An explicit construction of a pre-quantum line bundle for the moduli space of flat $G$-bundles over a Riemann surface is given, where $G$ is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups $LG$ and the author's previous work on the obstruction to pre-quantization of the moduli space of flat $G$-bundles.

We are interested in Poisson structures transverse to nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $s{{l}_{n}}$, we construct some families of nilpotent orbits with quadratic transverse structures.

A quasi-Poisson manifold is a $G$-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in ${{\Lambda }^{3}}\mathfrak{g}$ associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

We study the symplectic geometry of the moduli spaces ${{M}_{r}}={{M}_{r}}\left( {{\mathbb{S}}^{3}} \right)$ of closed $n$-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in $\text{SU}\left( 2 \right)$ by the diagonal conjugation action of $\text{SU}\left( 2 \right)$. Here the fusion product of $n$ conjugacy classes is a Hamiltonian quasi-Poisson $\text{SU}\left( 2 \right)$-manifold in the sense of $\left[ \text{AKSM} \right]$. An integrable Hamiltonian system is constructed on ${{M}_{r}}$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on ${{M}_{r}}$ relates to the symplectic structure obtained from gauge-theoretic description of ${{M}_{r}}$. The results of this paper are analogues for the 3-sphere of results obtained for ${{M}_{r}}\left( {{\mathbb{H}}^{3}} \right)$, the moduli space of $n$-gons with fixed side-lengths in hyperbolic 3-space $\left[ \text{KMT} \right]$, and for ${{M}_{r}}\left( {{\mathbb{E}}^{3}} \right)$, the moduli space of $n$-gons with fixed side-lengths in ${{\mathbb{E}}^{3}}\left[ \text{KM}1 \right]$.