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.
A root system is a collection R inside R n of vectors such that
Let alpha be in R, write lambda alpha for the line through alpha and write P alpha from R n to lambda alpha for the orthogonal projection to this line. Then for any beta in R, P alpha of beta equals a half n beta alpha times 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 little s l 2 C subalgebra S alpha which acts on little g, and weight diagrams of little s l 2 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 plus or minus alpha.
The figure below shows one of the roots beta of SU(3) orthogonally projecting to alpha over 2 for another root alpha.
Verifying (2) and (4)
We will now verify (2) and (4).
We would like to prove that a half n beta alpha times 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.
little g is a representation of S alpha equals the subalgebra spanned by X, H alpha and Y where H alpha equals 2 alpha sharp over K alpha sharp, alpha sharp.
What is the weight of Z in little g beta with respect to this action? We have little ad H alpha of Z equals beta of H alpha times Z so Z has weight beta of H alpha, so that beta of H alpha is an integer. Let's call this integer n beta alpha.
Calculating, we get beta of H alpha equals K of beta sharp, H alpha, which equals 2 K star beta, alpha over K star 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 hat alpha equals alpha over length alpha, then you take the dot product beta dot hat alpha to get the component of beta in this direction, then you multiply by hat alpha to get the corresponding vector in this direction: P alpha of beta equals beta dot hat alpha times hat alpha equals beta dot alpha over alpha dot alpha, times alpha.
Therefore we deduce P alpha of beta equals a half n beta alpha times alpha as required.
We want to show that the only roots on lambda alpha are plus or minus 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 minus beta, minus 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 a half, so we need alpha equals beta over 2 and beta equals 2 alpha.
Take the representation of S alpha given by summing the root spaces on this line, which is little g minus beta, direct sum little g minus alpha, direct sum the span of H alpha, direct sum little g alpha, direct sum little g beta
All these root spaces are 1-dimensional, so this root diagram is just the root diagram of Sym 4 C 2. This means that the representation is isomorphic to Sym 4 C 2. However, X in little g alpha acts on little g alpha as zero because little ad X X equals X bracket X equals 0. If you look at the representation Sym 4 C 2 then you see that Sym 4 of X from the weight space with weight 2 to the weight space with weight 4 is nonzero, so we get a contradiction.
Our goal is now to classify root systems, which will give us a classification of Lie algebras of compact semisimple groups.