The <special unitary group> is $SU(2)=\{U\in M_2(\mathbb C):U^\dagger U=I,\ \det U=1\}$. Differentiating these equations at the identity shows that its <Lie algebra> is
$$
\mathfrak{su}(2)=\{X:X^\dagger=-X,\ \operatorname{tr}X=0\}.
$$
For $T_i=-i\sigma_i/2$, the <Pauli matrix>[Pauli-matrix] identity $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$ gives
$$
[T_i,T_j]=\epsilon_{ijk}T_k,
$$
so the <structure constant of a Lie algebra>[structure constants] in this basis are $c^k{}_{ij}=\epsilon_{ijk}$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-302/3/a/solution