Tannaka Duality, Coclosed Categories and Reconstruction for Nonarchimedean Bialgebras

Abstract

The topic of this paper is a generalization of Tannaka duality to coclosed categories. As an application we prove reconstruction theorems for coalgebras (bialgebras, Hopf algebras) in categories of topological vector spaces over a nonarchimedean field K. In particular, our results imply reconstruction and recognition theorems for categories of locally analytic representations of compact p-adic groups, which was the major motivation for this work.

Appendix

1.1 Comodule structure on ${\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right)$.

In applications reconstructing bialgebra structure might be sufficient due to the uniqueness of the antipode. However in the recognition theorem one would like to have conditions, which define Hopf algebra structure on ${\mathrm{coend}}_{\mathcal {C}}\left( \mathbb {F}\right)$. Our conjecture is that one also can reconstruct Hopf algebra structure under assumptions weaker that rigidity. Currently we cannot prove it in full generality and this is the reason why in the title of this paper we only put “reconstruction for bialgebras”.

Suppose we have a Hopf algebra H in a category $\mathcal {C}$ and ${\mathrm{Comod}}_{\mathcal {C}_{0}}-H$ be the category right H-comodules, which are the objects of the rigid subcategory $\mathcal {C}_{0}\subset \mathcal {C}$. Then for every $X\in {\mathrm{Comod}}_{\mathcal {C}_{0}}-H$ we have a comodule structure on $X^{*}$ via the antipode of H and this gives the rigid structure on ${\mathrm{Comod}}_{\mathcal {C}_{0}}-H$. Thus the category ${\mathrm{Comod}}_{\mathcal {C}_{0}}-H$ is coclosed and for any $X,Y\in {\mathrm{Comod}}_{\mathcal {C}_{0}}-H$ we have an equality in $\mathcal {C}_{0}$

$$\begin{aligned} {\mathrm{coend}}_{\mathcal {\mathcal {{\mathrm{{\mathrm{Comod}}_{\mathcal {C}_{0}}}-H}}}}\left( X,Y\right) ={\mathrm{coend}}_{\mathcal {\mathcal {C_{{\mathrm{0}}}}}}\left( X,Y\right) =Y^{*}\otimes X. \end{aligned}$$

We anticipate that the first part of this equality holds for general coclosed categories. Here we only explain how to give ${\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right)$ a comodule structure.

Let now $\mathcal {C}_{0}\subset \mathcal {C}$ be a subcategory coclosed in $\mathcal {C}$.

Let $C\in {{\mathcal {C}}}$ be a coalgebra and $X\in {{\mathcal {C}}}_{0}$ is a right C-comodule. Then the coaction $\rho _{X}:X\rightarrow X\otimes _{\mathcal {C}}C$ induces the map

$$\begin{aligned} \rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{r}:=\varDelta _{X,Y,X\otimes _{\mathcal {C}}C,\rho _{X}}:{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) \rightarrow {\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) \otimes _{\mathcal {C}}C \end{aligned}$$

for every $Y\in \mathcal {C}_{0}$ via diagram

One can check that $\rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{r}$ satisfy the axioms of the right C-comodule coaction. In a similar way the coaction $\rho _{Y}:Y\rightarrow Y\otimes _{\mathcal {C}}C$ induces the map

$$\begin{aligned} {\tilde{\rho }}_{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}:{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) \rightarrow C\otimes _{\mathcal {C}}{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) \end{aligned}$$

via diagram

One can check that ${\tilde{\rho }}_{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}$ satisfy the axioms of the left $C^{cop}$-comodule coaction. In case $C=H$ is a Hopf algebra in $\mathcal {C}$ we can turn ${\tilde{\rho }}_{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}$ into a right H-comodule coaction

$$\begin{aligned} \rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}:=\left( \mathrm {id}_{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }\otimes _{\mathcal {C}}S_{H}\right) \circ \tau \circ {\tilde{\rho }}_{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}. \end{aligned}$$

Combining $\rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}$ and $\rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{r}$ (similar to the tensor product of $X^{*}$ and X in a rigid category), we define the map

$$\begin{aligned} \rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }:=\left( \mathrm {id}_{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }\otimes _{\mathcal {C}}m_{H}\right) \circ \left( \rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{l}\otimes _{\mathcal {C}}\mathrm {id}_{H}\right) \circ \rho _{{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) }^{r} \end{aligned}$$

which satisfy the axioms of the right H-comodule coaction. Under this comodule structure, the coevaluation map

$$\begin{aligned} {\mathrm{coev}}_{X,Y}:X\rightarrow Y\otimes _{\mathcal {C}}{\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right) \end{aligned}$$

becomes a morphism of right H-comodules. By dualizing the argument in [18], proposition 10.1, one can show that coaction morphisms are morphisms of H-comodules, making ${\mathrm{cohom}}_{\mathcal {C}}\left( X,Y\right)$ into ${\mathrm{cohom}}_{\mathcal {{\mathrm{Comod}}_{\mathcal {C}}-H}}\left( X,Y\right)$.