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Journal of the American Mathematical Society
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$th powers vanish in $ {\tilde H^ * }(\Omega X;{{\mathbf{Z}}_p})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0894-0347-1989-0991015-7
PII: S 0894-0347(1989)0991015-7
Article copyright: © Copyright 1989 American Mathematical Society