$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$
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!



Uses for (Framed) E2 algebras twisted by braided monoidal structure
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" ($E\_2$).


