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A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$.

A fusion category $\mathcal{C}$ is in fact given by one solution of the pentagonal equation on its fusion ring $\mathcal{R}$.

Starting with a fusion ring $\mathcal{R}$, it's in general quite difficult (or unattainable) of solving the pentagonal equation.

But if we start with two fusion rings $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ (so more constraints) :

Is there a simplified manner to find some related fusion categories in duality $\mathcal{C}$ and $\mathcal{C}^{*}$, and then semi-simple finite dimensional Hopf algebras in duality $\mathbb{A}$ and $\mathbb{A}^{*}$ ?

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  • $\begingroup$ How do you produce the fusion category from the Hopf algebra? Do you mean the representation category? $\endgroup$
    – Manuel Bärenz
    Commented Aug 13, 2017 at 8:40
  • $\begingroup$ @Turion: yes. Here we deal with the representation category and the corepresentation category. $\endgroup$
    – Sebastien Palcoux
    Commented Aug 13, 2017 at 17:43
  • $\begingroup$ But the representation category can always be endowed with the trivial associator from vector spaces, right? I guess I don't completely understand your question. $\endgroup$
    – Manuel Bärenz
    Commented Aug 14, 2017 at 10:45
  • $\begingroup$ I think fusion categories always representations of some quasi-Hopf algebras. But those are not coassociative, they only are if the associator is trivial. $\endgroup$
    – Manuel Bärenz
    Commented Aug 14, 2017 at 10:47
  • $\begingroup$ @Turion: I'm not an expert but I don't think so (at least in the sense I know, see here). Note that there are fusion rings which are not categorifiable. $\endgroup$
    – Sebastien Palcoux
    Commented Aug 14, 2017 at 19:50

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