I’m keeping the same notation as all the previous posts here on semisimple Lie algebras.

Consider the real vector space

\displaystyle E = \sum_{\alpha \in \Phi} \mathbb{R} \alpha \subset \mathfrak{h}^{\vee}.

I claim that the form {(\cdot, \cdot)} (obtained by the isomorphism {\mathfrak{h}^{\vee} \rightarrow \mathfrak{h}} induced by the Killing form and the Killing form itself) is actually an inner product making {E} into a euclidean space. To see this, we will check that {(\alpha, \alpha) > 0} for all {\alpha}. Indeed:

\displaystyle (\alpha, \alpha) = B(T_{\alpha}, T_{\alpha})

where {B} is the Killing form, by definition.

Now

\displaystyle B(T_{\alpha}, T_{\alpha}) = \mathrm{Tr}_{\mathfrak{g}} ( \mathrm{ad} T_{\alpha}^2) = \sum_{\beta \in \Phi} \mathrm{Tr}_{\mathfrak{g}_{\beta}} ( \mathrm{ad} T_{\alpha}^2) .

Now {T_{\alpha}} acts by the scalar {\beta(T_{\alpha}) = (\beta, \alpha)} on {\mathfrak{g}_{\beta}}, so after dividing by {(\alpha, \alpha)^2}, this becomes

\displaystyle (\alpha, \alpha)^{-1} = \sum_{\beta \in \Phi} \left( \frac{ (\beta, \alpha)}{(\alpha, \alpha ) } \right)^2.

But as we showed yesterday, {\frac{ (\beta, \alpha)}{(\alpha, \alpha )} \in \mathbb{Q}}, so the sum in question is actually positive. This proves one half of:

Proposition 1 {E} is a euclidean space and {\mathfrak{h}^{\vee} = E \oplus iE}. (more…)

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OK, now we’ve gotten some of the basic facts about the root space decomposition down. So, as usual {\mathfrak{g}} is a semisimple Lie algebra and {\mathfrak{h}} a Cartan subalgebra; we have the decomposition {\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}}, where {\Phi \subset \mathfrak{h}^{\vee}} is the root system. For each {\alpha \in \Phi}, we can choose a pair of vectors {X_{\alpha} \in \mathfrak{g}_{\alpha}< Y_{\alpha} \in \mathfrak{g}_{-\alpha}, H_{\alpha} \in \mathfrak{h}}. Then {X_{\alpha}, Y_{\alpha}, H_{\alpha}} generate a subalgebra {\mathfrak{s}_{\alpha} \subset \mathfrak{g}} which is isomorphic to {\mathfrak{sl}_2}. Here {\alpha(H_{\alpha})=2} and {H_{\alpha}} is a multiple of {T_{\alpha}}, which in turn is the dual to {\alpha} under the Killing form that identifies {\mathfrak{h} \simeq \mathfrak{h}^{\vee}}.

That was a lightning review of where we are; if you’ve missed something, check back at this post.

The notation {\mathfrak{s}_{\alpha}} suggests that the algebra should only depend on {\alpha} and not on the particular choice of {X_{\alpha}, Y_{\alpha}} (but {H_{\alpha}} is uniquely determined from {\alpha(H_{\alpha})=2} and {H_{\alpha} \in \mathbb{C} T_{\alpha}}). Indeed, this is the case, and it follows from

Proposition 1 When {\alpha \in \Phi}, {\mathfrak{g}_{\alpha}} is one-dimensional.

 

Choose any {\mathfrak{s}_{\alpha}} coming from suitable {X_{\alpha}, Y_{\alpha}} and {H_{\alpha}}. We have a representation of {\mathfrak{s}_{\alpha}} on

\displaystyle V := \bigoplus_{\mathbb{Z} \alpha} \mathfrak{g}_{\alpha}

(recall {\mathfrak{g}_0 = \mathfrak{h}}) and we can apply the representation theory of {\mathfrak{sl}_2} to it. (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…)