Classifying fusion categories $\otimes$-generated by an object of small Frobenius–Perron dimension

Abstract

The goal of this paper is to classify fusion categories ${\mathcal {C}}$ which are $\otimes$-generated by an object X of Frobenius–Perron dimension less than 2, with the additional mild assumption that the adjoint subcategory of ${\mathcal {C}}$ is $\otimes$-generated by the object $X\otimes X^*$. This classification has recently become accessible due to a result of Morrison and Snyder, showing that any such category must be a cyclic extension of a category of adjoint ADE type. Our main tools in this classification are the results of Etingof et al. (Quantum Topol 1(3);209–273, 2010. https://doi.org/10.4171/QT/6), classifying cyclic extensions of a given category in terms of data computed from the Brauer–Picard group, and Drinfeld centre of that category, and the results of Edie-Michell (Int. J. Math. 29(5):1850036, 2018. https://doi.org/10.1142/S0129167X18500362) which compute the Brauer–Picard group and Drinfeld centres of the categories of adjoint ADE type. Our classification includes the expected categories, constructed from cyclic groups and the categories of ADE type. More interestingly we have categories in our classification that are non-trivial de-equivariantizations of these expected categories. Most interesting of all, our classification includes three infinite families constructed from the exceptional quantum subgroups ${\mathcal {E}}_4$ of ${\mathcal {C}}( \mathfrak {sl}_4, 4)$, and ${\mathcal {E}}_{16,6}$ of ${\mathcal {C}}( \mathfrak {sl}_2, 16)\boxtimes {\mathcal {C}}( \mathfrak {sl}_3,6)$.

Appendix A. Fusion Rules for the categories \({\mathcal {E}}^{(n)}_4\) and \({\mathcal {E}}^{(n,\pm )}_{16,6}\)

In this appendix we compute the fusion rules for the categories ${\mathcal {E}}^{(n)}_4$ and ${\mathcal {E}}^{(n,\pm )}_{16,6}$. We begin with the ${\mathcal {E}}^{(n,\pm )}_{16,6}$ case, which is conceptually easier, but computationally harder.

Recall from Lemma 3.18 that the categories ${\mathcal {E}}^{(n,\pm )}_{16,6}$ are ${\mathbb {Z}}_{6}$-graded extensions of ${\text {Ad}}(D_{10}^{(n)})$. The 3-graded piece under this grading is either the bimodule $D_{10}^{\text {odd}}$ or $_{P\leftrightarrow Q}D_{10}^{\text {odd}}$. Thus ${\mathcal {E}}^{(n,\pm )}_{16,6}$ has a ${\mathbb {Z}}_{2}$-graded subcategory equivalent to $D_{10}^{(\pm n , \pm )}$, and ${\mathcal {E}}^{n,m,\pm }_{16,6}$ is thus a ${\mathbb {Z}}_{3}$-graded extension of $D_{10}^{(\pm n , \pm )}$.

The Brauer–Picard group of $D_{10}^{(\pm n , \pm )}$ was computed in [8], and the only order 3 bimodules are $_{P\leftrightarrow Q}E_7$ and $_{P\leftrightarrow Q}{\overline{E}}_7$. Thus, as a ${\mathbb {Z}}_{3}$-graded extension of $D_{10}^{(\pm n , \pm )}$, the 1 and 2 graded pieces consist of these two bimodules. This gives us fusion rules for tensoring a $D_{10}$ object with either a $_{P\leftrightarrow Q}E_7$ object, or a $_{P\leftrightarrow Q}\overline{E_7}$ object.

We now aim to determine how two $_{P\leftrightarrow Q}E_7$ objects tensor. Due to the grading, the tensor of two such object must live in the $_{P\leftrightarrow Q}\overline{E_7}$ piece. Let X and Y be $_{P\leftrightarrow Q}E_7$ objects, and Z a $D_{10}$ object, then associativity of the fusion rules gives us that

$$\begin{aligned} (X \otimes Z)\otimes Y \cong X \otimes (Z\otimes Y). \end{aligned}$$

Along with the fact that tensor product must preserve Frobenius–Perron dimensions, this allows us to completely determine the fusion rules for tensoring two objects in the 1-graded piece.

By considering Frobenius–Perron dimensions, and the grading of the category, we can completely determine the dual of each object. Each $D_{10}$ object is self-dual, and each $_{P\leftrightarrow Q}E_7$ is dual to the unique object in the $_{P\leftrightarrow Q}\overline{E_7}$ piece with the same Frobenius–Perron dimension.

Finally, an application of Frobenius reciprocity gives fusion rules for the entire category. The full fusion rules for the categories ${\mathcal {E}}^{(n,\pm )}_{16,6}$ can be found at the authors website http://cainedie.com/E166fusion.txt. To save space we only present in Fig. 1 the fusion graph for the $\otimes$-generating object of Frobenius–Perron dimension $2\cos (\frac{\pi }{18})$, living in the 1-graded piece.

