Milnor–Moore theorem explained
In algebra, the Milnor–Moore theorem, introduced by 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. Here we say
A is
connected if
is the field and
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:
U(\pi\ast(\OmegaX) ⊗ \Q)\congH\ast(\OmegaX;\Q),
where
denotes the
loop space of
X, compare with Theorem 21.5 from . This work may also be compared with that of . Here the multiplication on the right hand side induced by the product
\OmegaX x \OmegaX → \OmegaX
, and then by the Eilenberg-Zilber multiplication
C*(\OmegaX) x C*(\OmegaX) → C*(\OmegaX)
.
On the left hand side, since
is simply connected,
is a
-vector space; the notation
stands for the universal enveloping algebra.
References
- Milnor . John W. . John Milnor. Moore . John C. . On the structure of Hopf algebras . . 81 . 2 . 1965 . 211–264 . 10.2307/1970615 . 1970615. 0174052 .
- Web site: Lecture 3 on Hopf algebras. Spencer. Bloch. Spencer Bloch. 2014-07-18. https://web.archive.org/web/20100610114451/http://math.uchicago.edu/~mitya/bloch-hopf/hopf3.pdf. 2010-06-10. dead.
- Spencer Bloch, "Three Lectures on Hopf algebras and Milnor–Moore theorem". Notes by Mitya Boyarchenko.
- Book: Yves. Félix. Stephen. Halperin. Stephen Halperin. Jean-Claude. Thomas. Jean-Claude Thomas . Rational homotopy theory. Graduate Texts in Mathematics. 205. Springer-Verlag. New York . 2001. 1802847. 0-387-95068-0. 10.1007/978-1-4613-0105-9 . (Book description and contents at the Amazon web page)
External links
