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Let $G$ be a connected complex semi-simple Lie group and ${\mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${\mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld-Jimbo quantum group, we obtain a deformation quantization $\mathbb{C}_\hslash[(N \backslash G)^m]$ of a Poisson structure $\pi^{(m)}$ on products $(N \backslash G)^m$ of principal affine spaces of a connected and simply connected complex semisimple Lie group $G$. The Poisson structure $\pi^{(m)}$ descends to a Poisson structure $\pi_m$ on products $(B \backslash G)^m$ of the flag variety of $G$ which was introduced and studied by the Lu and the author. Any ample line bundle on $(B \backslash G)^m$ inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of $\mathbb{C}_\hslash[(N \backslash G)^m]$. We define the notion of a strongly coisotropic subalgebra in a Hopf algebra, and explain how strong coisotropicity guarantees that any homogeneous coordinate ring of a homogeneous space of a Poisson Lie group can be quantized in the sense of Ciccoli, Fioresi, and Gavarini., Comment: 28 pages, to be published in J. Noncommut. Geom