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?