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

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- Solution i

Solution

For simple roots

α_{i}

, the Cartan matrix is

A_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩ = \frac{2 ( α _{i} , α _{j} )}{( α _{j} , α _{j} )} .

Its diagonal entries are two, while its off-diagonal entries encode the angles and relative lengths in the Dynkin diagram.

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