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I am reading Quillen's paper on rational homotopy theory.

In appendix A of this paper Quillen defines a notion he calls complete Hopf algebras. These are certain cocommutative bialgebra structures (where the coalgebra structure is with respect to a completed tensor product) which are complete as algebras (with respect to a certain filtrations).

Question: Do complete Hopf algebras have an antipode (i.e. a map $S : H \to H$ satisfying the usual diagrammatic axiom)?

In more detail, here is the definition Quillen makes:

A complete Hopf algebra $A$ is a complete augemented algebra A endowed with a diagonal $\Delta : A \to A\widehat\otimes A$ which is a map of complete augmented algebras and which is coassociative, cocommutative and has the augmentation map $A \to k$ as a counit.

Here $k$ is a field of characteristic $0$ (or just $k = \mathbb{Q}$). A complete augmented algebra is an augmented algebra with an algebra filtration $... \subseteq F_2 \subset F_1 \subset F_0 = A$ such that $F_1A$ is the augmentation ideal, $grA$ is generated (as an algebra) by $gr_1A = F_1/F_2$ and such that $A$ is complete with respect to the filtration.

Maps of complete augmented algebras are maps of augmented algebras $f : A \to A'$ sending the $n$-th part of the filtration to the corresponding $n$-th part of the filtration, i.e. $f(F_n) \subset F'_n$

I would also appreciate good references for places to read more about complete Hopf algebras.

Edit: I believe the following might work. For an augmented algebra $A$ let us denote the augmentation ideal by $\overline A$. The coproduct $\Delta$ induces a reduced coproduct $\overline \Delta : \overline A \to \overline A \widehat \otimes \overline A$ by $\overline \Delta(x) = \Delta(x) - x\widehat\otimes 1 - 1\widehat \otimes x$. If an (ordinary) bialgebra $H$ is conilpotent then the formula \begin{eqnarray} S(x) = -x + \sum_{n \geq 1} (-1)^{n+1} \mu^n \circ \overline{\Delta}^{n-1}(x) \end{eqnarray} defines an antipode (here $\mu$ is the multiplication map). But the same formula makes sense if $H$ is a complete Hopf algebra (in the above sense) since we assume that $H$ is complete with respect to a filtration $\{ F_n \}_n$ where $\overline H^n \subseteq F_n$ for all $n$. So the sum in the definition converges.

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  • $\begingroup$ I doubt that you want $\mu^n$ and $\overline{\Delta}^{n-1}$ in the same addend; either of the exponents should be changed. That said, your proof looks good to me, though I haven't checked the topological details (one thing that worries me about these completions is that I have no idea how well they interact with tensor products). $\endgroup$ – darij grinberg Aug 12 '15 at 20:38

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