For a finite-dimensional <Lie algebra representation> $\phi:\mathfrak g\to\mathfrak{gl}(V)$, the <Trace form of a Lie algebra representation> is
$$
B_V(x,y)=\operatorname{tr}(\phi(x)\phi(y)).
$$
The <Killing form> is the trace form of the <Adjoint representation of a Lie algebra>:
$$
\kappa(x,y)=\operatorname{tr}(\operatorname{ad}_x\operatorname{ad}_y).
$$
A bilinear form $B$ is $\mathfrak g$-invariant when
$$
B([z,x],y)+B(x,[z,y])=0,
$$
equivalently $B([x,y],z)=B(x,[y,z])$. For a trace form this follows from cyclicity of trace:
$$
\operatorname{tr}([\phi(z),\phi(x)]\phi(y))
+\operatorname{tr}(\phi(x)[\phi(z),\phi(y)])=0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-102/2/a/solution