Let $ \mathcal C $ be a monoidal category. Then $ \mathcal C $ is both a left and right module category over itself. Moreover, the Drinfeld centre of $ \mathcal C $ can be defined as the category of functors from $ \mathcal C $ to itself which commute with these module category structures.
$$
Z(\mathcal C) = Fun\_{\mathcal C | \mathcal C}(\mathcal C, \mathcal C)
$$


Suppose that $\mathcal C = H\text{-mod}$ where $H$ is a Hopf algebra. Then a well-known result identifies $ Z(\mathcal C) = D(H)\text{-mod}$ where $D(H) $ is the Drinfeld double of $ H$.


I am interested in the generalization of this result to the setting of more general module categories. Let $ \mathcal C $ be a monoidal category and let $ \mathcal M $ be a $ \mathcal C$-module category. Then we can consider the category of functors from $ \mathcal M $ to itself, compatible with the module structure:
$$
\mathcal D = Fun\_{\mathcal C}(\mathcal M, \mathcal M)
$$


Assume that $ \mathcal C = H\text{-mod}$ and $ \mathcal M = A\text{-mod} $ where $H$ is a Hopf algebra, $ A$ is an algebra and $ A $ is also an $ H$-comodule (this comodule structure gives rise to the action of $ \mathcal C $ on $ \mathcal M $).


**Question: Under this setup, can we realize $ \mathcal D $ as the module category of some algebra constructed from $ H $ and $ A $?**


My question is motivated from the theory of lattice models in condensed matter physics. In the paper [Models for gapped boundaries and domain walls](http://de.arxiv.org/pdf/1104.5047.pdf) by Kitaev and Kong, the authors consider a Levin-Wen model with input $ \mathcal C, \mathcal M $ as above. In this model, the category of functors $ \mathcal D $ is the category of boundary excitations. Moreover, in section 4 of this paper, the author construct an algebra whose module category is claimed to be equivalent to $ \mathcal D $. (Actually they just claim that simple objects correspond.) So I was wondering if these ideas had been developed in the mathematics literature.



Generalization of Drinfeld double to comodule algebras
Such an algebra exists. As far as I am aware, the algebra was first described in chapter 6 of [The blob complex](https://arxiv.org/abs/1009.5025) by Morrison and Walker. In this paper, the algebra is construct from a diagrammatic calculus for the module and tensor category rather than from $H$ and $A$ directly.


Since then, these algebras have appeared under many names in condensed matter physics, for example the "dube algebra" in [Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation](https://arxiv.org/abs/1711.07982). From the physical perspective, the representations of these algebras (or sphere modules as they are called my Morrison and Walter) describe codimension 2 defects in the corresponding topological phase of matter.