Fig. 1

Fusion graph for the $\otimes$-generating object of ${\mathcal {E}}^{(n,\pm )}_{16,6}$ of Frobenius–Perron dimension $2\cos (\frac{\pi }{18})$

We now compute the ${\mathcal {E}}^{(n)}_4$ case. Recall the category ${\mathcal {E}}^{(n)}_4$ is a ${\mathbb {Z}}_{4}$-graded extension of a category of adjoint $A_7$ type, with the 1, 2, and 3 graded pieces being the bimodules $_{f^{(2)} \leftrightarrow f^{(4)} }A_7^\text {odd}$, $D_5^\text {even}$, and $_{f^{(2)} \leftrightarrow f^{(4)} }D_5^\text {even}$ respectively.

From [5] we know the Frobneius–Perron dimensions of the 12 simple objects of ${\mathcal {E}}^{(n)}_4$. They are

$$\begin{aligned}&\left\{ 1,\sqrt{2}+1,\sqrt{2}+1,1,\sqrt{\sqrt{2}+2},\sqrt{\sqrt{2}+2},\sqrt{2 \left( \sqrt{2}+2\right) },\sqrt{2},\sqrt{2}\right. \\&\quad \left. +2,\sqrt{\sqrt{2}+2},\sqrt{\sqrt{2}+2},\sqrt{2 \left( \sqrt{2}+2\right) }\right\} . \end{aligned}$$

With the first 4 objects living in the 0-graded piece, the next 3 objects living in the 1 graded piece, the next 2 objects living in the 2 graded piece, and the final 3 objects living in the 3-graded piece. For simplicity we call the 12 objects of this category the numbers ${\mathbf {1}}$ through $\mathbf {12}$.

Simply by considering Frobenius–Perron dimensions, along with the associativity check from the ${\mathcal {E}}_{16,6}$ case, allows us to completely determine fusion rules for all objects apart from fusion between the objects 5, 6 and 10, 11. Here we get four possible fusion rules:

Fusion rule	${\mathbf {5}}\otimes \mathbf {10}$	${\mathbf {5}}\otimes \mathbf {11}$	${\mathbf {6}}\otimes \mathbf {10}$	${\mathbf {6}}\otimes \mathbf {11}$
1	${\mathbf {1}} \oplus {\mathbf {2}}$	${\mathbf {3}} \oplus {\mathbf {4}}$	${\mathbf {3}} \oplus {\mathbf {4}}$	${\mathbf {1}} \oplus {\mathbf {2}}$
2	${\mathbf {1}}\oplus {\mathbf {3}}$	${\mathbf {2}} \oplus {\mathbf {4}}$	${\mathbf {2}} \oplus {\mathbf {4}}$	${\mathbf {1}} \oplus {\mathbf {3}}$
3	${\mathbf {3}} \oplus {\mathbf {4}}$	${\mathbf {1}} \oplus {\mathbf {2}}$	${\mathbf {1}} \oplus {\mathbf {2}}$	${\mathbf {3}} \oplus {\mathbf {4}}$
3	${\mathbf {2}} \oplus {\mathbf {4}}$	${\mathbf {1}} \oplus {\mathbf {3}}$	${\mathbf {1}} \oplus {\mathbf {3}}$	${\mathbf {2}} \oplus {\mathbf {4}}$

However a direct computation shows each of these four fusion rules are isomorphic. Thus we can completely determine the fusion rules for the categories ${\mathcal {E}}^{(n)}_4$. Again we only present the fusion graph for the generating object ${\mathbf {5}}$ of Frobenius–Perron dimension $\sqrt{2+\sqrt{2}}$ in Fig. 2. Full fusion rules can be found at the authors website http://cainedie.com/E4fusion.txt.

Fig. 2

Fusion graph of the object ${\mathbf {5}} \in {\mathcal {E}}^{(n)}_{4}$

Using the fusion rules of ${\mathcal {E}}_4$, we can directly compute the fusion ring automorphisms.

Lemma A.1

There exist three non-trivial fusion ring automorphisms of ${\mathcal {E}}^{(n)}_4$ given by

$$\begin{aligned}&1)\,\, {\mathbf {5}} \leftrightarrow {\mathbf {6}} , \mathbf {10} \leftrightarrow \mathbf {11} \\&2)\,\, {\mathbf {2}} \leftrightarrow {\mathbf {3}} , {\mathbf {5}} \leftrightarrow \mathbf {10} , {\mathbf {6}} \leftrightarrow \mathbf {11}, {\mathbf {7}} \leftrightarrow \mathbf {12} \end{aligned}$$

and

$$\begin{aligned} 3)\,\, {\mathbf {2}} \leftrightarrow {\mathbf {3}} , {\mathbf {5}} \leftrightarrow \mathbf {11} , {\mathbf {6}} \leftrightarrow \mathbf {10}, {\mathbf {7}} \leftrightarrow \mathbf {12} \end{aligned}$$

While this Lemma may seem out of place, it proves important in showing that the categories ${\mathcal {E}}_4^{(n)}$ realises two different ${\mathbb {Z}}_{4}$-graded extensions of the category ${\text {Ad}}(A_7^{(n)})$.