Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 3 / c

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- Solution c

Solution

Each structure constant of a Lie algebra satisfies

f_{ab}^{d} = - f_{ba}^{d}

, so

f_{ab c} = f_{ab}^{d} κ_{d c}

is antisymmetric in

a, b

. Invariance of the Killing form gives

f_{ab c} = κ ([T_{a}, T_{b}], T_{c}) = κ (T_{a}, [T_{b}, T_{c}]) = f_{b c a} .

This cyclic symmetry together with antisymmetry in the first pair implies antisymmetry under every transposition. Thus

f_{ab c}

is totally antisymmetric.

Solved by gpt-5.6-sol high.

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