# Lie Groups and Lie Algebras

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.

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... if they have the same dimension. –  Torsten Schoeneberg Jan 20 '14 at 11:48

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 $T^k\times\mathbb{R}^{n-k}$.