The weight lattice
Suppose we have a compact group . We know that contains a maximal torus . Let be the Lie algebra of .
Let be the complexification of . In our example, this amounts to allowing to be complex. We're really interested in the matrices , and these live inside
Given a complex representation , we know that where
is a map of the form In other words, lives in , the dual space of . So our weight diagrams are pictures inside .
For , our weight diagram consisted of a line with some dots on. The line was really a picture of . But what were the dots?
We know that if then , so .
to be the lattice
Then our weights live in the
For example, the triangular dot paper where we drew our weight diagrams of representations was a depiction of the weight lattice of .
For , and So the weight lattice is really just the integers inside .
For , the weight lattice was the triangular lattice we used to draw pictures on, where the and "axes" are separated by 120 degrees. In the next couple of videos, we will see that (for a class of Lie algebras called semisimple Lie algebras) the weight lattice comes equipped with a natural geometric structure called the Killing form, and it's with respect to this geometry that the angle is 120 degrees.
Explicitly, what is the set of matrices for our usual diagonal choice of maximal torus in ?