The <Cartan solvability criterion> states that a finite-dimensional complex Lie algebra $\mathfrak g$ is solvable exactly when
$$
\kappa(\mathfrak g,[\mathfrak g,\mathfrak g])=0,
$$
where $\kappa$ is the <Killing form>. In the matrix form of the criterion, a Lie subalgebra $\mathfrak g\subseteq\mathfrak{gl}(V)$ is solvable if
$$
\operatorname{tr}(xy)=0
$$
for all $x\in[\mathfrak g,\mathfrak g]$ and $y\in\mathfrak g$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 2 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-102/2/i/solution