Milnor–Moore theorem

In algebra, the Milnor–Moore theorem, introduced by John W. Milnor and John C. Moore (1965) 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 ${\displaystyle \dim A_{n}<\infty }$ for all n, the natural Hopf algebra homomorphism

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

from the universal enveloping algebra of the graded Lie algebra ${\displaystyle P(A)}$ of primitive elements of A to A is an isomorphism. Here we say A is connected if ${\displaystyle A_{0}}$ is the field and ${\displaystyle A_{n}=0}$ 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 ${\displaystyle xy-(-1)^{|x||y|}yx-[x,y]}$.

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:

${\displaystyle U(\pi _{\ast }(\Omega X)\otimes \mathbb {Q} )\cong H_{\ast }(\Omega X;\mathbb {Q} ),}$

where ${\displaystyle \Omega X}$ denotes the loop space of X, compare with Theorem 21.5 from Félix, Halperin & Thomas (2001). This work may also be compared with that of (Halpern 1958a, 1958b). Here the multiplication on the right hand side induced by the product ${\displaystyle \Omega X\times \Omega X\rightarrow \Omega X}$, and then by the Eilenberg-Zilber multiplication ${\displaystyle C_{*}(\Omega X)\times C_{*}(\Omega X)\rightarrow C_{*}(\Omega X)}$.

On the left hand side, since ${\displaystyle X}$ is simply connected, ${\displaystyle \pi _{\ast }(\Omega X)\otimes \mathbb {Q} }$ is a ${\displaystyle \mathbb {Q} }$-vector space; the notation ${\displaystyle U(V)}$ stands for the universal enveloping algebra.

References

• Milnor, John W.; Moore, John C. (1965). "On the structure of Hopf algebras". Annals of Mathematics. 81 (2): 211–264. doi:10.2307/1970615. JSTOR 1970615. MR 0174052.
• Bloch, Spencer. "Lecture 3 on Hopf algebras" (PDF). Archived from the original (PDF) on 2010-06-10. Retrieved 2014-07-18.
• Spencer Bloch, "Three Lectures on Hopf algebras and Milnor–Moore theorem". Notes by Mitya Boyarchenko.
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2001). Rational homotopy theory. Graduate Texts in Mathematics. Vol. 205. New York: Springer-Verlag. doi:10.1007/978-1-4613-0105-9. ISBN 0-387-95068-0. MR 1802847. (Book description and contents at the Amazon web page)
• Halpern, Edward (1958a), "Twisted polynomial hyperalgebras", Memoirs of the American Mathematical Society, 29: 61 pp, MR 0104225
• Halpern, Edward (1958b), "On the structure of hyperalgebras. Class 1 Hopf algebras", Portugaliae Mathematica, 17 (4): 127–147, MR 0111023
• May, J. Peter (1969). "Some remarks on the structure of Hopf algebras" (PDF). Proceedings of the American Mathematical Society. 23 (3): 708–713. doi:10.2307/2036615. JSTOR 2036615. MR 0246938.(Broken link)

External links

• Akhil Mathew (23 June 2012). "Formal Lie theory in characteristic zero".