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
for all n, 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.
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
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: where
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).
Here the multiplication on the right hand side induced by the product
On the left hand side, since
is simply connected,
stands for the universal enveloping algebra.
This abstract algebra-related article is a stub.