# How simplify the pentagonal equation from two fusion rings?

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 $\mathbb{A}^{*}$ ?

• How do you produce the fusion category from the Hopf algebra? Do you mean the representation category? – Manuel Bärenz Aug 13 '17 at 8:40
• @Turion: yes. Here we deal with the representation category and the corepresentation category. – Sebastien Palcoux Aug 13 '17 at 17:43
• 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. – Manuel Bärenz Aug 14 '17 at 10:45
• I think fusion categories always representations of some quasi-Hopf algebras. But those are not coassociative, they only are if the associator is trivial. – Manuel Bärenz Aug 14 '17 at 10:47
• @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. – Sebastien Palcoux Aug 14 '17 at 19:50