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

The special unitary group is $SU(2)=\{U\in M_2(\mathbb{C}):U^{†}U=I,\det U=1\}$. Differentiating these equations at the identity shows that its Lie algebra isFor $T_i=-i{\sigma }_i/2$, the Pauli-matrix identity $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$ givesso the structure constants in this basis are $c^k{}_{ij}={ϵ}_{ijk}$.

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 $\dim d_{\Lambda }=\Lambda +1$.

In the defining representation,Multiplying the three factors in the disentangling identity and comparing with $e^{i\theta {\sigma }_1/2}$ givesThis factorization is the coordinate patch where $\cos (\theta /2)\ne 0$.

Since $-\theta T_1=i\theta {\sigma }_1/2$, the same disentangling identity applies in every Lie algebra representation. The rightmost $E_{+}$ kills $|\Lambda ⟩$, the leftmost $E_{-}$ kills $⟨\Lambda |$, and the middle factor acts by its highest weight. Therefore

Because $d(T_1)$ commutes with itself and is anti-Hermitian,Applying the preceding highest-weight matrix coefficient yields