Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / c

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- Solution c

Solution

Write

g (x) = I + x^{i} T_{i} + O (x^{2})

in a matrix representation and

[T_{i}, T_{j}] = c^{k}_{ij} T_{k}

. Direct multiplication gives

g (y)^{- 1} g (x)^{- 1} g (y) g (x) = I + y^{i} x^{j} [T_{i}, T_{j}] + O (3),

so

w^{k} = c^{k}_{ij} y^{i} x^{j} + O (3) = - c^{k}_{ij} x^{i} y^{j} + O (3) .

The cancellation of the constant and linear terms explains why the group commutator first detects the Lie bracket at quadratic order.

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