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$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$ If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $A$ in $\C$, with multiplication operations $$A^{\otimes n} (:= A\otimes_\C A\otimes_\C \cdots\otimes_\C A)\to A.$$

Similarly, if $\C$ is a braided monoidal category (resp., a ribbon category), one can define a notion of $E_2$ DG algebra $A$ (resp., framed $E_2$ DG algebra $A$) "twisted" by $\C$, consisting of operations $A^{\otimes n}\to A$ compatible with braiding. (Note: I actually don't know a reference for this, but it follows from standard "homotopy field theory" arguments involving the Ran space.)

In particular, if $\C$ is a braided monoidal (or ribbon) category coming from an associator on a Lie algebra $\g$ (with choice of Casimir), there is a whole category of "associator-twisted" $\g$-equivariant $E_2$ (resp., framed $E_2$) algebras.

My question is whether algebras of this type have been encountered before. They feel very CFT-ish, and so I'm particularly curious about physics and knot theory applications. In particular, the framed variant should gove some kind of derived 2D TQFT-style invariants.

Any references would be useful. Thanks!

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    $\begingroup$ Related reference: "Planar algebras in braided tensor categories" by myself, David Penneys, James Tener arxiv.org/abs/1607.06041 $\endgroup$ Apr 24 '21 at 23:03
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    $\begingroup$ @AndréHenriques Thank you! The notion of "ribbon braided operad" seems very close to what I am looking for. Is there a relationship between this "untwisted" notion of planar algebra and E2 algebras or (Koszul dually) Hopf algebras? $\endgroup$ Apr 24 '21 at 23:52
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    $\begingroup$ Curious - does one really need to invoke Ran spaces? I would have thought naively that for any operad $O$ in say topological spaces and an $O$-monoidal ($\infty$-)category $C$ we can discuss $O$-algebras in $C$? $\endgroup$ Apr 25 '21 at 1:52
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    $\begingroup$ These algebras go under the name "braided commutative algebras," or E_2 algebras in an E_2 monoidal category $\endgroup$ Apr 25 '21 at 1:52
  • $\begingroup$ @DavidBen-Zvi Good point! Yes, I think you're right: I think you can for example say that an O-algebra in C is an algebra of O in the (category, object) pair category over a given O-monoidal C $\endgroup$ Apr 25 '21 at 20:09
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I don't know specific references (the papers - in reverse chronological order - of Liang Kong, Hao Zheng, Ingo Runkel, Christoph Schweigert and Jurgen Fuchs is where I'd start), but the notion is certainly very natural in TFT, in at least two (closely related) ways:

  • if you think of your braided tensor category $C$ as defining a 4d TFT (i.e. as an object in the Morita 4-category thereof), then $E_2$ algebras $A$ in $C$ are a special case of boundary conditions in the TFT (left modules over $A$ form a monoidal category over $C$, or Morita morphism to/from the unit). As such they are actually quite rare in say reps of quantum groups generically, and have to do with coisotropic subalgebras of $\mathfrak g$.

  • if you think of $C$ as the value on the circle of a 3d TFT (eg as defined by a rational vertex algebra), you get $E_2$ algebras $A$ in $C$ as the value on the circle of boundary conditions for the TFT. Or you can look at interfaces of 3d TFTs, giving $E_2$-functors of $E_2$-categories of which I think your setup is a special case by passing to $E_2$-A-modules (maybe I'm getting turned around though). Or less generally - a concrete example of your setup is $C=B-E_2-mod$ and $B\to A$ an $E_2$-morphism - as such this appears all over CFT, when you have a homomorphism of algebras of observables (eg a map of rational vertex algebras).

From the TFT POV the difference between $E_2$ and framed $E_2$ is the difference between framed TFT and oriented TFT - the latter naturally produces framed $E_2$-algebras ( as usual the terminology is awful, for TFT definitely seems better to talk about "oriented 2-disc algebras" (framed $E_2$) and "framed 2-disc algebras" ($E_2$).

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    $\begingroup$ What David is referring to in its first point is also known as an algebra over the Swiss-Cheese operad, in that case a pair of a monoidal category $M$ and a braided functor from $C$ to the Drinfeld center of $M$. If $A$ is an $E_2$-algebra in $C$ then $M=A-mod_C$ should have such a structure. As such it has to do do with quantization of shifted Lagrangian/coisotropic structures. In the particular case you mention this should indeed arise as quantizations of Lagrangians in the 2-shted symplectic stack $BG$. See eg Safronov-Melani arxiv.org/abs/1704.03201 $\endgroup$
    – Adrien
    Apr 25 '21 at 9:03
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    $\begingroup$ Examples of those typically arise in Etingof-Kazhdan quantization, e.g. if $G$ is the drinfeld double of a (quasi-)Poisson algebraic group $A$, then $O(A)$ is a braided commutative algebra in the Drinfeld category of $G$. As David says it boils down to the fact that the Lie algebra of $A$ is coisotropic in the Lie algebra of $G$ which, as shown by Pavel Safronov, is equivalent to the fact that $BA \rightarrow BG$ is shifted coisotropic. $\endgroup$
    – Adrien
    Apr 25 '21 at 9:28

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