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 $\dim A_n < \infty$, the natural Hopf algebra homomorphism

$U(P(A)) \to A$

from the universal enveloping algebra of the "graded" Lie algebra $P(A)$ 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].)

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

External links

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