Group valued moment maps for even and odd simple $G$-modules

Let $G$ be a complex simple Lie group, and $\mathfrak{g}$ its Lie algebra. It is well known that a finite-dimensional $G$-module $V$ carrying a nondegenerate invariant bilinear form gives rise to a Hamiltonian Poisson space with a quadratic moment map $\mu$. We show that under condition $\mathrm{Hom}_{\mathfrak{g}}({\textstyle{\bigwedge}}^3 V, S^3V)=0$ this space can be viewed as a quasi-Poisson space with the same bivector, and with the group valued moment map $\Phi = \exp \circ \mu$. Furthermore, we show that by modifying the bivector by the standard $r$-matrix for $\mathfrak{g}$ one obtains a space with a Poisson action of the Poisson-Lie group~$G$, and with the moment map in the sense of Lu taking values in the dual Poisson-Lie group~$G^\ast$.