Milnor–Moore theorem

In algebra, the Milnor–Moore theorem, introduced by John W. Milnor and John C. Moore (1965) classifies an important class of Hopf algebras, of the sort that often show up as cohomology rings in algebraic topology.

The theorem states: given a connected, graded, cocommutative Hopf algebra A over a field of characteristic zero with ${\displaystyle \dim A_{n}<\infty }$ for all n, the natural Hopf algebra homomorphism

${\displaystyle U(P(A))\to A}$

from the universal enveloping algebra of the graded Lie algebra ${\displaystyle P(A)}$ of primitive elements of A to A is an isomorphism. Here we say A is connected if ${\displaystyle A_{0}}$ is the field and ${\displaystyle A_{n}=0}$ for negative n. 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 all elements of the form ${\displaystyle xy-(-1)^{|x||y|}yx-[x,y]}$.

In algebraic topology, the term usually refers to the corollary of the aforementioned result, that for a pointed, simply connected space X, the following isomorphism holds:

${\displaystyle U(\pi _{\ast }(\Omega X)\otimes \mathbb {Q} )\cong H_{\ast }(\Omega X;\mathbb {Q} ),}$

where ${\displaystyle \Omega X}$ denotes the loop space of X, compare with Theorem 21.5 from (Félix, Halperin & Thomas 2001). This work may also be compared with that of (Halpern 1958a, 1958b).