Milnor–Moore theorem

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 ${\displaystyle \dim A_{n}<\infty }$ 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. (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 ${\displaystyle \Omega X}$ 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 1958) harv error: multiple targets (2×): CITEREFHalpern1958 (help).

References

• 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: 708–713. doi:10.2307/2036615. 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.

External links

• Akhil Mathew. "Formal Lie theory in characteristic zero".