The Ising center over two fields

We continue the Ising example, now paying attention to the field of definition. The expected simple objects and splitting phenomenon are described in Mäurer and Thiel (2024), §6.1 and Mäurer (2026), Theorem 2.5.6 and Appendix B.1.

Over $\mathbb Q(\sqrt2)$

using TensorCategories, Oscar
K, r2 = quadratic_field(2)
C = ising_category(K,r2)
Z = center(C)
S = simples(Z)
@assert length(S) == 5
@assert sort(int_dim.(End.(S))) == [1,1,1,2,4]
[(object(X), int_dim(End(X))) for X in S]

5-element Vector{Tuple{SixJObject, Int64}}:
 (𝟙, 1)
 (𝟙, 1)
 (𝟙 ⊕ χ, 1)
 (2⋅χ, 2)
 (4⋅X, 4)

The five underlying objects are $\mathbb 1$, $\mathbb 1$, $\mathbb 1\oplus\chi$, $2\chi$, and $4X$. Their endomorphism algebras have dimensions $1,1,1,2,4$ over $K$. Thus two of the simple central objects are not absolutely simple.

Enumeration uses randomized algebra algorithms. Ordering, matrix bases, and individual coefficients can vary. Select an object by its mathematical property instead of relying on a fixed position:

T = only([T for T in S if int_dim(End(T)) == 2])
@assert is_isomorphic(object(T), C[2] ⊕ C[2])[1]
@assert is_central(T)
@assert all(is_invertible, half_braiding(T))
matrix(half_braiding(T,C[3]))

[ 0   1//2]
[-2      0]

In this case the component on $X$ squares to minus the identity. Its eigenvalues require a square root of $-1$:

h = half_braiding(T,C[3])
@assert h ∘ h == -id(domain(h))
@assert half_braiding(T,one(C)) == id(object(T))
int_dim(End(T))

The component is a matrix over $K$ even though its eigenvalues are not in $K$. A nonzero noninvertible endomorphism cannot already exist in the division algebra $\operatorname{End}(T)$. After scalar extension that algebra can acquire idempotents; their images split the extended object.

Over $\mathbb Q(\zeta_{16})$

Use a specified embedding of $K$ into the cyclotomic field, so the meaning of $\sqrt2$ is unambiguous:

L, z = cyclotomic_field(16)
embedding = hom(K,L,z^2 + z^-2)
CL = extension_of_scalars(C,L; embedding=embedding)
ZL = center(CL)
SL = simples(ZL)
@assert length(SL) == 9
@assert all(T -> int_dim(End(T)) == 1, SL)
@assert is_split_semisimple(ZL)
squared_dimensions = sort([QQ(dim(T)^2) for T in SL])
@assert squared_dimensions == [1,1,1,1,2,2,2,2,4]
@assert sum(squared_dimensions) == 16
squared_dimensions

9-element Vector{QQFieldElem}:
 1
 1
 1
 1
 2
 2
 2
 2
 4

The squared dimensions comprise four entries equal to $1$, four equal to $2$, and one equal to $4$. Their sum is $16=\dim(\mathcal C)^2$. This is the familiar rank-nine center of a split Ising category. Over a field supporting a nondegenerate Ising braiding, the general equivalence $\mathcal Z(\mathcal C)\simeq\mathcal C\boxtimes\mathcal C^{\mathrm{rev}}$ explains the nine simples (Etingof et al., 2015; §8.20). This argument concerns a braided realization after extension; it does not supply a braiding over the original field $K$.

split(Z) offers automatic field search for supported cases, including this example. For number fields its current search uses minimal polynomials in simple endomorphism algebras; it does not handle arbitrary noncommutative division algebras. Here the explicit field $\mathbb Q(\zeta_{16})$ and embedding suffice.

From half-braidings to $F$- and $R$-symbols

D = six_j_category(ZL)
@assert length(simples(D)) == 9
@assert is_braided(D)
@assert pentagon_axiom(D)
@assert hexagon_axiom(D)
F, R = F_symbols(D), R_symbols(D)
(length(F), length(R))

(1296, 100)

The $F$- and $R$-symbols use the decomposition bases chosen during skeletonization, with the matrix conventions and dictionary layout described earlier.

Ordinary SixJCategory coordinates require split simples. Trying to extract such coordinates from the category with five simple objects over $K$ would discard its nontrivial division algebras; scalar extension must come first.

Continue with Relative centers, which retain half-braidings only against a specified tensor subcategory.