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 [math]\displaystyle{ \dim A_n \lt \infty }[/math] for all n, the natural Hopf algebra homomorphism
- [math]\displaystyle{ U(P(A)) \to A }[/math]
from the universal enveloping algebra of the graded Lie algebra [math]\displaystyle{ P(A) }[/math] of primitive elements of A to A is an isomorphism. Here we say A is connected if [math]\displaystyle{ A_0 }[/math] is the field and [math]\displaystyle{ A_n = 0 }[/math] 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 [math]\displaystyle{ xy- (-1)^{|x||y|}yx - [x,y] }[/math].
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:
- [math]\displaystyle{ U(\pi_{\ast}(\Omega X) \otimes \Q) \cong H_{\ast}(\Omega X;\Q), }[/math]
where [math]\displaystyle{ \Omega X }[/math] denotes the loop space of X, compare with Theorem 21.5 from (Félix Halperin). This work may also be compared with that of (Halpern 1958a, 1958b).
References
- Bloch, Spencer. "Lecture 3 on Hopf algebras". http://www.math.uchicago.edu/~mitya/bloch-hopf/hopf3.pdf.
- Félix, Yves; Halperin, Steve; Thomas, Jean-Claude (2001). Rational homotopy theory. Graduate Texts in Mathematics. 205. New York: Springer-Verlag. doi:10.1007/978-1-4613-0105-9. ISBN 0-387-95068-0.
- Halpern, Edward (1958a), "Twisted polynomial hyperalgebras", Memoirs of the American Mathematical Society 29: 61 pp
- Halpern, Edward (1958b), "On the structure of hyperalgebras. Class 1 Hopf algebras", Portugaliae Mathematica 17 (4): 127–147
- May, J. Peter (1969). "Some remarks on the structure of Hopf algebras". Proceedings of the American Mathematical Society 23 (3): 708–713. doi:10.2307/2036615. http://www.math.uchicago.edu/~may/PAPERS/9.pdf.
- Milnor, John W.; Moore, John C. (1965). "On the structure of Hopf algebras". Annals of Mathematics 81 (2): 211–264. doi:10.2307/1970615. https://doi.org/10.2307/1970615.
External links
- Akhil Mathew (23 June 2012). "Formal Lie theory in characteristic zero". http://amathew.wordpress.com/2012/06/23/formal-lie-theory-in-characteristic-zero/.
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