Let be a field of characteristic zero. The intuition is that in this case, a Lie algebra is the same data as a “germ” of a Lie group, or of an algebraic group. This is made precise in the following:
Theorem 1 There is an equivalence of categories between:
- Cocommutative Hopf algebras over which are generated by a finite number of primitive elements.
- Finite-dimensional Lie algebras.
- Infinitesimal formal group schemes over (with finite-dimensional tangent space), i.e. those which are thickenings of one point.
- Formal group laws (in many variables).