The <Adjoint representation of a Lie algebra> is
$$
\operatorname{ad}:\mathfrak g\longrightarrow\mathfrak{gl}(\mathfrak g),
\qquad
\operatorname{ad}_x(y)=[x,y].
$$
The <Killing form> is the symmetric invariant bilinear form
$$
\boxed{\kappa(x,y)=\operatorname{tr}(\operatorname{ad}_x\operatorname{ad}_y)}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-102/1/a/solution