nLab Milnor-Moore theorem

Idea

The universal enveloping algebra of a Lie algebra naturally becomes a Hopf algebra in which all the elements of the original Lie algebra are primitive elements. The Milnor-Moore theorem states conditions under which, conversely, any Hopf algebra generated by primitive elements is the universal enveloping algebra of the Lie algebra structure on these elements.

References

The result is due to

• John Milnor, John Moore, On the structure of Hopf algebras, Annals of Math. 81 (1965), 211-264. (pdf)

Discussion for the special case of abelian restricted Lie algebras (with an eye towards its use in the May spectral sequence) is in

• Peter May, The cohomology of restricted Lie algebras and of Hopf algebras, Journal of Algebra 3, 123-146 (1966) (pdf)

• Peter May, Some remarks on the structure of Hopf algebras, Proceedings of the AMS, vol 23, No. 3 (1969) (pdf)