I need examples (the more the better, even better if there is a systematic way of construction) of (weak-)bialgebras or (weak-)Hopf algebras $H$ with a finite dimensional representation $\rho$ and a finite dimensional corepresentation $v_{ij}$ such that
(1). The subalgebra generated by $v_{ij}$ has polynomial growth rate, i.e., the dimensional of the space of degree $n$ polynomials in $\{v_{ij}\}$ grows at most polynomially in $n$;
(2). In case $H$ is a Hopf algebra, I want it to be non-involutive, i.e., the antipode satisfies $S^2\neq id$;
(3). I want the tensor $\rho(v_{ij})$ to be unitary, i.e. $\sum_k \rho(v_{ki})^\dagger \rho(v_{kj})=\sum_k \rho(v_{ik})\rho(v_{jk})^\dagger=I~\delta_{ij}$, where $I$ is the identity matrix, and $\dagger$ denotes the Hermitian conjugate.
Note that $H$ trivially satisfies condition (1) if it is finite dimensional. But, according to a theorem of Larson and Radford, (2) rules out all finite dimensional semisimple Hopf algebras, of which finite dimensional $\mathbb{C}^*$-Hopf algebras is a special case. The compact quantum group $SU_q(2)$ with $q\in\mathbb{R}$ with fundamental 2-dimensional corepresentation $v_{ij}$ satisfies both (1) and (2), and almost (3) except that the representation $\rho$ is infinite-dimensional [the algebra $SU_q(2)$ has no finite dimensional representation].
Is there an example satisfying all above conditions (1),(2) and (3)?
