Hopf algebras up to homotopy

Abstract

Let ( A , d ) (A,d) denote a free r r -reduced differential graded R R -algebra, where R R is a commutative ring containing n − 1 {n^{ - 1}} for 1 ≤ n > p 1 \leq n > p . Suppose a “diagonal” ψ : A → A ⊗ A \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 ) (A,d) must equal U ( L , δ ) U(L,\delta ) for some free differential graded Lie algebra ( L , δ ) (L,\delta ) if A A is generated as an R R -algebra in dimensions below r p 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 r -connected CW complex X X of dimension ≤ r p \leq rp , that the Adams-Hilton model over R R is an enveloping algebra and that p th p\text {th} powers vanish in H ~ ∗ ( Ω X ; Z p ) {\tilde H^ * }(\Omega X;{{\mathbf {Z}}_p}) .