Commutative case

On $A = K^{D}$ with its basis of orthogonal idempotents, multiplicativity forces each $\cop(e_{i})$ to be a sum of basis tensors $e_{p} \otimes e_{q}$ with disjoint supports, so the coproduct is a partial operation $p \circ q = i$. Coassociativity makes it associative, the weak unit axiom says that if $p \circ q$ is defined then $(p \circ q) \circ r$ is defined exactly when $q \circ r$ is, and the counit and antipode are then unique.

Theorem (Yamanouchi for $C^{*}$-algebras; folklore over any field). Weak Hopf structures on $K^{D}$ are, up to isomorphism, the function algebras $K^{\mathcal{G}}$ of the groupoids $\mathcal{G}$ with $D$ arrows, and $\dim A_{t}$ is the number of objects.

A connected groupoid is $\mathrm{Pair}_{n} \times G$, so with $g(m)$ the number of groups of order $m$:

The catalog holds all 11 111 function algebras for $D \le 20$ and their duals, the groupoid algebras $K[\mathcal{G}]$; 10 722 of the duals are new. The counts of groups match OEIS A000001 and the self-dual entries A018819. The theorem covers split algebras only: $\mathbb{Q}[C_{3}] \cong \mathbb{Q} \times \mathbb{Q}(\zeta_{3})$ is commutative and semisimple but not $\mathbb{Q}^{3}$, and non-split forms are open.

On a split commutative algebra the groupoid is a complete invariant: written in the primitive-idempotent basis, $\Delta$ is its composition table. Two such algebras are isomorphic exactly when their groupoids are.