# reference request: What representations of the Lie algebra of a Lie group come from representations of the group itself?

I think this is a very classic math, but I can not find a complete answer in the literature.

Leave $$G$$ be a group of lies, $$mathfrak {g}$$ the lie algebra of $$mathfrak {g}$$. Suppose $$rho: mathfrak {g} to mathfrak {gl} (V)$$ It is an arbitrary representation of finite dimension of the Lie algebra. $$mathfrak {g}$$ in the real vector space $$V$$. I want to ask when there is a representation. $$rho_G: G a GL (V)$$ of the group Lie $$G$$, such that the tangent map of $$rho_G$$ to $$e$$ (the identity for $$G$$) is exactly $$rho$$.

I think when $$G$$ is simply connected, then each representation (finite-dimensional) of $$mathfrak {g}$$ It is induced by a representation of $$G$$ as a group of lies. But yes $$G$$ is simply supposed to be connected, or even more generally, what if $$G$$ Is it an arbitrary group of lies?