So I'm reading the part in Ana Cannas da Silva's book "Lectures on Symplectic Geometry" available (on her website) about hamiltonian group actions on a symplectic manifold. She starts by defining $\mathbb{R}$-actions and $S¹$ actions by saying that the vector field on M that they generate must be hamiltonian. Then she defines hamiltonian $T^n$-actions (p.154) by the requirement that the restriction of the action to each circle ${1} \times\ldots\times S¹\times\ldots\times\{1\}$ be hamiltonian (plus the requirement that each of the n corresponding hamiltonian functions be invariant une the action of the rest of $T^n$)

And then, finally, she defines a hamiltonian action of a general Lie group G as one having a "moment map". This is a natural generalisation because the existence of a moment map is equivalent (I believe!) to the fact that for each $X\in Lie(G)$, the vector field $X^* $ on M induced by $X$ is hamiltonian (i.e. $ X^*_p=\frac{d}{dt}(\exp(tX)\cdot p)(0)$).

For indeed, if that it that case, then on can just define the moment map $\mu : M \rightarrow Lie(G)^* $ by setting $<\mu(p),X>:=\mu^X(p)$, where $\mu^X$ is a hamiltonian function for $X^* $ chosen so that $\mu$ is G-equivariant with respect to the coadjoint action on $Lie(G)^* $. (Note that in the case where $G$ is commutative such as $G=T^n$, this last condition boils down to $\mu$ being $G$-invariant.)

The question: I am trying to prove that the ad-hoc definition implies the general definition in the $T^n$ case. The problem I am having is that we only know that n vector fields $ X_1^*,\ldots,X_n^* $ (one for each subcircle $ 1\times\ldots\times S¹\times\ldots\times 1 \subset T^n $) are hamiltonian. Knowing that the $X_i$'s form a basis of $Lie(T^n)$, does this imply that every $X\in Lie(T^n)$ induces a hamiltonian $X^* $. Does something like '$(X+Y)^* =X^* +Y^* $' hold?

More generally, given a Lie group $G$ acting on a symplectic manifold $M$, is it necessary to check that each $X\in Lie(G) $ induce a hamiltonian vector field on $M$, or is it sufficient to check this for a basis of $Lie(G) $ ?

Thanks.

So I'm reading the part in Ana Cannas da Silva's book "Lectures on Symplectic Geometry" available (on her website) about hamiltonian group actions on a symplectic manifold. She starts by defining $\mathbb R$-actions and $\mathbb S^1$ actions by saying that the vector field on $M$ that they generate must be hamiltonian. Then she defines hamiltonian $\mathbb T^n$-actions (p.154) by the requirement that the restriction of the action to each circle $\{1\} \times\ldots\times\mathbb S^1\times\ldots\times\{1\}$ be hamiltonian (plus the requirement that each of the $n$ corresponding hamiltonian functions be invariant under the action of the rest of $\mathbb T^n$)

And then, finally, she defines a hamiltonian action of a general Lie group G as one having a "moment map". This is a natural generalisation because the existence of a moment map is equivalent (I believe!) to the fact that for each $X\in\operatorname{Lie}(G)$, the vector field $X^* $ on $M$ induced by $X$ is hamiltonian (i.e. $ X^*_p=\frac{d}{dt}(\exp(tX)\cdot p)(0)$).

For indeed, if that it that case, then on can just define the moment map $\mu : M \rightarrow\operatorname{Lie}(G)^* $ by setting $\langle\mu(p),X\rangle:=\mu^X(p)$, where $\mu^X$ is a hamiltonian function for $X^* $ chosen so that $\mu$ is G-equivariant with respect to the coadjoint action on $\operatorname{Lie}(G)^* $. (Note that in the case where $G$ is commutative such as $G=\mathbb T^n$, this last condition boils down to $\mu$ being $G$-invariant.)

The question: I am trying to prove that the ad-hoc definition implies the general definition in the $\mathbb T^n$ case. The problem I am having is that we only know that n vector fields $ X_1^*,\ldots,X_n^* $ (one for each subcircle $ 1\times\ldots\times\mathbb S^1\times\ldots\times 1 \subset\mathbb T^n $) are hamiltonian. Knowing that the $X_i$'s form a basis of $\operatorname{Lie}(\mathbb T^n)$, does this imply that every $X\in\operatorname{Lie}(\mathbb T^n)$ induces a hamiltonian $X^* $. Does something like $\color{red}{(X+Y)^* =X^* +Y^*}$ hold?

More generally, given a Lie group $G$ acting on a symplectic manifold $M$, is it necessary to check that each $X\in\operatorname{Lie}(G) $ induce a hamiltonian vector field on $M$, or is it sufficient to check this for a basis of $\operatorname{Lie}(G) $ ?

Thanks.