Using the matrix-unit commutator and summing the adjoint indices gives
$$
\boxed{\kappa(X,Y)=2n\operatorname{tr}(XY)-2\operatorname{tr}(X)\operatorname{tr}(Y)}.
$$
Equivalently, in components this is $2nX^i{}_jY^j{}_i-2X^i{}_iY^j{}_j$. It vanishes on the scalar center, as expected because $\mathfrak{gl}_n$ is not semisimple.


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 2 / f / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-302/2/f/solution