Fusion categories and splitting

This page combines the linear, abelian, semisimple, monoidal, and rigid structures introduced separately on the preceding pages. The definitions in Etingof et al. (2015), §4.1 initially fix an algebraically closed field, without a characteristic-zero hypothesis, and (Etingof et al., 2015; §4.16) explains what changes over an arbitrary field. The weak fusion terminology used here follows Mäurer and Thiel (2024), §2.1.

Weak fusion, fusion, and multi variants

A weak fusion category over a field $k$ is a finite semisimple $k$-linear rigid monoidal category with finite-dimensional Hom spaces, $k$-bilinear tensor product, and simple tensor unit. The package uses weak multifusion category for the analogous structure in which the tensor unit need not be simple.

An object $S$ is scalar if the canonical map

\[\label{eq:simple-endomorphism-field-map} k\longrightarrow\operatorname{End}_{\mathcal C}(S)\]

is an isomorphism. A weak multifusion category is multifusion when all its simple objects are scalar, and a weak fusion category is fusion under the same condition. Equivalently, the latter is a split weak fusion category. Over an algebraically closed field, finite-dimensional division algebras are scalar, so the distinction between weak fusion and fusion disappears. The adjective “weak” concerns splitting over the coefficient field; it does not weaken associativity, semisimplicity, or rigidity.

The non-split setting is needed even when one starts with a split fusion category: its Drinfeld center need not be split over the same coefficient field. When the global dimension is nonzero, the center is nevertheless a weak fusion category, and passing to a splitting field produces a split fusion category (Mäurer and Thiel, 2024; Theorem 2.1 and §5). Thus non-split categories arise naturally during categorical constructions, even when the input category is split over its field of definition.

There is a separate issue over an imperfect field. Some treatments require a (multi)fusion category to be separable. For finite semisimple categories this condition is automatic over a perfect field, including every finite field, but can be stronger than semisimplicity over an imperfect field (Sanford, 2025; Definition 2.9 and the discussion on pp. 3–4). The package predicates and this manual use the semisimple convention of Mäurer and Thiel (2024), §2.1.

The corresponding structural vocabulary refines the combinations introduced earlier under Models and the category interface:

Generic predicates report declared structure or consequences of stronger declarations. A category-specific method may instead compute an invariant or enumerate simple objects, so a predicate can be expensive or unsupported over a particular coefficient field.

For non-split input, the generic is_tensor and is_ring predicates do not test the scalar-unit condition: they return true for a category declared weak fusion, even when the simple tensor unit has endomorphism algebra larger than $k$. Use is_fusion(C) or is_split_semisimple(C) when an algorithm requires scalar simple objects.

Non-split multiplicities

For a simple object $S$, Schur's lemma makes $D_S=\operatorname{End}_{\mathcal C}(S)$ a finite-dimensional division algebra over $k$. If $X$ is semisimple, its multiplicity of $S$ is

\[\label{eq:nonsplit-simple-multiplicity} [X:S]=\dim_{D_S}\operatorname{Hom}(S,X) =\frac{\dim_k\operatorname{Hom}(S,X)}{\dim_k D_S}.\]

Replacing $D_S$ by $k$ gives incorrect multiplicities. The division algebra need not be commutative. Over a finite field it is a field, but it need not be the coefficient field itself.

Scalar extension can require splitting idempotents to obtain all new simple objects. Changing coefficients and enumerating the new simples are distinct steps. For supported centers, the center tutorial explains extension_of_scalars and split; the construction table separately explains the role of karoubian_envelope.

These multiplicities are the structure constants of the Grothendieck ring. Over a non-splitting field they give a weak fusion ring.

Scalar extension and algorithmic splitting

Let $k\hookrightarrow K$ be a field extension. Extending the coefficients of a finite $k$-linear category first gives a category $\mathcal C\otimes_k K$ with the same objects and

\[\label{eq:categorical-scalar-extension-hom} \operatorname{Hom}_{\mathcal C\otimes_k K}(X,Y) =\operatorname{Hom}_{\mathcal C}(X,Y)\otimes_k K.\]

This operation can create idempotent endomorphisms whose images were not objects of the original category. The scalar extension relevant here is therefore the idempotent completion

\[\label{eq:categorical-scalar-extension-karoubi} \mathcal C\boxtimes_k K =\operatorname{Kar}(\mathcal C\otimes_k K).\]

An object of the completion may be represented by a pair $(X,e)$ with $e\in\operatorname{End}_{\mathcal C\otimes_k K}(X)$ idempotent; it represents the image of $e$. A field $K$ is a splitting field for $\mathcal C$ when every simple object of $\mathcal C\boxtimes_k K$ has endomorphism ring $K$. This construction, including its agreement with a Deligne scalar extension, is developed in (Mäurer and Thiel, 2024; §5.1).

The decomposition problem is reduced to finite-dimensional algebra. If

\[X\cong\bigoplus_i S_i^{\oplus m_i}, \qquad D_i=\operatorname{End}_{\mathcal C}(S_i),\]

then

\[\label{eq:endomorphism-algebra-decomposition} \operatorname{End}_{\mathcal C}(X) \cong\prod_i\operatorname{Mat}_{m_i}(D_i).\]

Direct-sum decompositions of $X$ correspond to systems of orthogonal idempotents in this algebra. Algorithmically, one computes $\operatorname{End}(X)$, decomposes it, and realizes suitable primitive idempotents as images in the category. Over a splitting field, the semisimple algebras $D_i\otimes_k K$ are products of full matrix algebras over $K$; images of primitive idempotents then give split simple summands.

