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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Hopf algebras up to homotopy

Author: David J. Anick
Journal: J. Amer. Math. Soc. 2 (1989), 417-453
MSC: Primary 16A24; Secondary 55P15
MathSciNet review: 991015
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Abstract: 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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Article copyright: © Copyright 1989 American Mathematical Society