What is the exact relationship between Lie groups and Lie algebras? I know it's not bijective because all commutative Lie groups have isomorphic Lie algebras.



Lie Groups and Lie Algebras
Up to isomorphism, there is one *simply connected* Lie group for every Lie algebra. Indeed, there is also a homomorphism of simply connected Lie groups for every homomorphism of the corresponding Lie algebras so one gets an equivalence of categories this way.


This pans out nicely in yr commutative example: the simply connected abelian groups are just the Lie algebras under addition. All other abelian Lie groups have a non-simply connected torus component and look like $T^k\times\mathbb{R}^{n-k}$.


To get the remaining Lie groups into the picture, recall that the universal cover of any Lie group is a simply connected Lie group.


