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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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The pairing is defined by evaluation. $\mathfrak g^*$ consists of functionals $\mathfrak g\to \mathbb R$. So $\langle \phi,X\rangle=\phi(X).$ Then the coadjoint representation is defined by the formula stated. If you don't get a good answer fleshing this out, I will expand on this, but I have to go right now. –  Grumpy Parsnip Jul 9 '12 at 12:36
    
@JimConant So given some basis $L_1, \ldots L_k$ on $\mathfrak{g}$ I will have $<L_i^*,L_j>= \delta_{ij}$ (where $\delta_{ij} $ is Kronecker delta)? I will try to work out an example tonight and post it on mathstack to see if I understood it correctly. –  Novo Jul 9 '12 at 16:14
    
Yes, that is correct. –  Grumpy Parsnip Jul 9 '12 at 16:39
    
In a bit late, but isn't there a minus sign missing in the definition of $\mathrm{Ad}_X^{\ast}$ provided in the original post? –  darij grinberg Mar 3 at 16:01

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