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 .
Let it be the graduated Lie algebra of the primitive elements in and the universal enveloping algebra of .
Then is the natural Hopf algebra homomorphism
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
injective and its image is of the primitive elements generated.
K theory
A special case arises when applied to the algebraic K-theory of a ring : the Hurewicz homomorphism
in the group of homology of the general linear group is injective and its image is of the primitive elements generated.
literature
- John Milnor , John Moore : On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965) pp. 211-264, online .