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As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping algebra $U(\frak{g})$? Secondly, if this is case, does it then extend to Drinfeld--Jimbo quantum group setting? That is for a quantized enveloping algebra $U_q(\frak{g})$, with corresponding dually paired coordinate algebra ${\cal O}(G)$, does there exist a coaction of ${\cal O}(G)$ on $U_q(\frak{g})$?

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