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Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ multiplicative with respect to the groupoid multilpication and such that the unit map $X\rightarrow \mathcal{G}$ is Lagrangian.

It is known that some Poisson manifolds $X$ can be integrated to symplectic groupoids $\mathcal{G}\rightrightarrows X$. Conversely, given a symplectic groupoid $\mathcal{G}\rightrightarrows X$ one can uniquely recover a Poisson structure on $X$ such that $p_1\colon\mathcal{G}\rightarrow X$ is Poisson and $p_2\colon\mathcal{G}\rightarrow X$ is anti-Poisson.

What is an object integrating a Poisson-Lie group $G$?

Clearly, this should give rise to a symplectic groupoid $\mathcal{G}\rightrightarrows G$ together with three Lagrangians in $\mathcal{G}\times\mathcal{G}\times\overline{\mathcal{G}}$ (here bar means the manifold with the opposite symplectic structure) lifting the two projections and the multiplication, and a Lagrangian $\mathcal{G}\times_G\mathcal{G}\rightarrow \mathcal{G}\times\mathcal{G}\times\overline{\mathcal{G}}$ from the definition of a symplectic groupoid.

I would guess that the corresponding object should be a symplectic 2-groupoid $\mathcal{G}_2\stackrel{\rightarrow}\rightrightarrows G\rightrightarrows \mathrm{pt}$, but I haven't figured out how to get the 4 Lagrangians.

Note that there is a known integration of doubles of Poisson groups to double symplectic groupoids and hence to symplectic 2-groupoids (http://arxiv.org/abs/1012.4103), but the one-simplices of the latter are something like $G\times G^\ast$ instead of just $G$.

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  • $\begingroup$ Ah, very nice. So, if $G$ is a Poisson group, $\mathrm{B} G$ is 1-shifted Poisson, hence $T^*_{\mathrm{B}G}[1]$ is a Lie algebroid. The fiber of $T^*_{\mathrm{B}G}[1]$ is $\mathfrak{g}^*$ which integrates to $G^\ast$, so I guess the integration to a symplectic 2-groupoid with 1-simplices the double is the "right" integration. One also ought to remember $\mathrm{pt}\rightarrow \mathrm{B}G$, which is possibly coisotropic, so it integrates to a Lagrangian. Thanks! $\endgroup$
    – Pavel Safronov
    Commented Jan 3, 2015 at 10:25
  • $\begingroup$ Good, that's farther than I had managed last night. (And it looks like, as I requested, my badly-formatted comment was deleted.) Infinitesimally, $\mathrm{pt} \to \mathrm B G$ is not just coisotropic, but sub-Poisson, so I expect that to remain true at the level of simplicial manifolds as well. Note that the double, being endowed with a metric, receives a 2-shifted symplectic structure on its classifying space; this should be analogous to the symplectic structure on the morphisms in the symplectic groupoid. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jan 3, 2015 at 16:28

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I think this question was already answered in the comments, but I just wanted to point out Jiang Hua Lu's thesis (which can be found on her website http://hkumath.hku.hk/~jhlu/publications.html) talks about this in some detail. In section 4.2, given a Poisson Lie group $G$, she constructs a double symplectic groupoid $\Gamma$ over $G$ and $G^*$, which simultaneously integrates the Poisson structure on both $G$ and the dual Poisson Lie group $\bar G^*$ (the Poisson structure on $G^*$ is negated).

It's perhaps interesting to note that $\Gamma$ has a natural interpretation as a moduli space of flat connections (Pavol Severa does this in his paper `Moduli spaces of flat connections and Morita equivalence of quantum tori' http://arxiv.org/abs/1106.1366). His description gives a nice TFT interpretation of the algebraic structures on $\Gamma$.

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