Tambara Yamagami Categories
Tambara and Yamagami classified a class of near-group categories of multiplicity one over algebraically closed fields in Daisuke Tambara, Shigeru Yamagami (1998). 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.
TensorCategories.TambaraYamagami
— FunctionTambaraYamagami(A::GAPGroup)
Construct $TY(A,τ,χ)$ over $ℚ̅$ where $τ = √|A|$ and $χ$ is a generic non-degenerate bilinear form.
TambaraYamagami(K::ring, A::GAPGroup)
Construct $TY(A,τ,χ)$ over $K$ where $τ = √|A|$ and $χ$ is a generic non-degenerate bilinear form.
TambaraYamagami(K::Ring, A::GAPGroup, τ::RingElem)
Construct $TY(A,τ,χ)$ over $K$ where $χ$ is a generic non-degenerate bilinear form.
TambaraYamagami(K::Ring, A::GAPGroup, τ::RingElem)
Construct $TY(A,τ,χ)$ over $K$ where $τ = √|A|$.
TambaraYamagami(K::Ring, A::GAPGroup, τ::RingElem, χ::BilinearForm)
Construct the Category $TY(A,τ,χ)$.
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 $A = \mathbb Z_2$.
TensorCategories.Ising
— FunctionIsing()
Construct the Ising category over $ℚ̅$.
Ising(F::Ring)
Construct the Ising category over $F$.
Ising(F::Ring, τ::RingElem)
Construct the Ising category with specific $τ = √2$.
Ising(F::Ring, q::Int)
Construct the braided Ising category over $F$ where q = ±1 defined the braiding defined by ±i.
Ising(F::Ring, τ::RingElem, q::Int)
Construct the Ising fusion category where $τ = √2$ a root and q ∈ {1,-1}
specifies the braiding if it exists.
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 Ramona Wolf (2021).
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 first two have multiplicity larger then 1 and are up to now not included. The third has multiplicity 1 and is also known as the Haagerup-Izumi category for $\mathbb Z_3$. It has six simple objects and fusion rules
\[\renewcommand*{\arraystretch}{1.5} \begin{array}{c||c|c|c|c|c|c} & \mathbb 1 & \alpha & \alpha^\ast & \rho & {}_{\alpha}\rho & {}_{\alpha^\ast}\rho \\ \hline \hline \mathbb 1 & \mathbb 1 & \alpha & \alpha^\ast & \rho & {}_{\alpha}\rho & {}_{\alpha^\ast}\rho \\ \hline \alpha & \alpha & \alpha^\ast & \mathbb 1 & {}_\alpha\rho & {}_{\alpha^\ast}\rho & \rho \\ \hline \alpha^\ast & \alpha^\ast & \mathbb 1 & \alpha & {}_{\alpha^\ast}\rho & \rho & {}{\alpha}\rho \\ \hline \rho & \rho & {}_{\alpha^\ast}\rho & {}_\alpha\rho & \mathbb 1 \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha^\ast \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho \\ \hline {}_\alpha\rho & {}_\alpha\rho & {}_{\alpha^\ast}\rho & \rho & \alpha \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \mathbb 1 \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha^\ast \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho \\ \hline {}_{\alpha^\ast}\rho & {}_{\alpha^\ast}\rho & \rho & {}_\alpha\rho & \alpha^\ast \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \alpha \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho & \mathbb 1 \oplus \rho \oplus {}_\alpha\rho \oplus {}_{\alpha^\ast}\rho \end{array} ´´´ We implement the category ``\mathcal H_3`` as an instance of `SixJCategory`. The other two will follow as soon as we know the proper ``6j``-symbols. \]
@docs HaagerupH3
# Fusion Categories From Truncated Hecke Categories
TODO: Explanation
@docs I2 I2subcategory
# Various Other Categories Given by ``6J``-Symbols
Here are some more examples to play around.
## Categorifications by Vercleyen and Singerland
In [vercleyen2023low](@cite) they found a huge number of fusion rings and some explicit categorifications of two categories that are neither Tambara-Yamagami nor Haagerup-Izumi categories.
### FR``{}_2^{82}``
They provide 97 different (but maybe equivalent) associators one can acces.
@docs fr_8122
### FR``{}_3^{94}``
We have a singel associator for this ring.
@docs fr_9143 ```