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I have trouble understanding the definition of the coadjoint representation of a Lie algebra.

Typically you first define a natural pairing between the Lie algebra and Lie coalgebra: \begin{equation} \langle, \rangle : \mathfrak{g}^* \times \mathfrak{g} \to \mathbb{R} \end{equation} I don't really understand how this is defined, the literature does not seem very explicit. How is this natural pairing defined?

Let $\mathrm{Ad}_X$ denote the adjoint representation. The coadjoint $\mathrm{Ad}^*_X$ representation is then given by \begin{equation} \langle Z, \mathrm{Ad}_X( Y) \rangle =\langle \mathrm{Ad}^*_X Z, Y \rangle \; \; \mathrm{with} \; \; Z \in \mathfrak{g}^*, \; \; X,Y \in \mathfrak{g} \end{equation} Do I compute this just by evaluation of the basis?

Would it be possible to supply me with a simple example?

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    @darijgrinberg Doubly late, but yes you're right.2018-12-19

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