Drinfeld centers and half-braidings

The center is implemented in terms of objects and morphisms of the input category. It does not require an $F$-symbol presentation. For the construction see Etingof et al. (2015), §7.13, with the convention translation stated below. The structural theorem, algorithm, and splitting step over arbitrary fields are developed in Mäurer and Thiel (2024), §§2.5 and 4–5 and Mäurer (2026), Chapter 2.

The direction of a half-braiding

A central object in the package is a pair $(Z,\gamma)$, with natural isomorphisms

\[\label{eq:center-half-braiding-component} \gamma_X:Z\otimes X\longrightarrow X\otimes Z.\]

With $a_{X,Y,Z}:(X\otimes Y)\otimes Z\to X\otimes(Y\otimes Z)$, the compatibility is

\[\label{eq:center-half-braiding-hexagon} \gamma_{X\otimes Y} =a^{-1}_{X,Y,Z}\circ(\mathrm{id}_X\otimes\gamma_Y) \circ a_{X,Z,Y}\circ(\gamma_X\otimes\mathrm{id}_Y) \circ a^{-1}_{Z,X,Y}.\]

The unit condition is $\gamma_{\mathbb 1}=\operatorname{id}_Z$ under the package's normalized unit identifications. This is the convention of Müger (2003), Definition 3.1, Mäurer and Thiel (2024), Equation (1.2) and Mäurer (2026), Definition 2.1.1: the central object $Z$ occurs first in the source. By contrast, Etingof et al. (2015), Definition 7.13.1 put the ambient object first and write

\[\label{eq:egno-half-braiding-component} \widetilde\gamma_X:X\otimes Z\longrightarrow Z\otimes X.\]

The conventions are related by $\widetilde\gamma_X=\gamma_X^{-1}$. This componentwise inversion also reverses the displayed braiding. The package uses $c_{(Z,\gamma),(W,\delta)}=\gamma_W$, whereas Etingof et al. (2015), Proposition 8.5.1, Equation (8.15) use $c_{(Z,\widetilde\gamma),(W,\widetilde\delta)}=\widetilde\delta_Z=(\delta_Z)^{-1}$. Thus, under the direct identification by inverse half-braidings, the two conventions give reverse braided structures.

A morphism $f:(Z,\gamma)\to(Z',\gamma')$ is a morphism $f:Z\to Z'$ satisfying

\[\label{eq:center-morphism-condition} (\mathrm{id}_X\otimes f)\circ\gamma_X =\gamma'_X\circ(f\otimes\mathrm{id}_X).\]

center(C) constructs the center parent without enumerating its simple objects. Explicit CenterObjects can therefore be formed whenever the ambient category supplies the operations needed for their half-braidings. If $\mathcal C$ is braided, center_embedding(C) constructs the canonical functor $X\mapsto(X,c_{X,-})$ into this center. The keyword reverse=true constructs the second canonical functor using $c_{-,X}^{-1}$. When the input is already modular, simples(center(C)) uses its braiding to construct the center simples directly. Otherwise it invokes the substantially stronger induction algorithm and decomposes endomorphism algebras. For input $X$, central induction has underlying object

\[\label{eq:center-induction-object} \bigoplus_{S\in\operatorname{Irr}(\mathcal C)} (S\otimes X)\otimes S^*.\]

Here $\operatorname{Irr}(\mathcal C)$ denotes the chosen representatives of the simple objects.

The enumeration algorithm applies this construction to induction generators and obtains simple central summands from their endomorphism algebras. The mathematical algorithm is proved for a pivotal fusion category in the split sense used in this manual, with $\dim(\mathcal C)\ne0$ (Mäurer and Thiel, 2024; Assumption 4.1). The present implementation's general induction branch uses chosen spherical morphisms in its projection formulas. Its supported input is therefore the narrower class of split spherical fusion categories, again with $\dim(\mathcal C)\ne0$ in the coefficient field. It is not an enumeration algorithm for arbitrary monoidal categories.

In positive characteristic these hypotheses require separate verification. For a pivotal fusion category over any field, the center is weak fusion exactly when $\dim(\mathcal C)\ne0$; if the center is split, it is modular (Mäurer and Thiel, 2024; Theorem 2.1). Thus semisimplicity of the input alone does not imply semisimplicity of its center, and the center need not be split even when the input is split.

Once the simple objects of $\mathcal Z(\mathcal C)$ over $k$ have been computed, one chooses a common splitting field for their endomorphism algebras. After scalar extension, primitive idempotents in these algebras give the split simple central summands. This is an application of the general splitting procedure, and is Algorithm 4 of Mäurer and Thiel (2024), §5.2. It is a separate stage from computing the center over $k$: a center can be fully computed while some of its simple objects remain non-split over that field.

For supported CenterCategory models, split(Z) searches for a common splitting field and returns the extended center together with the field embedding; the field found need not be minimal. If a target field $K$ and an embedding have already been chosen, extension_of_scalars(Z,K; embedding=...) extends the center and decomposes the known non-split simple central objects.

Numerical center computations use the same public interface, with structural equations interpreted in the working ball field.

Constructing half-braidings by hand

For $\operatorname{Vec}_{k}(C_2)$ in characteristic zero, the four simple central objects are indexed by a degree $g\in C_2$ and a character $\chi:C_2\to k^\times$. On an object homogeneous of degree $h$, the half-braiding is $\chi(h)$ times the usual interchange map.

Here is this construction over $\mathbb Q$. We do not enumerate the center to construct these objects.

