= Hamiltonian existence for a semisimple symplectic action
{c}
{title2=$[\mathfrak g,\mathfrak g]=\mathfrak g$}
For a symplectic left <Lie group action>, use actual fundamental fields $\xi_P$ and $\iota_{X_f}\omega=df$. They satisfy $[\xi_P,\eta_P]=-[\xi,\eta]_P$. <Cartan's magic formula> then gives $\iota_{[\xi,\eta]_P}\omega=d[\omega(\xi_P,\eta_P)]$. Thus every commutator generator has a globally exact contraction. A <semisimple Lie algebra> is perfect, so every generator has a global Hamiltonian. The separate <semisimple moment-map equivariance> result removes the constant two-cocycle obstruction and gives an equivariant <moment map> for a connected group.
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