Left-Invariant Vector Field

Let $G$ be a Lie group . Write $L_g:G\to G$ for left translation by $g$.

Definition (left invariance).
A smooth vector field $X$ on $G$ is left-invariant if for all $g,h\in G$,

where $(dL_g)_h$ is the differential (pushforward) of $L_g$ at $h$. Equivalently, $(L_g)_*X=X$ for every $g\in G$.

Proposition (determined by the value at the identity).
Let $e$ be the identity of $G$. The assignment

defined by

is a vector space isomorphism from $T_eG$ onto the space of left-invariant vector fields on $G$. Under the identification $T_eG=\mathfrak g$ from the Lie algebra of a Lie group , every element of $\mathfrak g$ corresponds to a unique left-invariant vector field.

Bracket compatibility.
The set of left-invariant vector fields is closed under the Lie bracket of vector fields. Transporting this bracket to $T_eG$ via the isomorphism above recovers the standard Lie algebra bracket on $\mathfrak g$.

Left-invariant vector fields also connect directly to the exponential map : for $v\in\mathfrak g$, the integral curve through $e$ of $X^L_v$ is the one-parameter subgroup $t\mapsto \exp_G(tv)$.

Examples

1. $(\mathbb R^n,+)$: constant vector fields.
   For the additive Lie group, left translations are $L_a(x)=a+x$ and $(dL_a)_x=\mathrm{id}$. Hence left-invariant vector fields are exactly constant-coefficient fields:

   $$X(x)=v \quad \text{for a fixed } v\in \mathbb R^n.$$

2. Matrix groups: $X(g)=gA$.
   Let $G\subseteq GL(n,\mathbb R)$ be a matrix Lie group and let $A\in\mathfrak g\subseteq M_n(\mathbb R)$. The associated left-invariant vector field is

   $$X(g)=gA,$$

   since $(dL_g)_e(A)=gA$ under the usual identification of tangent vectors with matrices.

3. Circle group $S^1$: rotation generator.
   Viewing $S^1\subset\mathbb C$, a left-invariant vector field is

   $$X(z)=iz,$$

   which is tangent to the circle and corresponds to the unit tangent vector $i\in T_1S^1$.