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Let $(\mathfrak{g}, [,], \delta)$ be a Lie bialgebra where $\delta$ is the cobracket. It is well-known that there exists a simply connected Poisson-Lie group $G$ such that $\mathfrak{g} = \mathrm{Lie}(G)$.

Question: Is there a general method to construct the Poisson bracket $\{,\}$ on $G$ starting from the Lie bialgebra structure on $\mathfrak{g}$? In particular, when $\mathfrak{g}$ is the Borel subalgebra $\mathfrak{b}_+$ of $\mathfrak{sl}_2$ (with generators $H$ and $E$ such that $[H, E] = 2E$) and $\delta(H) = 0,\delta(E) = E \wedge H$, is it possible to explicitly write down the Poisson bracket for $B_+$?

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    $\begingroup$ This is the restriction of the standard structure on $SL_2$ which is written down in many textbooks, e.g. see section 4.4.1 of cmls.polytechnique.fr/cmat/kosmann/lnp2.pdf and set $c=0$. $\endgroup$
    – Adrien
    Commented Aug 14 at 16:14
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    $\begingroup$ @Adrien Your explanation was very clear. Thank you very much! $\endgroup$
    – yohei ohta
    Commented Aug 17 at 4:10

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