Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 1 / d

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- Solution d

Solution

For a smooth Lie-group representation

D : G \to G L (V)

, define its derived representation

d D (X) = \frac{d}{d t}_{t = 0} D (e^{tX}) .

Differentiation makes

d D

linear. Applying

D

to the group commutator curve

e^{tX} e^{s Y} e^{- tX} e^{- s Y}

and taking the mixed derivative at

(0, 0)

gives

d D ([X, Y]) = [d D (X), d D (Y)] .

Thus

d D : g \to gl (V)

is a Lie algebra representation.

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