Hopf algebras up to homotopy

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 ${\tilde H^ * }(\Omega X;{{\mathbf {Z}}_p})$.