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 $\omega _{\beta }(\xi _{\mathfrak {g^{*}}}(v),\eta _{\mathfrak {g^{*}}}(v))=-\beta ([\xi ,\eta ])+\sum (\eta ,\xi )$.

Furthermore, we introduce a deformed Poisson bracket on (M, ω) with which some classical results of conservative mechanics still hold true in a new setting.