Milnor–Moore theorem

In algebra, the Milnor–Moore theorem, introduced in (Milnor–Moore 1965), states: given a connected graded cocommutative Hopf algebra A over a field of characteristic zero with $\dim A_n < \infty$ , 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 elements xy-yx - (-1)^|x||y|[x, y].)

In topology, the term usually refers to the corollary of the result explained above that for a simply connected space X, it holds true that $U(\pi _{\ast }(\Omega X)\otimes \mathbb {Q} )\cong H_{\ast }(\Omega X;\mathbb {Q} ),$ compare,^[1] theorem 21.5

This work may also be compared with that by E. Halpern [1958] listed below.

References

↑ Y. Felix, S. Halperin, J.-C. Thomas, "Rational Homotopy theory", Springer 2001.

Lecture 3 of Hopf algebras by Spencer Bloch
Y. Felix, S. Halperin, J.-C. Thomas, "Rational Homotopy theory", Springer 2001.
J. May, "Some remarks on the structure of Hopf algebras"
J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2), 81 : 2 (1965) pp. 211–264
E. Halpern, "Twisted Polynomial Hyperalgebras", Memoirs of the American Mathematical Society 1958; 61 pp; - See more at:
E. Halpern, "On the structure of hyperalgebras. Class 1 Hopf algebras", Portugaliae mathematica 17(4), 127-147 (1958).

External links

Formal Lie theory in characteristic zero, a blog post by Akhil Mathew

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[1] Y. Felix, S. Halperin, J.-C. Thomas, "Rational Homotopy theory", Springer 2001.