I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$.
How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$? Thank you very much.
Let $G$ be a Lie group acting on $M$. Then there's a Lie algebra homomorphism $\sigma: \mathfrak g \to \mathfrak X(M)$ by sending $A \in \mathfrak g$ to the vector field that generates the flow given by $\varphi_t(x) = \exp(tA)x$. I assume this is the way of passing from $\mathfrak g$ to $\mathfrak X(M)$ the authors refer to (without looking at the paper).