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In algebra, the Milnor–Moore theorem, introduced by John W. Milnor and John C. Moore (1965), states: given a connected, graded, cocommutative Hopf algebra A over a field of characteristic zero with \( \dim A_n < \infty \)for all n, the natural Hopf algebra homomorphism

\( U(P(A)) \to A \)

from the universal enveloping algebra of the graded Lie algebra P(A) of primitive elements of A to A is an isomorphism. (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 \( {\displaystyle xy-yx-(-1)^{|x||y|}[x,y]} \).)

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 \( \Omega X \) 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 1958).


Bloch, Spencer. "Lecture 3 on Hopf algebras" (PDF). Archived from the original (PDF) on 2010-06-10. Retrieved 2014-07-18.
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. MR 1802847.
Halpern, Edward (1958), "Twisted polynomial hyperalgebras", Memoirs of the American Mathematical Society, 29: 61 pp, MR 0104225
Halpern, Edward (1958), "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.
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.

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