Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 3 / b / Solution

The Cartan-Weyl basis obeysFor a highest-weight vector $|\Lambda ⟩$, $E_{+}|\Lambda ⟩=0$ and $H|\Lambda ⟩=\Lambda |\Lambda ⟩$. Successive nonzero lowerings give the weight basisFinite dimensionality requires the highest weight $\Lambda$ to be a nonnegative integer. The weights are $\Lambda ,\Lambda -2,\dots ,-\Lambda$, each with multiplicity one, so $\dim d_{\Lambda }=\Lambda +1$.