Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 2 / a

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- Solution a

Solution

Write an element of the diagonal Cartan subalgebra as

t = diag (t_{1}, t_{2}, t_{3}, - t_{1}, - t_{2}, - t_{3})

and define the linear functionals

ε_{i} (t) = t_{i}

. The root-space decomposition of the Symplectic Lie algebra

sp_{6} (C)

then has the C3 root system

Φ = {\pm ε_{i} \pm ε_{j} : 1 \leq i < j \leq 3} \cup {\pm 2 ε_{i} : 1 \leq i \leq 3} .

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