The call to braiding below supplies the ordinary interchange map of this particular symmetric ambient category. The center construction itself does not require the ambient category to be braided.

using TensorCategories, Oscar
G = cyclic_group(2)
C = graded_vector_spaces(QQ,G)
S = simples(C)
Z = center(C)
central = [
    CenterObject(Z, X,
        [braiding(X,Y) * (Y == one(C) ? QQ(1) : QQ(epsilon)) for Y in S])
    for X in S for epsilon in (1,-1)
]
@assert length(central) == 4
for X in central
    @assert is_central(X)
    @assert all(is_invertible, half_braiding(X))
    @assert half_braiding(X, one(C)) == id(object(X))
end
[int_dim(Hom(X,Y)) for X in central, Y in central]

4×4 Matrix{Int64}:
 1  0  0  0
 0  1  0  0
 0  0  1  0
 0  0  0  1

The displayed matrix is the identity: these objects are simple and pairwise nonisomorphic. Their completeness follows independently from the classification by the two degrees and two characters. We can also test the equation on a non-simple object:

X = central[end]
Y = S[1] ⊕ S[2]
W = S[2]
rhs = inv_associator(Y,W,object(X)) ∘
      (id(Y) ⊗ half_braiding(X,W)) ∘ associator(Y,object(X),W) ∘
      (half_braiding(X,Y) ⊗ id(W)) ∘ inv_associator(object(X),Y,W)
@assert half_braiding(X,Y⊗W) == rhs
matrix(half_braiding(X,Y))

[1    0]
[0   -1]

CenterObject(Z,X,gamma) stores components in the order simples(C); object forgets the half-braiding, half_braiding(X) returns the stored list, and half_braiding(X,Y) extends it to a general object. The constructor stores the components without validation. is_central checks the coherence equations; invertibility and the normalized unit condition $\gamma_{\mathbb 1}=\operatorname{id}_X$ are additional requirements on the supplied components. For split semisimple input, naturality on simple objects is automatic from scalar simple endomorphisms. With non-split simples, the current check does not separately test naturality against their non-scalar endomorphisms, so manually entered data require that verification as well.

The braiding in the center is $c_{(X,\gamma),(Y,\delta)}=\gamma_Y$. Thus two central objects with the same underlying object can be different. The forgetful functor is faithful and tensor, but its target is $\mathcal C$, not necessarily vector spaces.

Extracting structural data

Once a center is split, use six_j_category(Z) to construct a skeleton, or six_j_symbols(Z) to compute its associators. Use the same binary Hom bases when computing associators and braiding; independently chosen bases need not give a compatible pair of $F$- and $R$-symbols.

Numerical centers and skeletonization

A numerical center computation follows the same categorical construction as an exact one. The scalar comparisons and linear algebra are instead performed in the category's arbitrary-precision ball field, with the interpretation given under Numerical fusion categories. The orthonormal-basis construction described below and its numerical application to the Haagerup center are developed by Mäurer (2026), §§4.2.4–4.3 and 5.2.3.

Under the pivotal-fusion hypotheses above, semisimplicity of the center is equivalent to $\dim(\mathcal C)\ne0$. The center-specific predicate is_semisimple(Z) currently tests this condition by structural comparison when $Z=\mathcal Z(\mathcal C)$. If this dimension is a ball containing zero, test !contains_zero(dim(category(Z))) before relying on that predicate. The half-braiding and skeletal coherence checks use ball-aware comparisons; the dimension comparison in is_fusion(Z) does so for ArbField and AcbField.

C = numeric(anyonwiki(3,1,0,1,1,1,1),256)
Z = center(C)
simples(Z)
S = skeletonize(Z)

For unitary input, numerical skeletonization chooses an orthonormal basis in each binary fusion space with respect to the dagger inner product. It uses the same chosen bases to compute both the associator and the braiding, so the resulting $F$- and $R$-symbols belong to one common gauge. The skeletal category can then be checked with the ordinary structural predicates:

@assert is_unitary(S)
@assert is_modular(S)
@assert pentagon_axiom(S)
@assert hexagon_axiom(S)

Its numerical symbol dictionaries can be written with the numerical data-exchange format.

The complete pentagon test examines every quadruple of simple objects and can be expensive. randomized_pentagon_axiom(S,n) checks $n$ randomly selected quadruples and is useful during a long computation, but it is not a replacement for the complete test when final validation is feasible.

For a pivotal fusion category with $\dim(\mathcal C)\ne0$, the center satisfies

\[\label{eq:center-global-dimension} \dim\mathcal Z(\mathcal C)=(\dim\mathcal C)^2.\]

The method dim(Z) for a CenterCategory returns the right-hand side by construction. Once the center is split, check an enumeration by comparing this value with $\sum_{T\in\operatorname{Irr}(Z)}\dim(T)\dim(T^*)$; the center-specific is_fusion(Z) method makes this comparison after checking that the simples are split. Further checks are the fusion rules, unitarity when applicable, nondegeneracy of the $S$-matrix, and the pentagon and hexagon equations. These checks probe different parts of the construction. The displayed identity uses the pivotal categorical dimension; sphericality is not required (Mäurer and Thiel, 2024; Theorem 2.1). Independently, in the usual characteristic-zero fusion setting, $\operatorname{FPdim}(\mathcal Z(\mathcal C))=\operatorname{FPdim}(\mathcal C)^2$ (Etingof et al., 2015; Theorem 7.16.6).

Continue with The Ising center over two fields, which follows a non-split center through scalar extension and extraction of a single compatible skeleton. Precomputed database centers are described under AnyonWiki; a saved skeletal center does not retain the original explicit half-braidings.