Background
Let $f: M \to N$ be a smooth map between smooth manifolds.
Two vector fields $X$ in $M$ and $Y$ in $N$ are said to be $f$-related if for all $p \in M$, $(f_*)_p(X_p) = Y_{f(p)}$; equivalently, if for every smooth function $g: N \to \mathbb{R}$, one has
$$(Yg) \circ f = X(g \circ f).$$
One immediate consequence of this definition is that if $X_i$ and $Y_i$ are $f$-related, for $i=1,2$, then so are their Lie brackets $[X_1,X_2]$ and $[Y_1,Y_2]$.
A number of basic results about Lie groups in a differential-geometric context follow from this observation; for example,
- the left-invariant/right-invariant vector fields form a Lie subalgebra,
- the differential of a smooth map between Lie groups is a Lie algebra homomorphism,
- the Lie algebra of a Lie subgroup is a Lie subalgebra (really a special case of the above),
- etc
Now, perhaps unwisely, while I was preparing a problem sheet for an undergraduate summer project, I added a problem on f-relatedness, one of whose parts was to show the following.
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, acting on the
leftright on a smooth manifold $M$. If $X \in \mathfrak{g}$, let $X'$ denote the corresponding fundamental vector field on $M$. Show that $[X',Y'] = [X,Y]'$, where the left-hand side is the Lie bracket of vector fields on $M$ and the right-hand side is the vector field on $M$ corresponding to the Lie algebra bracket of $X,Y \in \mathfrak{g}$.
One can prove this directly, of course, but I (mistakenly?) thought there ought to be a slick proof using $f$-relatedness. After one of my students complained that he could not find a proof using $f$-relatedness, I realised (somewhat embarrassingly) that neither could I!
The naive thing does not work: one can exhibit $X'_m = (\alpha_m)_* X$, where $\alpha_m : G \to M$ is the map sending $g$ to $g\cdot m$. This does not work because the map used to relate $X$ and $X'$ depends on $m$.
Hence the following
Question
Is the vector field $X'$ on $M$ $f$-related to the left-invariant vector field on $G$ corresponding to $X \in \mathfrak{g}$?
In other words, the question is to exhibit $f$. It does not seem that it's as simple as a map $G\to M$, but perhaps there is some trick using the action $\alpha: G \times M \to M$ in some way.
Best Answer
I hate to throw cold water on the party, but surprisingly, the formula that the OP was trying to prove ($[X',Y']=[X,Y]'$) is actually false when $G$ acts on the left. This formula is correct if $G$ acts on the right on $M$; but if $G$ acts on the left, then the correct formula is $[X',Y']=-[X,Y]'$.
Here's how to prove it. Suppose first that $G$ acts smoothly on $M$ on the right. Fix $p\in M$, and consider the orbit map $\alpha^{(p)}: G\to M$ defined by $\alpha^{(p)}(g) = p\cdot g$. Then I claim that for each $X\in \mathfrak g$, the fundamental vector field $X'$ is $\alpha^{(p)}$-related to $X$. To see this, note that the group law $p\cdot gg' = (p\cdot g)\cdot g'$ translates to $$\alpha^{(p)}\circ L_g (g') = \alpha^{(p\cdot g)}(g')$$ (where $L_g$ is left multiplication by $g$). Let $g\in G$ be arbitrary and set $q=p\cdot g=\alpha^{(p)}(g)$. Because $X$ is left-invariant, $$ X'_q = d\bigl(\alpha^{(q)}\bigr)_1(X_1) = d\bigl(\alpha^{(p)}\bigr)_g\circ d(L_g)_1(X_1) = d\bigl(\alpha^{(p)}\bigr)_g(X_g)$$ (where the first equality is essentially the definition of $X'$, and the second follows from the previous equation by taking differentials at the identity). This proves the claim.
Because brackets of $\alpha^{(p)}$-related vector fields are themselves $\alpha^{(p)}$-related, it follows that $[X',Y']_p = ([X,Y]')_p$, and since this is true for every $p$ it is true globally.
Now if $G$ acts on $M$ on the left, the above argument doesn't work; if $\tilde\alpha^{(p)}$ denotes the orbit map for the left action, then $X'$ is not $\tilde\alpha^{(p)}$-related to $X$. Instead, we can create a right action by setting $p\cdot g = g^{-1}\cdot p$. Letting $\alpha^{(p)}$ denote the orbit map for this new right action, we have $\tilde\alpha^{(p)}=\alpha^{(p)}\circ\iota$, where $\iota: G\to G$ is inversion; and the argument above shows that $X$ and $X'$ are $\alpha^{(p)}$-related. Since $d\alpha^{(p)}$ preserves brackets and $d\iota$ reverses them, it follows that $d\tilde\alpha^{(p)}$ reverses brackets.
The problem with Eric's argument has to do with the identification between $\operatorname{Lie}(\operatorname{Diff}(M))$ and $\mathfrak X(M)$. Essentially the same argument as above shows that the natural map $\operatorname{Lie}(\operatorname{Diff}(M)) \to \mathfrak X(M)$ is actually a Lie algebra anti-isomorphism (if we consider $\operatorname{Diff}(M)$ as an infinite-dimensional Lie group acting on $M$ on the left).
The problem with Victor Protsak's argument is that it doesn't actually show that $(X,0)$ (considered as a vector field on $G\times M$) is $\alpha$-related to $X'$ -- to show this, you'd have to prove that $(d\alpha)_{(g,m)}(X,0)=X'$ for every $g$ in the group, not just $g=1$.
I had to sort all this out recently because I'm adding a section on infinitesimal group actions in the second edition of my book Introduction to Smooth Manifolds. When I tried to prove the same formula the OP was trying to prove, I was surprised to find that it led to contradictions. Eventually I came up with the argument I sketched here, and then found at least one other book that confirms the result. I'm at home now and don't have access to my books, but if you'd like I'll find the reference and post it tomorrow.