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. ${\displaystyle H_{*}(BGL(R);k)}$

literature

• John Milnor , John Moore : On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965) pp. 211-264, online .

Individual evidence

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