Tambara Yamagami Categories

Tambara and Yamagami classified a class of near-group categories of multiplicity one over algebraically closed fields in [2]. Let $A$ be an abelian group. Choose a square root $\tau \in k$ of $\vert A\vert$ and a non-degenerate symmetric bilinear form $\chi \colon A \times A \to k^\times$. Then the Tambara-Yamagimi category $TY(A,\chi,\tau)$ has objects $A \cup \{m\}$ with fusion rules

\[\begin{align*} a ⊗ b = a+b,~~a \otimes m = m \otimes a = m,~~ m \otimes m = \sum\limits_{a \in G} a \end{align*}\]

for $a,b \in G$. The non-trivial associativity constraints are given by

\[\begin{align*} a_{a,m,b} = \chi(a,b)\mathrm{id}_m & a_{m,a,m} = \bigoplus\limits_{b\in A}\chi(a,b)\mathrm{id}_b & a_{m,m,m} = \bigoplus\limits_{a,b \in A} \frac{1}{\tau\chi(a,b)}\mathrm{id}_m. \end{align*}\]

Those categories can be constructed with a generic symmetric bilinear form or with a custom bilinear form and over arbitrary fields.

tambara_yamagami(A::Group)

tambara_yamagami(K::ring, A::Group)

tambara_yamagami(K::Ring, A::Group, τ::RingElem)

tambara_yamagami(K::Ring, A::Group, τ::RingElem)

tambara_yamagami(K::Ring, A::Group, τ::RingElem, χ::BilinearForm)

Tambara-Yamagami categories are implemented as an instance of SixJCategory and hence all functionality follows from there.

Ising Category

The Ising fusion category is a special example of a Tambara-Yamagami category with $A = \mathbb Z_2$.

ising_category()

ising_category(F::Ring)

The Haagerup Subfactor

The fusion categories stemming from the Haagerup subfactor are a well known and important example of a fusion category. Details can be found in [3].

In the Morita equivalence class of the Haagerup subfactor lie three categories. We call them $\mathcal H_1,\mathcal H_2$ and $\mathcal H_3$. The third has multiplicity 1 and is also known as the Haagerup-Izumi category for $\mathbb Z_3$. It has six simple objects and teh same fusion rules as $\mathcal H_2$:

\[\begin{array}{c||c|c|c|c|c|c} & \mathbb 1 & \alpha & \alpha^\ast & \rho & {}_{\alpha}\rho & {}_{\alpha^\ast}\rho \\ \hline \hline \mathbb 1 & \mathbb 1 & \alpha & \alpha^\ast & \rho & {}_{\alpha}\rho & {}_{\alpha^\ast}\rho \\ \hline \alpha & \alpha & \alpha^\ast & \mathbb 1 & {}_\alpha\rho & {}_{\alpha^\ast}\rho & \rho \\ \hline \alpha^\ast & \alpha^\ast & \mathbb 1 & \alpha & {}_{\alpha^\ast}\rho & \rho & {}{\alpha}\rho \\ \hline \rho & \rho & {}_{\alpha^\ast}\rho & {}_\alpha\rho & \mathbb 1 \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha^\ast \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho \\ \hline {}_\alpha\rho & {}_\alpha\rho & {}_{\alpha^\ast}\rho & \rho & \alpha \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \mathbb 1 \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha^\ast \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho \\ \hline {}_{\alpha^\ast}\rho & {}_{\alpha^\ast}\rho & \rho & {}_\alpha\rho & \alpha^\ast \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \mathbb 1 \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho \end{array}\]

All three can be accessed via

haagerup_H1()

haagerup_H2()

haagerup_H3()

Fusion Categories From Truncated Hecke Categories

TODO: Explanation

I2(m::Int)
I2(m::Int, K::Ring)

I2subcategory(m::Int)
I2subcategory(m::Int, R::Ring)

Various Other Categories Given by $6J$-Symbols

Here are some more examples to play around.

Categorifications by Vercleyen and Singerland

In [4] they found a huge number of fusion rings and some explicit categorifications that are neither Tambara-Yamagami nor Haagerup-Izumi categories.

FR${}_2^{82}$

They provide 97 different (but maybe equivalent) associators one can access.

cat_fr_8122(n::Int64)

FR${}_3^{94}$

We have a singel associator for this ring.

cat_fr_9143()