We now define root system which abstracts many of the nice properties of root diagrams in a collection of axioms. These all hold for the root diagrams of compact semisimple groups; we have already verified some of them and the rest will be verified in this video. We will be able to prove a classification theorem for root systems, which will then translate back to give a classification theorem for Lie algebras of compact semisimple Lie groups.

Definition:

A root system is a collection $R\subset {\mathbf{R}}^{n}$
of vectors such that

Let $\alpha \in R$
, write ${\lambda}_{\alpha}$
for the line through $\alpha $
and write ${P}_{\alpha}:{\mathbf{R}}^{n}\to {\lambda}_{\alpha}$
for the orthogonal projection to this line. Then for any $\beta \in R$
, $${P}_{\alpha}(\beta )=\frac{1}{2}{n}_{\beta \alpha}\alpha $$
for some integer ${n}_{\beta \alpha}$
.

The reflection in the hyperplane orthogonal to ${\lambda}_{\alpha}$
preserves the root system $R$
. This generates a group of symmetries called the Weyl group of $R$
. We can verify this axiom easily for root diagrams because for each root we have the $\U0001d530\U0001d529(2,\mathbf{C})$
subalgebra ${S}_{\alpha}$
which acts on $\U0001d524$
, and weight diagrams of $\U0001d530\U0001d529(2,\mathbf{C})$
representations are symmetric about the origin, which gives the reflection symmetry as claimed (just as for $SU(3)$
.

The only roots on ${\lambda}_{\alpha}$
are $\pm \alpha $
.

Example:

The figure below shows one of the roots $\beta $
of $SU(3)$
orthogonally projecting to $\alpha /2$
for another root $\alpha $
.

Verifying (2) and (4)

We will now verify (2) and (4).

Verifying (2)

We would like to prove that ${P}_{\alpha}(\beta )=\frac{1}{2}{n}_{\beta \alpha}\alpha $
. The integer ${n}_{\beta \alpha}$
will be a weight (these are basically the only integers cropping up naturally in this subject). In fact, it will be the weight of ${H}_{\alpha}$
acting on $\beta $
.

$\U0001d524$
is a representation of ${S}_{\alpha}=\mathbf{C}\cdot X\oplus \mathbf{C}\cdot {H}_{\alpha}\oplus \mathbf{C}\cdot Y$
where ${H}_{\alpha}=2{\alpha}^{\mathrm{\u266f}}/K({\alpha}^{\mathrm{\u266f}},{\alpha}^{\mathrm{\u266f}})$
.

What is the weight of $Z\in {\U0001d524}_{\beta}$
with respect to this action? We have $${\mathrm{ad}}_{{H}_{\alpha}}Z=\beta ({H}_{\alpha})Z$$
so $Z$
has weight $\beta ({H}_{\alpha})$
, so that $\beta ({H}_{\alpha})\in \mathbf{Z}$
. Let's call this integer ${n}_{\beta \alpha}$
.

Calculating, we get $$\beta ({H}_{\alpha})=K({\beta}^{\mathrm{\u266f}},{H}_{\alpha})=\frac{2{K}^{*}(\beta ,\alpha )}{{K}^{*}(\alpha ,\alpha )}.$$

In Euclidean geometry, if I give you vectors $\alpha $
and $\beta $
and tell you to project $\beta $
orthogonally onto the line through $\alpha $
, you first convert $\alpha $
into a unit vector $\widehat{\alpha}=\alpha /|\alpha |$
, then you take the dot product $\beta \cdot \widehat{\alpha}$
to get the component of $\beta $
in this direction, then you multiply by $\widehat{\alpha}$
to get the corresponding vector in this direction: $${P}_{\alpha}(\beta )=(\beta \cdot \widehat{\alpha})\widehat{\alpha}=\frac{\beta \cdot \alpha}{\alpha \cdot \alpha}\alpha .$$

Therefore we deduce ${P}_{\alpha}(\beta )=\frac{1}{2}{n}_{\beta \alpha}\alpha $
as required.

Verifying (4)

We want to show that the only roots on ${\lambda}_{\alpha}$
are $\pm \alpha $
. Let's pick one of our roots and rotate so that the line through this root is horizontal. Suppose there are more roots on this line (besides the two opposite roots). In other words, suppose that $-\beta $
, $-\alpha $
, $\alpha $
and $\beta $
are roots on this line (and that they arise in this order).

We've just shown that $\beta $
is a half-integer multiple of $\alpha $
and that $\alpha $
is a half-integer multiple of $\beta $
. The only half-integer strictly between $0$
and $1$
is $1/2$
, so we need $\alpha =\beta /2$
and $\beta =2\alpha $
.

Take the representation of ${S}_{\alpha}$
given by summing the root spaces on this line, which is $${\U0001d524}_{-\beta}\oplus {\U0001d524}_{-\alpha}\oplus \mathbf{C}\cdot {H}_{\alpha}\oplus {\U0001d524}_{\alpha}\oplus {\U0001d524}_{\beta}$$

All these root spaces are 1-dimensional, so this root diagram is just the root diagram of ${\mathrm{Sym}}^{4}({\mathbf{C}}^{2})$
. This means that the representation is isomorphic to ${\mathrm{Sym}}^{4}({\mathbf{C}}^{2})$
. However, $X\in {\U0001d524}_{\alpha}$
acts on ${\U0001d524}_{\alpha}$
as zero because ${\mathrm{ad}}_{X}X=[X,X]=0$
. If you look at the representation ${\mathrm{Sym}}^{4}({\mathbf{C}}^{2})$
then you see that ${\mathrm{Sym}}^{4}(X):{W}_{2}\to {W}_{4}$
is nonzero, so we get a contradiction.

Outlook

Our goal is now to classify root systems, which will give us a classification of Lie algebras of compact semisimple groups.