Example:
If G = SU(3) then little t is the set of diagonal matrices with diagonal entries i theta_1, i theta_2, minus i (theta_1 plus theta_2)
The weight lattice
The weight lattice
Suppose we have a compact group G. We know that G contains a maximal torus T. Let little t be the Lie algebra of T.
Let little h be little t tensor C inside little g tensor C be the complexification of little t. In our example, this amounts to allowing theta_1 and theta_2 to be complex. We're really interested in the matrices H_(theta), diagonal theta_1, theta_2, minus (theta_1 plus theta_2), and these live inside little h R equals i times little t inside little h.
Given a complex representation R from big G to big G L V, we know that V splits as a direct sum of subspaces W_(lambda) where W_(lambda) is the subspace of v in V such that R of exp i Theta equals e to the i lambda of Theta times v for all Theta in little h R.
lambda is a map from little h R to R of the form lambda of Theta equals lambda_1 theta_1 plus dot dot dot plus lambda_n theta_n In other words, lambda lives in little h R star, the dual space of little h R. So our weight diagrams are pictures inside little h R.
Example:
For SU(2), our weight diagram consisted of a line with some dots on. The line was really a picture of little h R, the set of diagonal matrices theta, minus theta, with theta in R. But what were the dots?
We know that if exp of i Theta equals the identity then R of exp i Theta equals the identity, so e to the i lambda of Theta equals 1.
Definition:
Define little h Z inside little h R to be the lattice of Thetas in little h R such that exp of i Theta equals the identity. Then our weights live in the
For example, the triangular dot paper where we drew our weight diagrams of SU(3) representations was a depiction of the weight lattice of SU(3).
Example:
For SU(2), little h Z equals the set of theta such that exp if i times the 2-by-2 matrix theta, 0, 0, minus theta equals the identity, which equals the set of points 2 pi n with n an integer and little h Z star equals the set of lambdas such that lambda times 2 pi n is in 2 pi Z for all n in Z, which is just Z inside R So the weight lattice is really just the integers inside R.
Example:
For SU(3), the weight lattice was the triangular lattice we used to draw pictures on, where the lambda_1 and lambda_2 "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.
Pre-class exercise
Exercise:
Explicitly, what is the set of matrices \mathfrak{h}_{\ZZ} for our usual diagonal choice of maximal torus in SU(3)?