Differentiating $\operatorname{Ad}_{\exp(tX)}Y$ at zero gives the <Adjoint representation of a Lie algebra>
$$
\operatorname{ad}_X(Y)=[X,Y].
$$
The Jacobi identity implies $[\operatorname{ad}_X,\operatorname{ad}_Y]=\operatorname{ad}_{[X,Y]}$, proving that it is a Lie-algebra representation.

Solution

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

 Solution

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