Milnor–Moore theorem

Template:Short description In algebra, the Milnor–Moore theorem, introduced by Template:Harvs 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 $\dim A_{n} < \infty$ for all n, 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. Here we say A is connected if $A_{0}$ is the field and $A_{n} = 0$ 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 $x y - (- 1)^{| x | | y |} y x - [x, y]$ .

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:

U (π_{*} (Ω X) \otimes ℚ) ≅ H_{*} (Ω X; ℚ),

where $Ω X$ denotes the loop space of X, compare with Theorem 21.5 from Template:Harvtxt. This work may also be compared with that of Template:Harvs. Here the multiplication on the right hand side induced by the product $Ω X \times Ω X \to Ω X$ , and then by the Eilenberg-Zilber multiplication $C_{*} (Ω X) \times C_{*} (Ω X) \to C_{*} (Ω X)$ .

On the left hand side, since $X$ is simply connected, $π_{*} (Ω X) \otimes ℚ$ is a $ℚ$ -vector space; the notation $U (V)$ stands for the universal enveloping algebra.