1. Suppose you have a Lie group. The generators of this group, along with their commutation relations, form a Lie algebra. The maximal commuting subset of this is called a Cartan Sub-Algebra (CSA).
2. Suppose you have a CSA with 20 generators.
3. Given some vector space V, each of the 20 generators can be associated with 20 matrices. If you change V, you change the 20 matrices
4. If a vector A of the vector space V is an eigenvector of every element of the CSA (let’s call them M1, M2, M3 … M19, M20), then M_i acting on A produces 20 eigenvalues. These 20 numbers, together are called a weight, L. So L = L(e1, e2, e3 … e19, e20). L(M_i) = e_i. M_i acting on A = e_i * A. A is called a weight vector.
5. In theory, if you are given some random vector space D, you won’t know how many weights, how many L’s, exist in advance in this D-representation of the CSA. In practice, you work with vector spaces that you know something about, so it is possible to predict how many weights you will encounter in this vector space’s representation of the CSA.
6. So in vector space V there could be 5 weights, L1 L2 L3 L4 L5. In another vector space W there could be 8 weights, N1 N2 N3 N4 N5 N6 N7 N8. In another vector space Y there could be 13 weights, P1-P13.
7. Ok, now, let’s call the original Lie algebra T and the CSA S. T is, in fact, a vector space. If you represent S using the T-vector space, you have what is called the adjoint representation of this Lie algebra. The adjoint action, instead of being matrix multiplication, is the commutator: Adj_H(X) = [H,X]. H is an element of the CSA S. X is a vector in the Lie algebra T, the T-vector space (since T is a vector space).
8. Now, for all H in S, a non-zero X in T and [H,X] = number(H)*X, then X is an eigenvector of all CSA (S) elements, H. You can organize these numbers by saying that L(H) = number. L is called a root. L eats a CSA element H and outputs a number. If an element H of the CSA satisfies L_j(H) = integer for all roots L, then H is called a coweight.