Geodesic
Hopf algebras up to homotopy
Journal of the American Mathematical Society, Tome 02 (1989) no. 3, pp. 417-453

Voir la notice de l'article provenant de la source American Mathematical Society

Let $(A,d)$ denote a free $r$-reduced differential graded $R$-algebra, where $R$ is a commutative ring containing ${n^{ - 1}}$ for $1 \leq n p$. Suppose a “diagonal” $\psi :A \to A \otimes A$ exists which satisfies the Hopf algebra axioms, including cocommutativity and coassociativity, up to homotopy. We show that $(A,d)$ must equal $U(L,\delta )$ for some free differential graded Lie algebra $(L,\delta )$ if $A$ is generated as an $R$-algebra in dimensions below $rp$. As a consequence, the rational singular chain complex on a topological monoid is seen to be the enveloping algebra of a Lie algebra. We also deduce, for an $r$-connected CW complex $X$ of dimension $\leq rp$, that the Adams-Hilton model over $R$ is an enveloping algebra and that $p\text {th}$ powers vanish in ${\tilde H^ * }(\Omega X;{{\mathbf {Z}}_p})$. 
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Anick, David J. Hopf algebras up to homotopy. Journal of the American Mathematical Society, Tome 02 (1989) no. 3, pp. 417-453. doi: 10.1090/S0894-0347-1989-0991015-7

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