Left-Invariant Vector Field

Let $G$ be a Lie group . A vector field $X$ on $G$ is left-invariant if for every $g\in G$,

where $L_g$ is left translation and $(L_g)_*$ denotes the pushforward on vector fields.

Equivalently, for all $g,h\in G$,

Determined by the value at the identity

A left-invariant vector field is completely determined by its value $X_e\in T_eG$. Conversely, every $v\in T_eG$ defines a unique left-invariant vector field by

Thus the space of left-invariant vector fields is naturally identified with the Lie algebra $\mathfrak{g}=T_eG$.

Lie bracket compatibility

If $\widetilde X,\widetilde Y$ are left-invariant, then so is $[\widetilde X,\widetilde Y]$, and

defines the Lie bracket on $\mathfrak{g}$.

Flows of left-invariant vector fields yield one-parameter subgroups and are closely related to the exponential map .