The Cartan matrix of type $A_2$ is $\left( \begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right)$. What is the Cartan matrix of type $A_1 \times A_1$? Thank you very much.
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Do you know what the Cartan matrix of $A_1$ is? Then take a block matrix. – Dietrich Burde Feb 26 '17 at 11:47
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It's $$ \left( \begin{matrix} 2 & 0 \\ 0 & 2 \end{matrix} \right). $$
Recall the formula for the entries of the Cartan matrix, $a_{ij}=2(\alpha_i,\alpha_j)/(\alpha_i,\alpha_i)$. In $A_1\times A_1$, the two simple roots are orthogonal (the corresponding vertices in the Dynkin diagram have no edge), so the inner product in the numerator is zero for the off diagonal entries.
Ben West
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