I was reading section 4.5.1 of https://arxiv.org/pdf/1801.00123.pdf but I got stuck at the following because I don't know much about Hopf algebras.

Let $G$ be a complex Lie group and let $\mathbb{C}[G]$ be a ring of regular functions on $G$. Consider the completion at identity $\mathbb{C}[[G]] := \lim_{\leftarrow} \mathbb{C}[G]/I^n$, where $I := \{f\in \mathbb{C}[G] | f(1) = 0\}$. Then the Universal Enveloping Algebra $U(\mathfrak{g})$ is identified as a Hopf algebra with the continuous dual \begin{equation} \mathbb{C}[[G]]^* = \{ \psi \in Hom_{\mathbb{C}}(\mathbb{C}[G],\mathbb{C}) | \psi(I^n) = 0, n >> 0 \}. \end{equation}

So how do I see that $U(\mathfrak{g})$ can be identified as Hopf algebra to $\mathbb{C}[[G]]^*$?

Thank you.


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