Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 3

Every diagonal element of the stated Cartan subalgebra has the formLet ${\epsilon }_i$ extract $t_i$. The roots are the B2 root system

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ChooseHere ${\alpha }_1$ is long and ${\alpha }_2$ is short. The Dynkin diagram consists of two vertices joined by a double edge, with its arrow pointing from ${\alpha }_1$ toward the shorter root ${\alpha }_2$:

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The Weyl reflection $s_{{\alpha }_1}$ swaps ${\epsilon }_1$ and ${\epsilon }_2$, whereas $s_{{\alpha }_2}$ changes the sign of ${\epsilon }_2$. Hence

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The highest root isSince $\beta$ is long and the ${\epsilon }_i$ are orthonormal, ${\beta }^{\vee }=\beta$. The coroot pairing gives

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The displayed calculation identifies the root system of ${\mathfrak{so}}_5(\mathbb{C})$ as $B_2$. The symplectic Lie algebra ${\mathfrak{sp}}_4(\mathbb{C})$ has root system $C_2$. After exchanging the two simple-root labels, the $B_2$ and $C_2$ Cartan matrices agree, so their Dynkin diagrams define the same complex simple Lie algebra. The classification of finite-dimensional complex simple Lie algebras therefore givesThis is the Isomorphism between so5 and sp4.

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