Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a *quadratic category* with $(G,\tau,m)$ if its Grothendieck ring has basis $\{X\_g, Y\_g \ | \ g \in G\}$ and fusion rules (let $e$ be the neutral element of $G$):


* $X\_gX\_h = X\_{gh}$,
* $X\_gY\_e = Y\_g = Y\_eX\_{g^{\tau}}$,
* $Y\_e^2 = X\_e + m\sum\_{g \in G} Y\_g$.


Note that *quadratic* just requires the action of $G$ on the non-group part of the basis to be transitive, as stated in the second line above, in particular, we can have $Y\_{g\_1}=Y\_{g\_2}$ with $g\_1 \neq g\_2$.


Let $\mathcal{C}$ be a spherical quadratic category with $(G,\tau,m)$. By [1, Theorem 2.2], if $G$ is an odd group and $m$ is an odd number, then $G$ is abelian and $g^{\tau} = g^{-1}$ for any $g \in G$.


The Haagerup category is a quadratic category with $(C\_3, -1, 1)$.


**Question**: Is there a complex fusion category, quadratic with $(C\_3, 1, 1)$?


According to above statements, such a fusion category would be a *twisted* Haagerup category without spherical structure, in fact without pivotal structure by applying [2, Corollary 2.14].


Recall that the existence of a pivotal structure on every fusion category is a well-known conjecture [3, Conjecture 2.8]. If this conjecture is true then above question has a negative answer, but if this conjecture is false then above category could be a first counter-example. Anyway, it should be possible to answer above question independently of this conjecture.




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*References*


[1]: Izumi, Masaki The classification of $3^n$ subfactors and related fusion categories. Quantum Topol. 9 (2018), no. 3, 473–562.  

[2]: Ostrik, Victor. Pivotal fusion categories of rank 3. Mosc. Math. J. 15 (2015), no. 2, 373-396, 405.  

[3]: Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581-642.



A twisted Haagerup category without pivotal structure
**Quick answer**: no.


Andrew Schopieray pointed out to me the PhD thesis of Josiah E. Thornton [2].




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Let $\mathcal{F}$ be a fusion ring with basis $B=\{ b\_1, \dots, b\_r \}$. Let $G$ be the group of invertible elements $b\_i$ of $B$ (i.e. $\textrm{FPdim}(b\_i)=1$). The fusion ring $\mathcal{F}$ is called *quadratic* if the action of $G$ on $B \setminus G$ is transitive. If $G=B$ such a fusion ring is called *pointed*, and if $|B \setminus G|=1$ it is called *near-group*. A fusion category with a quadratic Grothendieck ring is called (in the literature) a *quadratic category* or a *generalized near-group category*.


**Theorem.**
A categorification $\mathcal{C}$ of a quadratic fusion ring must admit a spherical (so pivotal) structure.  

*Proof.* By [2, Theorem IV.3.6.], $\mathcal{C}$ must be $\varphi$-pseudo-unitary (i.e. pseudo-unitary up to Galois automorphism), and then spherical (so pivotal) by [1, Proposition 2.16].




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*References*


[1] V. Drinfeld, S. Gelaki, D. Nikshych, V. Ostrik, *On braided fusion categories. I.*, Selecta Math. (N.S.) 16 (2010), no. 1, 1--119.


[2] J.E. Thornton, *Generalized near-group categories*, Thesis (Ph.D.)–University of Oregon (2012) 72 pp.


