# Generalization of Drinfeld double to comodule algebras

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 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.

Such an algebra exists. As far as I am aware, the algebra was first described in chapter 6 of The blob complex 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. 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.

• I'm not sure whether to be more bothered that someone decided "dube" was a word, or that without needing to read the paper I knew that it must be a shortening of "defect tube." – Noah Snyder Sep 12 '18 at 22:44
• I once had some coauthors who advocated for "fube algebra", meaning fermionic version of the tube algebra. "Fube" did not make it into the final version of that paper. – Kevin Walker Sep 13 '18 at 19:21

I think Davydov's papers Centre of an algebra and Full centre of an H-module algebra might be what you are looking for.

• These paper looks interesting, but don't seem relevant for my question (unless I am missing something). – Joel Kamnitzer Sep 15 '18 at 1:19
• @JoelKamnitzer If I understood your question, it was "Given an algebra object $A$ in a nice monoidal category $\mathcal C$, how can I understand the the category of $A$-$A$ bimodules? In particular, can I understand it as the category of modules for some algebra $A'$ in some monoidal category $\mathcal C'$?" I think that you can take $\mathcal C' = \mathcal Z(\mathcal C)$, and, under some conditions on $A$, I think you can take $A' = Z(A)$. – Theo Johnson-Freyd Sep 16 '18 at 3:36

Section 1 (in particular Prop 1.23) of On module categories over finite-dimensional Hopf algebras by Andruskiewitsch-Mombelli come close to an answer to your question: namely, they show that if $H$ is a finite dimensional Hopf algebra, and $K,L$ f.d. comodules algebra algebra over $H$, then $Rep\ H$-linear right exact functor from $K-mod$ to $L-mod$ are $K-L$-bmodules in $H$-comodules.

They show this directly, but there is a general explanation. Note that your question has to be special to Hopf algebra (as opposed to general tensor categories) because the relation between $Rep\ H$ and $K-mod$ is somehow "external". One point of view I find illuminating, and hopefully might be useful to you if you want variant of this result, is the following: a comodule algebra $K$ is an $H^*$-module algebra hence becomes an algebra internal to the tensor category $C=Rep\ H^*$, so that you can talk about the category $K-mod_C$ of equivariant/internal $K$-modules. But it turns out $C$ and $D=Rep\ H$ are related by some sort of categorical Koszul duality, and this is where Hopf-ness is used crucially, because $C$ and $D$ are augmented by their fiber functor, ie Vect is a module over those.

Long story short, the assignment $$M \longmapsto Fun_C(Vect,M)$$ gives a 2-functorial equivalence between (appropriate adjectives) $C$-module categories and $D$-module categories. Now the above result follows from:

• this equivalence maps $K-mod_C$, to just $K-mod$ as a $D$-module category, hence to answer your question we might as well compute $C$-module endofunctors of $K-mod_C$
• but now we are in an "internal" situation, hence $C$-module functor from $K-mod_C$ to itself are given by internal $K$-bimodule (this is Eilenberg-Watts theorem ), i.e. $K$-bimodule in $H$-comodules.
• Thanks for the reference and explanation. So I guess the short answer to my question is that $$Fun_{H-mod}(A-mod, A-mod) = (A, A) \text{-bimodules internal to H-comod}$$. Is the right hand side naturally modules over some algebra? – Joel Kamnitzer Sep 15 '18 at 1:35
• Right, sorry if this was unclear. Also, I should have said that composition of functors corresponds to tensor product over $A$ so this takes care of the monoidal structure as well. As for whether it's modules over some algebra, it's definitely true if H-comod is braided (in which case you get modules over ths smash product $H^*\rtimes (A \tilde \otimes A^{op})$ where $\tilde \otimes$ is the braided tensor product and $op$ is also defined using the braiding. I'm unsure of the general case, though I might just missing something. – Adrien Sep 15 '18 at 8:45