I talked about the Lie algebra ${\mathfrak{sl}_2}$ a while back.  Now I’m going to do it more properly, and using the tools developed.  This is going to feature prominently in some of the proofs in the sequel.

Now, let’s see how all this works for the familiar case of ${\mathfrak{sl}_2}$, with its usual generators ${H,X,Y}$. This is a simple Lie algebra in fact. To see this, let’s consider the ideal ${I}$ of ${\mathfrak{sl}_2}$ generated by some nonzero vector ${aX + bH + cY}$; I claim it is all of ${\mathfrak{sl}_2}$.

Consider the three cases ${a \neq 0, b \neq 0, c \neq 0}$:

First, assume ${a}$ or ${c}$ is nonzero. Bracketing with ${H}$, and again, gives

$\displaystyle -2aX + 2 c Y \in I , \ (-2)^2 a X + 2^2 cY \in I, \ (-2)^3 a X + 2^3 cY \in I.$

Using a vanderMonde invertibility of this system of linear equations, we find that either ${X}$ or ${Y}$ belongs to ${I}$. Say ${X}$ does, for definiteness; then ${H = [X,Y] \in I}$ too; from this, ${Y = -\frac{1}{2} [H,Y] \in I}$ as well. Thus ${I = \mathfrak{sl}_2}$.

If ${a=c=0}$, then from ${b \neq 0}$, we find ${H \in I}$, which implies ${X = \frac{1}{2}[H,X] \in I}$ and similarly for ${Y}$. Thus ${I= \mathfrak{sl}_2}$.

I claim now that the algebra ${\mathbb{C} H}$ is in fact a Cartan subalgebra. Indeed, it is easily checked to be maximal abelian. Moreover, since ${H}$ acts by a diagonalizable operator on the faithful representation on ${\mathbb{C}^2}$, it follows that ${H \in \mathfrak{sl}_2}$ is (abstractly) semisimple. (more…)

Recall that in the representation theory of ${\mathfrak{sl}_2}$, one considered an element ${H}$ and its action on a representation ${V}$. We looked for its largest eigenvalue and the corresponding highest weight vector.

There is something along the same lines to be done here for arbitrary semisimple Lie algebras, though it is much more complicated (and interesting).   I’m only going to scratch the surface today.

Let ${\mathfrak{g}}$ be a semisimple Lie algebra and ${\mathfrak{h}}$ a Cartan subalgebra. Then ${\mathfrak{h}}$ is to play the role of ${H}$ in ${\mathfrak{sl}_2}$; the ${X,Y}$ matrices in ${\mathfrak{sl}_2}$ are now replaced by the root space decomposition

$\displaystyle \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}.$

We know that ${\mathfrak{h}}$ acts on a representation ${V}$ of ${\mathfrak{g}}$ by commuting semisimple transformations, so we can write

$\displaystyle \mathfrak{h} = \bigoplus_{\beta \in \mathfrak{h}^{\vee}} V_{\beta}$

where ${V_{\beta} := \{ v \in V: hv = \beta(h) v \ \forall h \in \mathfrak{h} \}}$. These are called the weight spaces, and the ${\beta}$ are called weights.

Now

$\displaystyle g_{\alpha} V_{\beta} \subset V_{\alpha + \beta }$

by an analog of the “fundamental calculation,” proved as follows. Let ${h \in \mathfrak{h}, x \in \mathfrak{g}_{\alpha}, v \in V_{\beta}}$. Then

$\displaystyle h (x v) =xh(v) + [h,x] v = x (\alpha(h)) v + \beta(h) x v = (\alpha + \beta)(h) xv.$ (more…)