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

Consider the real vector space

I claim that the form (obtained by the isomorphism induced by the Killing form and the Killing form itself) is actually an inner product making into a euclidean space. To see this, we will check that for all . Indeed:

where is the Killing form, by definition.

Now

Now acts by the scalar on , so after dividing by , this becomes

But as we showed yesterday, , so the sum in question is actually positive. This proves one half of:

Proposition 1is a euclidean space and .

It is clear that ; indeed, the roots span , because if were orthogonal to all the roots, then would act by zero on . (I probably should have said this before explicitly.)

I claim now that if we can write

for and the vectors linearly independent, then each . This will prove the final claim of the proposition, that the sum is direct. In fact, we get equations of the form

from which we can conclude, since all coefficients are rational, that .

** as a root system **

I now will go into some abstract generality on root systems. First, I’ve got to say a few things on reflections in a euclidean vector space —that is, a real one with an inenr product . Given a nonzero vector , the **reflection** around is defined by

A reflection is characterized by fixing the hyperplane perpendicular to and sending onto . Conversely, any endomorphism of that does that is the reflection.

A **root system** in is a finite set not containing zero that spans . We require:

1. The only multiples of any are

2. is invariant under the reflections .

3. If , then is an integer.

I will develop some of the theory of root systems in the sequel. John Armstrong at The Unapologetic Mathematician has been doing a series on this, starting here. Incidentally, if you haven’t seen his blog, you ought to check it out. It’s a free tour up the slopes of Mount Bourbaki, designed to show the interested nonmathematician (or aspiring mathematician) that staggering precipice one initially sees when encountering the subject conceals a gentle and mostly walkable path up the other side of the mountain. (OK, so my recent reading of *Climbing Mount Improbable *may have made me prone to abusing this particular metaphor.) I’m going to overlap with him slightly, but probably no more than I absolutely have to in order to do the basic representation theory.

First, we need a bit of motivation:

Proposition 2If is the set of roots of a semisimple Lie algebra with a fixed Cartan subalgebra and is defined as above, then is a root system.

It is clear from the definition that spans ; finiteness follows because is finite-dimensional. We proved 3 and 1 yesterday and actually computed in terms of maximal strings. The one to check is 2: if , we must prove .

To see this, consider the maximal string of roots as in yesterday; then . Here is units down from ; by this logic, which is units up from the bottom should belong to . Now though, so this proves the proposition.

I’ve been slightly disingenuous here. The choice of a root system seems to depend very heavily on the choice of the Cartan subalgebra, which is not unique. However, it is a theorem (which I may or may not give a proof of soon) that Cartan subalgebras are unique up to conjugation.

Anway, it turns out that you can recover a Lie algebra from a root system, so the classification of simple Lie algebras becomes reduced to the classification of simple root systems, which was successfully done—there are four “classical” infinite families and five exceptional cases.

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