# 10. The exponential map

We define the *exponential map* $ \exp \colon\mathfrak{g} \to G$ of a Lie group and prove it is a diffeomorphism on a neighbourhood of $0 \in \mathfrak{g}$. We then investigate Lie group actions on smooth manifolds, and then discuss the *adjoint representation*, which comes in two flavours:

$$\mathrm{Ad} \colon G \to \mathrm{GL}( \mathfrak{g}), \qquad \text{and} \qquad \mathrm{ad} = D \mathrm{Ad}(e) \colon \mathfrak{g} \to \mathfrak{gl}( \mathfrak{g}).$$

At the end of the lecture there is a short (non-examinable discussion) of the *infinite-dimensional Lie group*** **$ \mathrm{Diff}(M)$ of diffeomorphisms of a compact manifold $M$. We explain why from this point of view the Lie bracket on $ \mathfrak{X}(M)$ should really have the opposite sign… ¯\_(ツ)_/¯

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