Tube algebra

The rank, blocks and dimensions of the center from the tube algebra, for every category, with its checks, timings and the comparison with the exact centers. Its stages follow this description; the checks, the tolerances and the notes on them are at the bottom of the page.

The Ocneanu tube algebra has basis $\tau_{a,x,y,m}$ over admissible quadruples with $N_{xa}^mN_{ay}^m = 1$ , dimension $D = \sum N_{xa}^mN_{ay}^m$ , and product

\tau_{a,x,y,m}\cdot\tau_{a',y,y',m'} = \sum_{b,n}\,(F^{-1})^{xaa'}_{n;mb}\;F^{aya'}_{n;mm'}\;(F^{-1})^{aa'y'}_{n;bm'}\;\tau_{b,x,y',n} .

$\operatorname{Rep}(\operatorname{Tube}(\mathcal{C}))\simeq\mathcal{Z}(\mathcal{C})$ as linear categories. The dimension of the center of the algebra is the rank; the Wedderburn blocks $n_\alpha$ are the dimensions of the simple modules; the quantum dimensions are the weighted traces

d_\alpha = \frac{1}{n_\alpha}\sum_i (e_\alpha)_i \sum_j d_{x_j}\, C_{ij}^{\,j}

of the primitive central idempotents $e_\alpha$ .

Quantity	Status	Why
$\operatorname{rank}(\mathcal{Z}(\mathcal{C}))$	delivered	dimension of the center of the algebra
blocks $(n_\alpha)$	delivered	Wedderburn decomposition
quantum dimensions $(d_\alpha)$	delivered	weighted trace of the idempotents, signed for a Galois conjugate
fusion rules, braiding, $S$ , $T$ , $F$ , $R$ of $\mathcal{Z}(\mathcal{C})$	not a tube quantity	the equivalence with $\mathcal{Z}(\mathcal{C})$ is linear only: $\operatorname{Tube}(\mathcal{C})$ has no comultiplication, so its module category has no tensor product. These come from the ladder.

Stages

Five stages, each taking as input what the previous one returns. Stage 0 collects the three quantities read off the presentation. Checks are shown where they run; a failing check is recorded, never repaired.

Stage 0 — derived data

Inputfusion coefficients

N

F

-matrices

F^{d}_{abc}

Outputduals

a\mapsto a^*

, spherical dimensions

d

, block inverses

F^{-1}

1 $a^* \gets$ the $b$ with $N_{ab}^{0} = 1$
2 $d \gets$ the character of $K(\mathcal{C})$ with $|d_a| = 1/|(F^{a}_{aa^*a})_{00}|$ positive one if it exists
3 $F^{-1} \gets$ inverse of each admissible sub-block of $F$ never $\overline{F}$

Stage 1 — tube basis

Inputfusion coefficients

N

Outputbasis

\mathcal{B}

\operatorname{Tube}(\mathcal{C})

, its size

D

1 $\mathcal{B} \gets$ all quadruples $(a,x,y,m)$ with $N_{xa}^{m}\ge 1$ and $N_{ay}^{m}\ge 1$
2 $D \gets |\mathcal{B}|$

Stage 2 — structure constants

Input

N

F

F^{-1}

, basis

\mathcal{B}

Outputstructure constants

C_{ij}^{\ell}

i,j,\ell\in\mathcal{B}

1 $C \gets 0$
2for each pair $i,j\in\mathcal{B}$ with $y_i = x_j$ :
3for each $b$ with $N_{a_i a_j}^{b}\ge 1$ and $n$ with $N_{x_i b}^{n}\ge 1$ , $N_{b y_j}^{n}\ge 1$ :
4 $\ell \gets (b, x_i, y_j, n)$
5 $f_1 \gets (F^{-1})^{n}_{x_i a_i a_j}[m_i, b]$ , $f_2 \gets F^{n}_{a_i y_i a_j}[m_i, m_j]$ , $f_3 \gets (F^{-1})^{n}_{a_i a_j y_j}[b, m_j]$
6 $C_{ij}^{\ell} \gets C_{ij}^{\ell} + f_1 f_2 f_3$
7check $L_a L_b = L_{a\cdot b}$ for 8 random $a, b$ , relative to $\|C\|_\infty^2$ associativity
8check $\mathbf{1} = \sum_x \tau_{(0,x,x,x)}$ is a two-sided unitunit

Stage 3 — center and rank

Inputstructure constants

C

, tolerance

\varepsilon = 10^{-8}

Outputorthonormal basis

Q

Z(A)

, rank

\rho

1 $L_i \gets$ matrix of left multiplication by $\tau_i$ , $R_i \gets$ matrix of right multiplication by $\tau_i$
2 $\mathbf{M} \gets$ the $D^2\times D$ matrix with blocks $L_i - R_i$ , $i\in\mathcal{B}$
3 $Q \gets$ right singular vectors of $\mathbf{M}$ with singular value $< \varepsilon\,\|\mathbf{M}\|$ null space
4 $\rho \gets$ number of columns of $Q$ $\rho = \operatorname{rank}(\mathcal{Z}(\mathcal{C}))$

Stage 4 — primitive central idempotents

Inputstructure constants

C

, basis

Q

Z(A)

Outputprimitive central idempotents

e_1,\dots,e_\rho

1 $e_1,\dots,e_\rho \gets$ simultaneous diagonalization of $Q^\dagger L_{Q_1} Q,\dots,Q^\dagger L_{Q_\rho} Q$ multiplication by $Z(A)$ on $Z(A)$
2scale each $e_\alpha$ so that $e_\alpha^2 = e_\alpha$
3for $t = 1, 2, 3$ : $e_\alpha \gets 3e_\alpha^2 - 2e_\alpha^3$ for every $\alpha$ Newton on $e^2 = e$ ; $10^{-8}$ to $10^{-16}$
4check $e_\alpha e_\beta = \delta_{\alpha\beta} e_\alpha$ idempotency, orthogonality
5check $\sum_\alpha e_\alpha = \mathbf{1}$ completeness
6check $e_\alpha \tau_i = \tau_i e_\alpha$ for all $i$ centrality

Stage 5 — block dimensions and quantum dimensions

Inputstructure constants

C

, spherical dimensions

d

, idempotents

e_1,\dots,e_\rho

Outputblock dimensions

(n_\alpha)

, quantum dimensions

(d_\alpha)

\mathcal{Z}(\mathcal{C})

1 $w_i \gets \sum_j C_{ij}^{j}$ for every $i$ $w_i = \operatorname{Tr} L_{\tau_i}$
2 $T_i \gets \sum_j d_{x_j}\, C_{ij}^{j}$ for every $i$ trace weighted by the string label of $\tau_j$
3for each $\alpha = 1,\dots,\rho$ :
4 $n_\alpha \gets \sqrt{\sum_i (e_\alpha)_i\, w_i}$ must round to an integer
5 $d_\alpha \gets \frac{1}{n_\alpha}\operatorname{Re}\sum_i (e_\alpha)_i\, T_i$
6check $\sum_\alpha n_\alpha^2 = D$ block_sum_equals_dim
7check $\sum_\alpha d_\alpha^2 = \bigl(\sum_a d_a^2\bigr)^2$ global_dim

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