Remote Access Journal of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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})$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 16A24, 55P15

Retrieve articles in all journals with MSC: 16A24, 55P15

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society