 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-2/2/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For

H = diag (r, s, - r, - s)

, put

L_{1} (H) = r

L_{2} (H) = s

. Then

L_{3} = - L_{1}

L_{4} = - L_{2}

. The matrix unit identity

[H, E_{ij}] = (L_{i} - L_{j}) (H) E_{ij}

reads off the root spaces directly. The complete list is

α L_{1} - L_{2} L_{2} - L_{1} L_{1} + L_{2} - L_{1} - L_{2} 2 L_{1} - 2 L_{1} 2 L_{2} - 2 L_{2} g_{α} C (E_{12} - E_{43}) C (E_{21} - E_{34}) C (E_{14} + E_{23}) C (E_{32} + E_{41}) C E_{13} C E_{31} C E_{24} C E_{42}

Φ = {\pm 2 L_{1}, \pm 2 L_{2}, \pm (L_{1} + L_{2}), \pm (L_{1} - L_{2})} .

All eight root spaces have dimension one. The zero-weight space is the two-dimensional Cartan subalgebra

C (E_{11} - E_{33}) \oplus C (E_{22} - E_{44})

; zero itself is not a root.

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