$\mathrm{Mat}_{2}$

Theorem. $\mathrm{Mat}_{2}(K)$ carries exactly one weak bialgebra structure up to isomorphism, that of the pair groupoid on two objects.

Written directly the axioms are 800 cubic equations in 68 unknowns, and no Gröbner basis finished. Multiplicativity makes $\Delta$ an algebra map $\mathrm{Mat}_{n} \to \mathrm{Mat}_{n^{2}}$, classified up to conjugacy by a multiplicity $k \in \{0, \dots, n\}$; $k = 0$ has no counit, and $k = n$ would make $\mathrm{Mat}_{n}$ a bialgebra, which is impossible for $n \ge 2$. What remains is

20 unknowns and 548 equations; the Gröbner basis takes 19 s (51 s over $\mathbb{Q}$). The solution variety has dimension 4, the gauge freedom; primary decomposition gives one component; the pair groupoid is a smooth point with a 4-dimensional orbit.

On the solution variety $\pi$ is an associative product and $\iota$ a coassociative coproduct on $K^{n}$ satisfying the Frobenius compatibilities, which come from the weak unit axiom. The uniqueness itself follows from the counital subalgebras: $A_{t}$ is a unital subalgebra of $\mathrm{Mat}_{2}$ with $A_{s}$, of the same dimension, in its commutant, and only the diagonal leaves $\Delta(1)$ other than $0$ or $1 \otimes 1$. Conversely, every separable algebra checked, commutative or not, is the $A_{t}$ of such a structure.

Weak Hopf structures on $\mathrm{Mat}_{n}$. For any weak Hopf structure on $\mathrm{Mat}_{n}(K)$, $K$ algebraically closed of characteristic 0, the counital subalgebra has dimension $n$, and if it is commutative the structure is the pair groupoid. So the structure is unique exactly when $n \le 3$. Every semisimple algebra of dimension $n$ occurs as $H_{t}$ (Nikshych–Vainerman), and for $n = m^{2} \ge 4$ there are infinitely many structures, uncountably many over $\mathbb{C}$.