Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 2 / b / i

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- Solution i

Solution

Conjugation differentiates to the Adjoint representation

Ad_{g} X = \frac{d}{d t}_{0} g exp (tX) g^{- 1} .

Because conjugations compose,

Ad_{g h} = Ad_{g} Ad_{h}

and

Ad_{e} = I

, so this is a group representation.

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