= Hamiltonian Lie algebra homomorphism
{c}
{title2=$f\mapsto X_f,\quad[X_f,X_g]=X_{\{f,g\}}$}
On a <symplectic manifold>, use $\iota_{X_f}\omega=-df$ and $\{f,g\}=\omega(X_f,X_g)=X_f(g)$ for the <Poisson bracket>. <Cartan's magic formula> gives $\mathcal L_{X_f}\omega=0$. Taking the <interior product of a differential form> with $[X_f,X_g]$ then yields $-d\{f,g\}$, proving the homomorphism. Its kernel consists of locally constant functions, since $X_f=0$ precisely when $df=0$. On a connected manifold these are the constants. The quotient of smooth functions by this kernel is therefore the <Lie algebra> of <Hamiltonian vector fields>. A different contraction sign requires a compatible bracket sign.
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