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 over algebraically closed field of characteristic zero is due to

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

Discussion on the generalizations on the ground field is in

• MO question

• Daniel Quillen, Rational homotopy theory, Annals of Mathematics 90, No. 2 (1969) 205–295

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)

For dendriform algebras

• María Ronco, Eulerian idempotents and Milnor–Moore theorem for certain non-cocommutative Hopf algebras_, J. Alg. 254 (2002) 152–172

A homotopification:

• Imma Gálvez-Carrillo, María Ronco, Andy Tonks: On differential Hopf algebras and $B_\inf$ algebras [arXiv:2409.06632]

category: algebra