Help to understand the definition of a left-invariant vector field

Question

I have some trouble to understand the definition of a left-invariant vector field in the context of a lie group. We defined it as follow:

A vector field $X$ on a lie group $G$ is a called left-invariant, if for every $x \in G$ the following diagram is commutative:

We defined only the map $l_X: G \to G, \, g \mapsto xg$, to be the left multiplication with $x$. Now what is than the map $(l_X)_*$? I am confused about the way this should be defined.... Maybe a help would be to try to do an exercice about this notion. In the script there is one given as

X is a left invariant vector fied $\iff$ $(l_X)_*(X)=X \; \forall x \in G$

This should be easy to show, but bc I don't understand the definition I am not even able to start the proof of this statement. Many thanks for some help!

Answer 1

A left-invariant vector field is essentially a translation-invariant vector field. Let us look at such on $\mathbb{R}^3.$

A translation on $\mathbb{R}^3$ is a map $\tau_a:\mathbb{R}^3\to\mathbb{R}^3,\ p\mapsto p+a,$ where $a\in\mathbb{R}^3.$

A translation of a function $f\in C^\infty(\mathbb{R}^3)$ is defined by $(\tau_a f)(p) = f(\tau_{-a} p) = f(\tau_a^{-1} p)=(f\circ\tau_a^{-1})(p),$ i.e. $\tau_a f := f\circ\tau_a^{-1}.$

A translation of a vector field $X\in T\mathbb{R}^3$ is defined by $(\tau_a X)f = X(\tau_{-a}f) = X(\tau_a^{-1}f)=X(f\circ\tau_a).$

A vector field is translation invariant if $\tau_a X=X$ for all $a\in\mathbb{R}^3,$ i.e. if $X(f\circ\tau_a) = Xf$ for all $a\in\mathbb{R}^3$ and all $f\in C^\infty(\mathbb{R}^3).$

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We can easily generalize the above definitions to any manifold $M$ and a group $G$ acting on it (notation: $g\triangleright p$ for $g\in G,$ $p\in M$):

For $f\in C^\infty(M)$ we set define the action of $g$ on $f$ by $(g \triangleright f)(p):=f(g^{-1}\triangleright p).$

For a vector field $X\in TM$ we define $(g \triangleright X)f:=X(g^{-1} \triangleright f).$

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In the case of a Lie groups $G$ we have $M=G$ and two natural ways to define $g\triangleright p$:

1. left action, i.e. $g\triangleright p = gp$,
2. right action, i.e. $g\triangleright p = pg$.

You can probably guess which one is used in the definition of left-invariant vector fields.

The notation $l_x$ in your post corresponds to $x\triangleright\cdot$ and $((l_x)_*X)(f) = X(x^{-1}\triangleright f) = X(l_x^{-1}f)=(X\circ l_x^{-1})f,$ i.e. $(l_x)_*X = X\circ l_x^{-1}.$