Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 3 / e / Solution

The Trace form of a Lie algebra representation is invariant becauseChoose a basis orthonormal for the positive-definite form $-\kappa$ of the compact simple algebra. Any invariant bilinear form determines an endomorphism commuting with the irreducible adjoint action; Schur lemma makes it scalar. Thus $H(T_a,T_b)=c{\delta }_{ab}$. Since $d(T_a)$ is anti-Hermitian,The inequality is strict because the kernel of the nontrivial irreducible representation is an ideal and hence zero. Therefore $c=-\mu$ with $\mu >0$.

Solved by gpt-5.6-sol high.