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Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf

I was wondering if anybody knows of a nice relationship between the categories of left modules over a Hopf Algebra and left modules over the underlying e1 structure of an $E_2$ algebra.

In my mind, one should be able to use the $E_2$ structure to turn left modules into bi-modules in a systematic way and then use that the category of bi-modules is monoidal.

We know that the category of left modules over a Hopf algebra is monoidal. How are these monoidal categories related?

Also, a big related question I have is: can we pick an $E_3$-algebra related to the Hopf/$E_2$-algebra such that the category of left modules is braided and gives knot invariants similar to the Jones polynomial and other braided category invariants (from tangles)?

(Note: if any clarification is needed in my question please let me know and I will edit it.)

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    $\begingroup$ I couldn't understand the question about $\mathbb{E}_3$. It is indeed true that the $\infty$-category of left modules over an $\mathbb{E}_3$-algebra is braided monoidal. For the Jones polynomial you're looking at the $\mathbb{E}_3$-algebra obtained by quantizing the $\mathbb{P}_3$-algebra $\mathrm{C}^\bullet(\mathfrak{g})$ (the $\mathbb{P}_3$-structure uses the Casimir element on $\mathfrak{g}$). $\endgroup$
    – Pavel Safronov
    Commented Oct 13, 2018 at 9:15
  • $\begingroup$ Could you tell me a little more about this? (Do you have a reference) I'm primarily interested in knot theory and trying to understand how one can recover quantum knot invariants by procedures similar to this. My advisor thinks the E3 algebra should be something related (quasi-isomorphic) to the E3 deformation complex of U(sl2) (or some other enveloping hopf algebra). Does this sound right? $\endgroup$
    – Matthew Levy
    Commented Oct 13, 2018 at 13:25
  • $\begingroup$ I believe this is currently being investigated by Costello--Francis--Gwilliam. The $\mathbb{P}_3$-algebra $\mathrm{C}^\bullet(\mathfrak{g})$ is Koszul dual to $\mathrm{U}(\mathfrak{g})$ and the $\mathbb{E}_3$-algebra is Koszul dual to the corresponding quantum group. $\endgroup$
    – Pavel Safronov
    Commented Oct 13, 2018 at 16:15

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Let $A$ be a brace algebra and $B$ the Koszul dual bialgebra. There is a natural adjunction $$ \Omega\colon \mathrm{CoMod}_B\rightleftarrows \mathrm{LMod}_A $$ where, for instance, the functor $\mathrm{LMod}_A\rightarrow\mathrm{CoMod}_B$ sends $M\mapsto M\otimes B$ equipped with some standard differential.

For $\mathrm{LMod}_A$ the natural notion of weak equivalence is that of quasi-isomorphism and in $\mathrm{CoMod}_B$ a map $C_1\rightarrow C_2$ is a weak equivalence if $\Omega C_1\rightarrow \Omega C_2$ is a quasi-isomorphism. The localization of $\mathrm{LMod}_A$ with respect to quasi-isomorphisms is the derived category of $A$-modules and the localization of $\mathrm{CoMod}_B$ with respect to $\Omega$-quasi-isomorphisms is the coderived category of $B$-comodules. The above adjunction induces an adjoint equivalence between the corresponding localizations (see https://arxiv.org/abs/0905.2621 for details on all of these constructions).

$\mathrm{CoMod}_B$ indeed has a natural monoidal structure. However, $\mathrm{LMod}_A$ does not have any natural monoidal structure (there is no strict way to turn left modules over a brace algebra into right modules). There is, however, a monoidal structure on the derived category of $A$-modules. It is obtained using the Dunn--Lurie additivity equivalence $\mathcal{A}\mathrm{lg}_{\mathbb{E}_2}\cong \mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg})$.

Brace bar-cobar duality gives an adjunction $\mathrm{Alg}(\mathrm{CoAlg})\rightleftarrows \mathrm{Alg}_{\mathrm{Brace}}$. After inverting corresponding weak equivalences you obtain another equivalence of $\infty$-categories $\mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg})\cong \mathcal{A}\mathrm{lg}_{\mathbb{E}_2}$. As far as I know, nobody has compared Dunn--Lurie additivity for $\mathbb{E}_2$ with brace bar-cobar duality. Assuming such a comparison, the monoidal structure on the coderived category of $B$-comodules will indeed coincide with the monoidal structure on the derived category of $A$-modules.

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  • $\begingroup$ Just to clarify: the category of left modules over A (the brace algebra) does not strictly have a monoidal structure, but it does “up to homotopy”? We can take two left A-modules M,N and put a differential on M \otimes Bar(A) \otimes N that takes into account the quasi-isomorphism from A to A^op. I have worked out some of this. $\endgroup$
    – Matthew Levy
    Commented Nov 11, 2018 at 9:30
  • $\begingroup$ I don't have much useful to add to the discussion, however, I do strongly believe that brace bar-cobar duality is a manifestation of Dunn additivity. I would love to have that made explicit eventually. $\endgroup$
    – user35570
    Commented Feb 26, 2020 at 17:35

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