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The Center Construction

Let $\mathcal C$ be a monoidal category. The Drinfeld center of $\mathcal C$ is a category whose objects are tuples $(X,\gamma)$ such that $X \in \mathcal C$ and $\{\gamma_Z \colon X \otimes Z \to Z \otimes X \mid Z \in \mathcal C\}$ is a natural isomorphism such that

\begin{tikzcd} & (Y \otimes X) \otimes Z \ar[r, "a_{Y,X,Z}"] & Y \otimes (X \otimes Z) \ar[dr, "\id_Y \otimes \gamma_X(Z)"] & \\ (X \otimes Y) \otimes Z \ar[ur, "\gamma_X(Y) \otimes \id_Z"] \ar[dr, "a_{X,Y,Z}"] & & & Y \otimes (Z \otimes X) \\ & X \otimes (Y \otimes Z) \ar[r, "\gamma_X(Y \otimes Z)"] & (Y \otimes Z) \otimes X \ar[ur, "a_{Y,Z,X}"] & \end{tikzcd}

commutes for all $Y,Z \in \mathcal C$ and $\gamma_{\mathbb 1} = \mathrm{id}_X$.

Computing the Center

The Drinfeld center can be computed explicitly for reasonably small fusion categories using the algorithm described in [6]. Any fusion category implementing the corresponding interface is supported.

Example

I = ising_category()
C = center(I)
simples(C)

# output
5-element Vector{CenterObject}:
 Central object: 𝟙
 Central object: 𝟙
 Central object: 𝟙 ⊕ χ
 Central object: 2⋅χ
 Central object: 4⋅X

Centers of the AnyonWiki

Currently we are working at the task to compute all centers of multiplicity free fusion categories up to rank seven. At the moment all centers to rank 4 are available and some of rank five. Access them via 

C = anyonwiki_center(3,1,0,2,1,1,1)

print_multiplication_table(C)

# output
8×8 Matrix{String}:
 "(𝟙, γ)"        "(X2, γ)"       …  "(X2 ⊕ X3, γ)"
 "(X2, γ)"       "(𝟙, γ)"           "(𝟙 ⊕ X3, γ)"
 "(𝟙 ⊕ X2, γ)"   "(𝟙 ⊕ X2, γ)"      "(𝟙 ⊕ X3, γ) ⊕ (X2 ⊕ X3, γ)"
 "(X3, γ1)"      "(X3, γ1)"         "(𝟙 ⊕ X3, γ) ⊕ (X2 ⊕ X3, γ)"
 "(X3, γ2)"      "(X3, γ2)"         "(𝟙 ⊕ X3, γ) ⊕ (X2 ⊕ X3, γ)"
 "(X3, γ3)"      "(X3, γ3)"      …  "(𝟙 ⊕ X3, γ) ⊕ (X2 ⊕ X3, γ)"
 "(𝟙 ⊕ X3, γ)"   "(X2 ⊕ X3, γ)"     "(X2, γ) ⊕ (𝟙 ⊕ X2, γ) ⊕ (X3, γ1) ⊕ (X3, γ2) ⊕ (X3, γ3)"
 "(X2 ⊕ X3, γ)"  "(𝟙 ⊕ X3, γ)"      "(𝟙, γ) ⊕ (𝟙 ⊕ X2, γ) ⊕ (X3, γ1) ⊕ (X3, γ2) ⊕ (X3, γ3)"

F- and R-Symbols

We can export the $F$- and $R$-symbols as Dicts via the methods

TensorCategories.numeric_F_symbols — Function
numeric_F_symbols(C::SixJCategory; precision = 2048)
numeric_F_symbols(C::SixJCategory, e::AbsSimpleNumFieldEmbedding; precision = 2048)

Return a Dictionary of the F-symbols of $C$ evaluated under the embedding $e$.

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TensorCategories.numeric_R_symbols — Function
numeric_R_symbols(C::SixJCategory; precision = 2048)
numeric_R_symbols(C::SixJCategory, e::AbsSimpleNumFieldEmbedding; precision = 2048)

Return a Dictionary of the R-symbols of $C$ evaluated under the embedding $e$.

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AbstractAlgebra.is_isomorphic — Method
is_isomorphic(X::CenterObject, Y::CenterObject)

Check if X≃Y. Return (true, m) where mis an isomorphism if true, else return (false,nothing).

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Base.inv — Method
inv(f::CenterMorphism)

Return the inverse of fif possible.

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Base.one — Method
one(C::CenterCategory)

Return the one object of C.

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Base.zero — Method
zero(C::CenterCategory)

Return the zero object of C.

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Hecke.cokernel — Method
cokernel(f::CenterMorphism)

Return a tuple (C,c) where Cis the cokernel object and cis the projection.

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Hecke.dual — Method
dual(X::CenterObject)

Return the (left) dual object of X.

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Hecke.is_central — Function
is_central(Z::Object)

Return true if Z is in the categorical center, i.e. there exists a half-braiding on Z.

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Oscar.morphism — Method
morphism(f::CenterMorphism)

Return the image under the forgetful functor.

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TensorCategories.associator — Method
associator(X::CenterObject, Y::CenterObject, Z::CenterObject)

Return the associator isomorphism (X⊗Y)⊗Z → X⊗(Y⊗Z).

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TensorCategories.adjusted_dual_basis — Method
adjusted_dual_basis(V::AbstractHomSpace, U::AbstractHomSpace, S::Object, W::Object, T::Object)

Compute a dual basis for the spaces Hom(S, W⊗T) and Hom(S̄⊗W, T̄)

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