Does there exist a nontrivial triangular weak Hopf algebra?

Question

Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair ($H,\mathcal{R}$) where $H$ is a WHA and $\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(1)$ is called the universal $R$-matrix of $H$ which satisfies the following conditions: \begin{eqnarray} \Delta^{op}(h)\mathcal{R} &=& \mathcal{R}\Delta(h),\quad\forall h\in H,\nonumber\\ (\mathrm{id} \otimes \Delta)\mathcal{R} &=& \mathcal{R}_{13}\mathcal{R}_{12}, \nonumber\\ (\Delta \otimes \mathrm{id})\mathcal{R} &=& \mathcal{R}_{13}\mathcal{R}_{23}, \end{eqnarray} where $\Delta^{op}=\tau\circ \Delta$ denotes the comultiplication opposite to $\Delta$, and $\mathcal{R}_{12} = \mathcal{R}\otimes 1$, $\mathcal{R}_{23} = 1\otimes \mathcal{R}$. Here $\tau(x\otimes y)=y\otimes x$ denotes the swap operator in $\mathrm{End}(H\otimes H)$. In addition, it is required that there exists $\bar{\mathcal{R}}\in \Delta(1)(H\otimes H)\Delta^{op}(1)$ such that \begin{equation} \mathcal{R}\bar{\mathcal{R}} = \Delta^{op}(1), \qquad \bar{\mathcal{R}}\mathcal{R} = \Delta(1). \end{equation}

A QWHA is called triangular if $$\bar{\mathcal{R}}=\tau(\mathcal{R}).$$

Question. Does there exist a finite-dimensional triangular semisimple weak Hopf algebra that is not a Hopf algebra?

One motivation for this question is that triangular WHAs naturally produce weakly-involutive $R$-matrices defined in my other question: let $\rho$ be a finite dimensional representation of $H$ on a vector space $V$, and define $$R= \Pi\cdot[(\rho\otimes \rho)\mathcal{R}],$$
where $\Pi(v\otimes w)=w\otimes v$ is the swap operator in $\mathrm{End}(V\otimes V)$. Then it is straightforward to check that $R$ is a weakly-involutive $R$-matrix, with $R^2=P=(\rho\otimes \rho)\Delta(1)$.

Answer 1

Not sure whether this is what you want, but take $H=M_n(K)$, the full $n\times n$-matrix algebra over a field $K$. Then $H$ is a WHA with ordinary matrix multiplication and with comultiplication defined on their elementary matrices $E_{ij}$ (having $1$ in entry $(i,j)$ and $0$ elsewhere) by $$\Delta(E_{ij})=E_{ij}\otimes E_{ij},\qquad \epsilon(E_{ij}) = 1, \qquad S(E_{ij})=E_{ji}, \qquad \forall i,j.$$ Then $1=\sum_{i=1}^n E_{ii}$ and $\Delta(1)=\sum_{i=1}^n E_{ii}\otimes E_{ii} \neq 1\otimes 1$. Take $\mathcal{R}=\Delta(1)$, then $(H,\mathcal{R})$ is a finite dimensional triangular semisimple weak Hopf algebra that is not a Hopf algebra.