Ladder algebra

The fusion rules, $F$ - and $R$ -symbols and modular data of the center from the ladder algebra, for every category, with its checks, timings and the comparison with the exact centers. Its stages follow this description; the checks and the notes on the braiding and the modular data are at the bottom of the page.

The weak Hopf algebra of Kitaev-Kong in the formulation of Bai-Zhang (arXiv:2503.06731), spanned by pairs of morphisms $u\otimes s$ with $u: y'\to a\,y\,b$ and $s: a\,x\,b\to x'$ ,

A = A_{\mathcal{C}}^{\mathcal{C}\boxtimes\mathcal{C}^{\mathrm{rev}}} = \bigoplus_{a,b,x,x',y,y'} \mathcal{C}(y',\, a\,y\,b)\otimes\mathcal{C}(a\,x\,b,\, x') .

Unlike the tube algebra it carries a comultiplication $\Delta$ , a counit, an antipode and a quasi-triangular structure $\mathcal{R}$ , and that is what makes its representation category equivalent to the center as a braided tensor category,

\operatorname{Rep}(A)\;\simeq\;\mathcal{Z}(\mathcal{C})\quad\text{braided}.

Stages 1 to 7 give the rank, the blocks and the fusion rules of $\mathcal{Z}(\mathcal{C})$ ; stage 8 gives its $F$ -symbols from Clebsch-Gordan maps, its $R$ -symbols as the Schur scalars of $\mathcal{R}$ , and from those the twists and the $S$ -matrix. Pentagon and hexagon residuals are measured on the symbols as computed; nothing is refined.

Quantity	Status	From
$\operatorname{rank}(\mathcal{Z}(\mathcal{C}))$ , blocks $(n_\alpha)$	delivered	stage 6: the primitive central idempotents of $A$
fusion rules $N_{\alpha\beta}^{\gamma}$ of $\mathcal{Z}(\mathcal{C})$	delivered	stage 7: decomposing $\rho_\alpha\otimes\rho_\beta$ through $\Delta$
$F$ -symbols of $\mathcal{Z}(\mathcal{C})$	delivered	stage 8a: change of basis between Clebsch-Gordan embeddings; pentagon measured
$R$ -symbols of $\mathcal{Z}(\mathcal{C})$	delivered	stage 8b: Schur scalars of the quasi-triangular $\mathcal{R}$ ; hexagon measured
twists $\theta_\alpha$ , $S$ -matrix	delivered	stage 8c: from the fusion rules and the diagonal $R$ -symbols; compared with the exact centers
stage 8 for large $\dim(A)$	stages 1 to 7 only	above the stage-8 size limit the rank, blocks and fusion rules are still computed

Stages

Stage	Builds	Checks
1	basis of A	its dimension against the fusion rules
2	product	associativity, unit
3	comultiplication $\Delta$ , counit	coassociativity, counit axiom, weak bialgebra
4	antipode $S$	counital projections, $\Delta S$ coalgebra homomorphism, $S^2$ group-like
6	primitive central idempotents: rank and blocks	the four idempotent identities, block sums, two block methods agree
7	fusion rules of $\mathcal{Z}(\mathcal{C})$ from $\rho_\alpha\otimes\rho_\beta$ through $\Delta$	commutativity, unit, non-negativity, N-matrices commute
8a	$F$ -symbols: change of basis between the two families of Clebsch-Gordan embeddings $V_\delta\to V_\alpha\otimes V_\beta\otimes V_\gamma$	pentagon, as computed
8b	$R$ -matrix $\mathcal{R}$ of Bai-Zhang Theorem 3.1, then the $R$ -symbols as its Schur scalars	quasi-triangularity against the product and Δ; hexagon with the raw F; monodromies
8c	modular data $(d, \theta, S)$ from the fusion rules and the diagonal $R$ -symbols	$S$ unitary, symmetric, $S^2$ a permutation, vacuum row, $\|\theta\| = 1$ , Verlinde reproduces the stage-7 fusion rules

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