The Flow of a fundamental vector field preserves the symplectic form…

/ Alexander

Today I was reading the Michèle Audin’s book, there is a statement she might had skipped:

If Lie group G has a Symplectic action on Symplectic Manifold (M,\omega), that is g^*\omega=\omega, then, for any X\in\mathfrak{g}, the flow g^t of the fundamental vector fields X' generated by X preserves the Symplectic form.

Here the fundamental vector field is defined as X'=(df_x)_1X, where f_x is the orbit map: f_x: G\longrightarrow X, with f_x(g)=g.x, and 1 is the identity in Lie group G.

This statement is not so obviously for me, so I took a couple of hours to think about it, here is my proof.

Actually, the action of the flow g^t on M is same as the action of \exp tX (and that leads to the desired consequence immediately), where the \exp is the expoential from Lie algebras to Lie Groups, indeed, for any x\in M, the result of (\exp tX)(x):\mathbb{R}\longrightarrow M forms a parameterized curve on M (if I may, I will denote it by \gamma(t)), if we can prove it satisfies the initial condition (\exp 0X)x=x, and the differential equation \frac{d(\exp tX)(x)}{dt}=X'(\gamma(t)), then the two actions will be same.

The initial condition is obviously satisfied, notice that the curve can be written as \gamma(t)=f_{x}\circ \exp tX, and if we denoted by L_{\exp(-tX)}, the action of the left multiplication, hence by the chain rule we have:

\begin{aligned}\frac{d(f_x\circ\exp tX)}{dt}&=\frac{d(f_{(\exp tX)(x)}\circ L_{\exp(-tX)}\circ\exp(tX))}{dt}\\&=(df_{\exp(tX)(x)})_{1}\left(\left.\frac{d\exp(tX)}{dt}\right|_{t=0}\right)\\&=(df_{\exp(tX)(x)})X\end{aligned}

That is obviously the value X'(\gamma(t)), there is another interesting view if we denoted by \exp(tX'):M\longrightarrow M, the flow induced by the fundamental vector field X' (actually, this Is a widely used notion for the flow of a vector field, for example, da Silva’s Lectures on Symplectic Geometry), then the result we proved is a really beautiful and harmonic formula:

\exp tX'=\exp tX

The \exp on the left hand side is the flow on Manifolds, the \exp on the right hand side is the flow on Lie Groups, they were attached perfectly via the fundamental vector field.

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