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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Hopf algebras up to homotopy
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by David J. Anick PDF
J. Amer. Math. Soc. 2 (1989), 417-453 Request permission

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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 2 (1989), 417-453
  • MSC: Primary 16A24; Secondary 55P15
  • DOI: https://doi.org/10.1090/S0894-0347-1989-0991015-7
  • MathSciNet review: 991015