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…)

Let ${\mathfrak{g}}$ be a semisimple Lie algebra over ${\mathbb{C}}$ and ${\mathfrak{h}}$ a Cartan subalgebra.

Given ${\alpha \in \mathfrak{h}^{\vee}}$, we can define a subspace of ${\mathfrak{g}}$

$\displaystyle \mathfrak{g}_{\alpha} = \{ x \in \mathfrak{g}: (\mathrm{ad} H)x = \alpha(H) x , \ \forall H \in \mathfrak{h} \}.$

The nonzero ${\alpha}$ that occur with ${\mathfrak{g}_{\alpha} \neq 0}$ are called roots, and they form a set ${\Phi}$. Because ${\mathfrak{h}}$ acts on ${\mathfrak{g}}$ by commuting diagonalizable operators (by semisimplicity of the elements of ${\mathfrak{h}}$), it follows by simultaneous diagonalization, that

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

Recall that ${\mathfrak{g}_0 = \mathfrak{h}}$, because a Cartan subalgebra is maximal abelian.

This is called the root space decomposition. A simple but important property is that ${[\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}] \subset \mathfrak{g}_{\alpha + \beta}}$; this is checked because the ${\mathrm{ad} H}$ are derivations.

The root space decomposition is highly useful in studying simple representations of ${\mathfrak{g}}$.

I shall collect here a few facts about it.

Proposition 1 ${\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}}$ are orthogonal under the Killing form unless ${\alpha + \beta = 0}$.

This follows by a familiar argument, in view of ${[\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}] \subset \mathfrak{g}_{\alpha + \beta}}$. (more…)

This post is the second in the series on ${\mathfrak{sl}_2}$ and the third in the series on Lie algebras. I’m going to start where we left off yesterday on ${\mathfrak{sl}_2}$, and go straight from there to classification.  Basically, it’s linear algebra.

Classification

We’ve covered all the preliminaries now and we can classify the ${\mathfrak{sl}_2}$-representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible ${V}$.

So, we know that ${H}$ acts diagonalizably on ${V}$, which means we can write

$\displaystyle V = \bigoplus_\lambda V_\lambda$

where ${Hv_\lambda = \lambda v_{\lambda}}$ for each ${\lambda}$, i.e. ${V_\lambda}$ is the ${H}$-eigenspace.

${\mathfrak{sl}_2}$ is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over ${\mathbb{C}}$ of trace zero, with the Lie bracket defined by:

$\displaystyle [A,B] = AB - BA.$

The representation theory of ${\mathfrak{sl}_2}$ is important for several reasons.

1. It’s elegant.
2. It introduces important ideas that generalize to the setting of semisimple Lie algebras.
3. Knowing the theory for ${\mathfrak{sl}_2}$ is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, ${\mathfrak{sl}_2}$ is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.