Consider the Lie group defined as

where

Its Lie algebra is the set of all matrices with real entries such that

A general element of is of the form

and so is a 21-dimensional real vector space. We would like to describe its root system.

A basis for this space is given below.

We complexify the Lie algebra by tensoring with to get . A basis for the complexified Lie algebra is given below. We use the notation to indicate the matrix whose -th entry is the complex conjugate of the -th entry of .

If we choose as a Cartan subalgebra, then we get the following picture of the root system of :

Definition: We say that a vector in a representation of has weight if

The arrows in our picture of the root system indicate how the elements of change the weights. For example, when acts on a vector with weight , it changes the weight to . changes the weight to .

If the representation comes from a representation of with trivial central character, then the sum must be even. So either all the weight are even or there are precisely two that are odd. Indeed, we have

In particular,

(since the center acts trivially.)

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April 1, 2015 at 5:23 am

AlexThis is the most difficult article about math I have ever seen, but great article too.