Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) \otimes U(\mathfrak{g})$ be the comultiplication. It is said that \begin{align} \langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle, \end{align} where $\langle, \rangle$ is the natural pairing on $\mathfrak{g} \otimes \mathfrak{g}^*$. Are there some references about this formula? Thank you very much.
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asked May 17, 2015 at 7:53
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4$\begingroup$ If $\mathfrak{g}$ is a Lie algebra, then $\mathfrak{g}^*$ is a Lie coalgebra. Your formula is the definition of the cobracket on $\mathfrak{g}^*$ (it has nothing to do with the comultiplication on $U(\mathfrak{g})$). $\endgroup$– Pavel SafronovCommented May 17, 2015 at 9:25
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1$\begingroup$ $\delta: \mathfrak g^* \to \bigwedge^2 \mathfrak g^*$ is also the Lie-Poisson structure on $\mathfrak g^*$. $\endgroup$– Peter MichorCommented May 17, 2015 at 9:44
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