Roots and Weights

Now we are going to generalize the analysis of the representations of the SU(2) algebra to an arbitrary simple Lie algebra. The idea is simple. First, we do what we always try to do in quantum mechanics — find the largest possible set of commuting hermitian observables and use their eigenvalues to label the states. In this case, our observables will be the largest set of hermitian generators we can find that commute with one another, and can therefore be simultaneously diagonalized. Their eigenvalues will be the analog of J3. The rest of the generators will be analogous to the raising and lowering operators in SU(2). We will find that every raising operator corresponds to an SU(2) subgroup of the Lie algebra, and then we can use what we know about 577(2) to learn about the larger algebra.