Thus the basic splitting procedure for a finite semisimple category is:

1. compute representatives $S_i$ of its simple objects and their algebras $D_i=\operatorname{End}(S_i)$;
2. choose one extension $k\hookrightarrow K$ that splits all the $D_i$;
3. extend objects, morphisms, and structural maps from $k$ to $K$; and
4. decompose each $D_i\otimes_k K$ and take the images of primitive idempotents, retaining one representative of every resulting simple.

The procedure is not specific to any particular construction of a category. Its implementation does depend on the chosen model: the model must support scalar extension, computation and decomposition of endomorphism algebras, and images of idempotents. The currently available operations and their precise scope are listed under Products, scalar extension, and related constructions.

Dagger structures and unitarity

For categories realized over $\mathbb C$, a dagger sends $f:X\to Y$ to $f^\dagger:Y\to X$, is conjugate-linear and involutive, reverses composition, and is compatible with tensor products. A unitary fusion category has a compatible positive dagger structure. After orthonormal bases have been chosen in the fusion spaces, its associator matrices are unitary. If the category also has a compatible braiding, its braiding matrices are unitary as well. In arbitrary bases these structural maps need not be represented by unitary matrices (Bonderson, 2007; Chapter 2).

For the later skeletal $F$-symbol model, implemented by SixJCategory, dagger(f) takes conjugate transposes of the stored matrix blocks. The predicate is_unitary(C) tests the supplied pivotal structure, dimensions, and associator matrices in these stored bases; it does not test the braiding matrices or search for a change of gauge. Exact coefficients in an abstract number field do not by themselves specify complex conjugation or positivity; the current method therefore returns false over $\mathbb Q$, ordinary number fields, and finite fields. Use a chosen complex realization when asking this predicate about characteristic-zero algebraic data. The numerical fusion-category chapter states the precise working-precision interpretation.

Premodular and modular categories

Following Mäurer and Thiel (2024), §2.4, a premodular weak fusion category is a braided spherical weak fusion category. For representatives $X_1,\ldots,X_r$ of its simple objects, the implementation uses the unnormalized entries

\[\label{eq:categorical-s-matrix} S_{ij}=\operatorname{Tr}\!\left( c_{X_i,X_j}\circ c_{X_j,X_i} \right).\]

The displayed endomorphism acts on $X_j\otimes X_i$. By cyclicity of the spherical trace, this is the usual trace of the double braiding on $X_i\otimes X_j$.

It is modular when this matrix is invertible. For split fusion categories over an algebraically closed field of characteristic zero, this is the standard definition (Etingof et al., 2015; Definitions 8.13.1, 8.13.2, and 8.13.4). The weak terminology extends the same matrix condition to non-splitting fields.

For a weak fusion category with non-scalar $\operatorname{End}(\mathbb 1)$, categorical traces take values in that endomorphism field. The current matrix routines require the traces and twists below to be representable by scalars in the declared coefficient field. The generic is_modular predicate imposes the stronger split-fusion hypothesis in any case.

In supported finite models, the package methods use the following conventions:

• smatrix(C) returns the unnormalized trace matrix $S$ in the ordering simples(C);
• normalized_smatrix(C) returns $S/\sqrt{\dim(\mathcal C)}$ and therefore requires a choice of square root;
• tmatrix(C) is the diagonal matrix of twist scalars, and therefore requires each represented twist on the chosen simples to be a scalar multiple of the identity over the coefficient field; and
• the generic is_modular(C) method first requires is_fusion(C), is_braided(C), and is_spherical(C) to report true, and then tests whether smatrix(C) is invertible.

The later Drinfeld-center construction, implemented by CenterCategory, has a construction-specific method. If $\mathcal C$ is a pivotal fusion category and $\dim(\mathcal C)\ne0$, then its Drinfeld center is a premodular weak fusion category, and it is modular when it splits (Mäurer and Thiel, 2024; Theorem 2.1). The method therefore tests that the computed center is fusion and spherical rather than recomputing the determinant of its $S$-matrix.

The split restriction is part of the current public predicate. A non-split weak fusion category may be modular in the mathematical sense above even though is_modular(C) returns false. When smatrix(C) is represented as a matrix over the coefficient field, modularity in the weak sense can instead be tested by checking that its determinant is nonzero. The anyon and CFT bridge compares the package's normalization with common physics conventions.

Positive characteristic

The characteristic can change both semisimplicity and splitting. The example uses the concrete representation model, whose full constructor description appears later in the catalogue:

using TensorCategories, Oscar
G = cyclic_group(3)
C = representation_category(GF(2), G)
@assert is_semisimple(C) && is_weak_fusion(C)
@assert !is_split_semisimple(C)
D = representation_category(GF(3), G)
@assert !is_semisimple(D)
(is_weak_fusion(C), is_weak_fusion(D))

(true, false)

The first case is semisimple by Maschke's theorem. The irreducible factor $x^2+x+1$ over $\mathbb F_2$ gives a two-dimensional simple with endomorphism field $\mathbb F_4$. In characteristic $3$, the same group's representation category is not semisimple.

A fusion category in positive characteristic can have a nonsemisimple center when its global dimension vanishes. Existence of the center alone does not imply modularity or applicability of every center algorithm (Mäurer and Thiel, 2024; Theorem 2.1 and §2.5).

Continue with Grothendieck rings.