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

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 $$A^{otimes n} (:= Aotimes_C Aotimes_C cdotsotimes_C A)to A.$$

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!