Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf algebras because the (skew)-primitives behave differently in both Hopf algebras. It is well-known that one can detect skew-primitives in the categories of comodules, indeed, skew-primitives are determined by extensions of one dimensional comodules and vice-versa.

However, can one detect skew-primitives in the monoidal structure of the representation category of $H$? Note that Tannaka-Krein reconstruction might not be a good approach as the Hopf algebras are infinite dimensional.

Edit (22/11/2017): Let $H$ be a Hopf algebra and $x\in H$ a skew-primitive element. Let $K$ be the Hopf-subalgebra generated by $x$. The embedding $K\hookrightarrow H$ induces a monoidal (restriction) functor $$\text{mod}(H)\rightarrow \text{mod}(K).$$ Is there any hope of classifying such functors given that you know the structure of $H$ and $K$? Perhaps, skew-primitives can be translated to such easy restriction functors?

Edit (23/11/2017): Suppose $F:\text{mod}(H')\xrightarrow{\sim} \text{mod}(H)$ as monoidal categories, then by Eilenberg-Watts' theorem $F\cong -\otimes_{H'}N $ where $N$ is a $H'$-$H-$bimodule. (We didn't use that $F$ is monoidal to invoke this theorem, the fact that $F$ is monoidal yields extra structure on $N$, but I'm not sure how much structure). Thus composing $F$ with the embedding $\text{mod}(H)\hookrightarrow \text{mod}(K)$ yields a monoidal functor $\text{mod}(H')\rightarrow \text{mod}(K)$. Is this functor still a 'restriction functor'? If so, that would be interesting since $H'$ might not have skew-primitive elements and such functors might not exist.

Edit (29/11/2017): It occured to me to look at the paper "Various Structures Associated to the Representation Categories of Eight-Dimensional Nonsemisimple Hopf Algebras" by Wakui in which all 8-dimensional Hopf algebras over an algebraically closed field are classified up to monoidal equivalence of their representation categories. Interestingly enough $\text{Rep}(A_{C_4,\omega'''})\cong \text{Rep}((A_{C_4}'')^*)$ as monoidal categories, but the Hopf algebras $A_{C_4,\omega'''}$ and $(A_{C_4}'')^*$ don't even have the same amount of grouplike elements. The skew-primitives however are the same. Anyway, this example suggests that we cannot recognize skew-primitives in a representation category as we can't even detect grouplike elements. (Not a proof, but a feeling). Am I correct on this?

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