpha \perp \mathfrak g_\beta.$$

But the form is non-degenerate. Can you see from here that the restriction of the form to each $\mathfrak g_\alpha \times \mathfrak g_{-\alpha}$ is non-degenerate, and how that implies what you want?