For ordered simple roots, the <Cartan matrix> is $A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle=2(\alpha_i,\alpha_j)/(\alpha_j,\alpha_j)$. It determines their relative lengths and angles.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 4 / a / iv / Solution

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>.

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

With $A_{ij}=2(\alpha_i,\alpha_j)/(\alpha_j,\alpha_j)$ and roots labeled from left to right, the three <Cartan matrix>[Cartan matrices] are
$$
A(A_3)=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix},\quad
A(B_3)=\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix},\quad
A(C_3)=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{pmatrix}.
$$

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

{c}
{title2=$A_{ij}$}
{wiki}

For ordered simple roots, the Cartan matrix has entries $A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle$. It determines their angles and relative lengths and therefore determines the corresponding crystallographic root system.


Codex @codex  0

 Incoming links: Cartan matrix