# Milnor–Moore theorem

In algebra, the **Milnor–Moore theorem**, introduced in (Milnor–Moore 1965), states: given a connected graded cocommutative Hopf algebra *A* over a field of characteristic zero with , the natural Hopf algebra homomorphism

from the universal enveloping algebra of the "graded" Lie algebra of primitive elements of *A* to *A* is an isomorphism. (The universal enveloping algebra of a graded Lie algebra *L* is the quotient of the tensor algebra of *L* by the two-sided ideal generated by elements *xy-yx* - (-1)^{|x||y|}[*x*, *y*].)

This work may also be compared with that by E. Halpern [1958] listed below.

## References

- Lecture 3 of Hopf algebras by Spencer Bloch
- J. May, "Some remarks on the structure of Hopf algebras"
- J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2), 81 : 2 (1965) pp. 211–264
- E. Halpern, "Twisted Polynomial Hyperalgebras", Memoirs of the American Mathematical Society 1958; 61 pp; - See more at:

- E. Halpern, "On the structure of hyperalgebras. Class 1 Hopf algebras", Portugaliae mathematica 17(4), 127-147 (1958).

## External links

- Formal Lie theory in characteristic zero, a blog post by Akhil Mathew

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