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Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal enveloping algebra of $\mathfrak{g}$. Then, by the quantum duality principle, $U_h(\mathfrak{g})$ is a quantization of $\mathscr{C}(G^*)$.

Question

Is it possible to explicitly give an isomorphism between $U_h(\mathfrak{g})$ and $\mathscr{C}_h(G^*)$ in a concrete Poisson Lie group $G$? $G$ can be anything, but I would like to know more about the case $G=\mathrm{SL}(2,\mathbb{C})$ if possible.

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  • $\begingroup$ What is your definition of $\mathscr{C}_h(G^*)$ ? The only definitions I know are either very abstract or use the duality principle. IMHO the simplest example is: take $G$ with the zero Poisson structure. Its dual is then $\mathfrak g^*$, with Poisson structure given by the Lie bracket of $\mathfrak g$ and group structure given by the underlying vector space. Then the quantum formal series Hopf algebra associated with $U(\mathfrak g)[[\hbar]]$ is the subalgebra generated by $\hbar \mathfrak g$, which is indeed a quantization of the formal Poisson group associated with $\mathfrak g^*$. $\endgroup$
    – Adrien
    Commented Jul 5, 2023 at 15:47
  • $\begingroup$ Thank you for your reply. I was using $\mathcal{C}_h(G^*)$ in the sense of the quantization of $\mathcal{C}(G^*)$, but then $\mathcal{C}(G^*)=U_h(\mathfrak{g})$, so my question makes no sense! The example you gave looks very concrete and good. I would like to understand the relation between the Drinfeld double and the Heisenberg double of the Poisson Lie group and the Drinfeld double and the Heisenberg double of the Hopf algebra through concrete examples. I will try to write down these relations with your examples. $\endgroup$
    – yohei ohta
    Commented Jul 6, 2023 at 6:14
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    $\begingroup$ IN that case the Drinfeld double is (as a Lie bialgebra) the semidirect product $\mathfrak g \ltimes \mathfrak g^*$ where $\mathfrak g$ acts by the coadjoint action. As a group this is the semi-direct product $G \ltimes \mathfrak g^*$ with a certain Poisson structure. One quantization of this is $ O(G)\ltimes U(\mathfrak g)[[\hbar]]$ where the action is again the adjoint one, and the coproduct is the obvious one. The Heisenberg double in that case is just the cotangent bundle $T^*G$ with its symplectic structure. $\endgroup$
    – Adrien
    Commented Jul 6, 2023 at 6:22
  • $\begingroup$ A quantization is $D(G)[[\hbar]]$ where $D(G)$ is the algebra of differential operators on $G$, which you can identify with the Hopf algebraic Heisenberg double, which is $U(\mathfrak g) \ltimes O(G)$ but this time with respect to the action of $U(\mathfrak g)$ via, say, left invariant differential operators. $\endgroup$
    – Adrien
    Commented Jul 6, 2023 at 6:23

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