# Milnor-Moore's theorem

The Milnor-Moore theorem , named after John Milnor and John Moore , is a theorem from the mathematical field of the theory of Hopf algebras . Under certain conditions he establishes a connection between such a Hopf algebra and the Lie algebra of the primitive elements contained in it.

## formulation

Let it be a graduated co-commutative Hopf algebra over a field of the characteristic and it applies and for all . ${\ displaystyle A = \ bigoplus _ {n \ in N} A_ {n}}$ ${\ displaystyle k}$ ${\ displaystyle \ operatorname {char} (k) = 0}$${\ displaystyle A_ {0} \ cong k}$${\ displaystyle \ dim (A_ {n}) <\ infty}$${\ displaystyle n}$

Let it be the graduated Lie algebra of the primitive elements in and the universal enveloping algebra of . ${\ displaystyle P (A)}$${\ displaystyle A}$${\ displaystyle U (P (A))}$${\ displaystyle P (A)}$

Then is the natural Hopf algebra homomorphism

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

an isomorphism .

## H-spaces

The following application is often referred to as the Milnor-Moore Theorem.

Let it be a path-connected homotopy-associative H-space . Then there is the Hurewicz homomorphism${\ displaystyle X}$

${\ displaystyle \ pi _ {*} (X) \ otimes k \ to H _ {*} (X; k)}$

injective and its image is of the primitive elements generated. ${\ displaystyle H _ {*} (X; k)}$

## K theory

A special case arises when applied to the algebraic K-theory of a ring : the Hurewicz homomorphism ${\ displaystyle R}$

${\ displaystyle K _ {*} (R) \ otimes k \ to H _ {*} (BGL (R); k)}$

in the group of homology of the general linear group is injective and its image is of the primitive elements generated. ${\ displaystyle H _ {*} (BGL (R); k)}$

## Individual evidence

1. Milnor-Moore, Theorem 5.18
2. Milnor-Moore, Appendix