For simple roots $\alpha_i$, the <Cartan matrix> is
$$
A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle
=\frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}.
$$
Its diagonal entries are two, while its off-diagonal entries encode the angles and relative lengths in the <Dynkin diagram>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-302/4/a/i/solution