Base fields and exact computation

For a linear category, the coefficient field or ring is part of the input. It specifies the available scalars, how equality is decided, and which algebra algorithms can be used. The interface calls it the base_ring, even when the mathematics requires a field. A method accepting ::Ring does not imply that its algorithm works over every ring.

Exact scalars and number fields

Ordinary division of Julia integers produces floating-point numbers. For exact rational arithmetic, work in QQ:

julia> a = QQ(1)/3;

julia> 3*a == 1
true

julia> parent(a) == QQ
true

A number field is a finite extension of $\mathbb Q$. OSCAR presents it by an algebraic generator and a polynomial relation. For example, to use a square root of $2$ exactly, construct $K=\mathbb Q(s)$ with $s^2=2$. The supplied Ising constructor is used here only to show that a category retains this field; its categorical operations appear in A first computation.

using TensorCategories, Oscar
K, s = quadratic_field(2)
@assert s^2 == 2
C = ising_category(K, s)
@assert base_ring(C) == K
base_ring(C)

Real quadratic field defined by x^2 - 2

The element $s$ is an algebraic generator; inside the abstract field it is not the positive square root. That distinction requires an embedding into the complex numbers.

Useful coefficient domains include QQ, number fields, algebraic_closure(QQ) for algebraic numbers, and GF(p) for a prime field. The field returned as the first component of abelian_closure(QQ) contains the abelian algebraic extensions, not all algebraic numbers. The second component constructs its distinguished roots of unity. Support for an operation can be narrower than this list.

Algebraic closure and fields of definition

The usual characteristic-zero theory of fusion categories works over an algebraically closed field $k$. In particular, the endomorphism algebra of every simple object is then $k$, so semisimple categories satisfy the split condition defined below. Etingof et al. (2015), §4.16 use this setting for most of their treatment and discuss arbitrary fields separately.

OSCAR can work exactly over the algebraic closure $\overline{\mathbb Q}$:

julia> Qbar = algebraic_closure(QQ)
Algebraic closure of rational field

This is an exact field of algebraic numbers, not a floating-point model of $\mathbb C$. For a multifusion category over an algebraically closed field of characteristic zero, it is large enough in principle: the category descends to an algebraic number field (Etingof et al., 2015; Corollary 9.1.8).

For computations, however, it is usually preferable to retain a reasonably small number field containing the structural coefficients. This keeps the field of definition visible and generally gives smaller exact linear-algebra problems. It also exposes the different complex realizations of the same algebraic data. The algebraic closure remains useful when roots must be chosen or objects must be split, but not every package algorithm supports every exact field equally well.

Complex embeddings of number fields

An abstract number field does not by itself choose a copy inside $\mathbb C$. A complex embedding $\iota\colon K\hookrightarrow\mathbb C$ chooses a complex root of the defining polynomial as the image of the generator. Thus $\mathbb Q(s)$ with $s^2=2$ has two complex embeddings, sending $s$ to $\sqrt2$ and $-\sqrt2$. OSCAR returns their values as certified complex balls; the predicate overlaps tests whether two such enclosures intersect. It enumerates the embeddings with complex_embeddings:

embeddings = complex_embeddings(K)
@assert length(embeddings) == 2
@assert overlaps(embeddings[1](s), -embeddings[2](s))
length(embeddings)

2

The exact field element $s$ and its value $\iota(s)$ play different roles. TensorCategories.jl stores structural coefficients in the exact field. A complex embedding selects their numerical realization. This choice can determine which realization is unitary and which signs or phases appear in $F$- and $R$-symbols.

K(3) constructs a scalar in K. Independently constructed isomorphic fields do not automatically identify their chosen roots. Specify the intended embedding when extending scalars.

The schematic call extension_of_scalars(C, L; embedding=iota) uses a chosen map iota from the old field into L and applies it to every coefficient of the structural maps. The concrete construction of L and iota depends on the coefficient fields. An embedding, a change of basis, and a change of pivotal structure are different operations.

Galois conjugation

Let the structural coefficients lie in a number field $K$. Applying a field embedding to every coefficient preserves the polynomial pentagon and hexagon equations. The resulting solution is called a Galois conjugate. Its fusion multiplicities are unchanged, while its embedded $F$- and $R$-symbols, pivotal dimensions and twists when present, and unitarity properties can change. Different embeddings become restrictions of automorphisms after passing to a normal closure; the field $K$ itself need not be Galois. Thus Galois conjugation is more general than ordinary complex conjugation. Galois-conjugate data need not define equivalent complex fusion categories. This coefficientwise action is used in the proof of Etingof et al. (2015), Proposition 9.6.5.

The standard rank-two example is the pair of Fibonacci and Yang–Lee realizations: they have the same fusion rule $\tau\otimes\tau=\mathbb 1\oplus\tau$, but the two roots of the defining quadratic equation give a unitary realization and its nonunitary Galois conjugate (Rowell et al., 2009; pp. 3–4). The Fibonacci catalogue entry shows how this choice appears in the package.

These operations are distinct: choosing an embedding selects a conjugate realization of the coefficients, whereas enlarging the coefficient field can also create new direct-sum decompositions of objects.

Splitting over the chosen field

For a simple object $S$, Schur's lemma says that $D_S=\operatorname{End}_{\mathcal C}(S)$ is a division algebra. The Hom-finiteness assumption makes $D_S$ finite-dimensional over the base field $k$. In the finite semisimple setting, split means that the canonical map $k\to D_S$ is an isomorphism for every simple $S$. This is automatic over an algebraically closed field, but not over a number field. The familiar identification of simple endomorphisms with scalars therefore requires the split hypothesis (Etingof et al., 2015; §4.16, pp. 87–88).

A category can be split over a number field even though that field is not algebraically closed. When it is not split, a simple object can decompose after a suitable scalar extension. The division algebras $D_S$ are the main source of the additional phenomena over non-algebraically closed fields (Sanford, 2025).

The distinction is computationally important. The scalar $F$-symbol model assumes split simples and chosen bases of their fusion spaces. The general category interface can also represent non-split categories, but multiplicities, decomposition, scalar extension, and center computations must then retain the simple endomorphism algebras. Treating them as copies of the base field gives wrong multiplicities.

The Ising category above is split, but its center need not split over the same field. This is a result about that field, not an incomplete center calculation. The center tutorial follows scalar extension and splitting. The terminology used by the package is summarized under Fusion categories and splitting. Its section on scalar extension and algorithmic splitting explains how the package passes to a common splitting field and obtains the new simple objects from idempotents in the extended endomorphism algebras. The terminology follows Mäurer and Thiel (2024), §2.1. The center algorithm and the subsequent splitting of central objects are developed in (Mäurer and Thiel, 2024; §§4–5).

TensorCategories.jl also supports coefficient fields of positive characteristic, such as GF(p). This is an important feature: the characteristic is part of the mathematical input and can change semisimplicity, splitting, and the behavior of categorical constructions. In positive characteristic a category can fail to be semisimple, independently of whether its simple objects are split. Enlarging the coefficient field within the same characteristic does not repair such a failure of semisimplicity. The characteristic example later in the manual shows both phenomena for representations of a cyclic group.

Continue with Numerical computations.