Symplectic Affine Action and Momentum with Cocycle

Abstract

Let G be a Lie group, \(\mathfrak {g}\) its Lie algebra, and \(\mathfrak {g^{*}}\) the dual of \(\mathfrak {g}\). Let Φ be the symplectic action of G on a symplectic manifold (M, ω). If the momentum mapping \(\mu :M\rightarrow \mathfrak {g^{*}}\) is not Ad∗-equivariant, it is a fact that one can modify the coadjoint action of G on \(\mathfrak {g^{*}}\) in order to make the momentum mapping equivariant with respect to the new G-structure in \(\mathfrak {g^{*}}\), and the orbit of the coadjoint action is a symplectic manifold. With the help of a two cocycle \(\sum :\mathfrak {g}\times \mathfrak {g}\rightarrow \mathbf {R}\), \((\xi ,\eta )\mapsto \sum (\xi ,\eta )=d\hat {\sigma }_{\eta }(e)\cdot \xi \) associated with one cocycle \(\sigma :G\rightarrow \mathfrak {g^{*}};~~\sigma (g)=\mu (\phi _g(m))-Ad^*_g\mu (m)\), we show that a symplectic structure can be defined on the orbit of the affine action \(\Psi (g,\beta ):=Ad_{g}^{*}\beta +\sigma (g)\) of G on \(\mathfrak {g^{*}}\), the orbit of which is a symplectic manifold with the symplectic structure \(\omega _{\beta }(\xi _{\mathfrak {g^{*}}}(v),\eta _{\mathfrak {g^{*}}}(v))=-\beta ([\xi ,\eta ])+\sum (\eta ,\xi )\).