= Milnor–Moore theorem
{c}
{title2=$U(\pi_*(\Omega Y)\otimes\mathbb Q)\cong H_*(\Omega Y;\mathbb Q)$}
For <simply connected> $Y$, rational <Hurewicz homomorphism> identifies $\pi_*(\Omega Y)\otimes\mathbb Q$ with the <primitive elements of a Hopf algebra> of $H_*(\Omega Y;\mathbb Q)$ and induces an isomorphism $U(\pi_*(\Omega Y)\otimes\mathbb Q)\cong H_*(\Omega Y;\mathbb Q)$ of graded Hopf algebras. The bracket on <homotopy> is the <Samelson product>. The underlying algebra theorem says that a connected graded cocommutative <Hopf algebra> over <characteristic zero> is the <universal enveloping algebra> of its primitive <Lie algebra>. These statements let a free <tensor algebra> of loop-space <homology> detect a <free graded Lie algebra> of <homotopy>.
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