Fundamental vector field

If a group $G$ is acting over a manifold $M$, every vector $V\in \mathfrak{g}$ gives rise to a vector field $V^{\mathrm{♯}}$ in $M$ in the following way: for every $p\in M$ we consider the curve ${\alpha }_{pV}(t)=p\cdot e^{tV}$. And we take $V^{\mathrm{♯}}(p)=\frac{d}{dt}(p\cdot e^{tV}){|}_{t=0}$. It is called the fundamental vector field.

They reflects the Lie algebra action.

When $M$ is $G$ itself, the fundamental vector field generated by $V$ is the right invariant vector field $R_V$. It is the same as the left invariant vector field generated by $A{d}_{g^{-1}}V$ where $Ad$ is the adjoint representation. See also Maurer-Cartan form#MC form and left and right actions.