Let $H$ be a finite dimensional semisimple Hopf algebra over $\mathbb{C}$. Can one choose a basis $\{h_1, \dots, h_n \}$ of $H$, where $h_1 = 1$, such that if we write $$ \Delta(h_i) = \sum_{1 \leq j,k \leq n} \alpha_{ijk} h_j \otimes h_k $$ for some $\alpha_{ijk} \in \mathbb{C}$, then $\alpha_{ij1} = 0$ for all $i,j$, except $i=1$ (since obviously $\Delta(1) = 1 \otimes 1$)? That is, if we write $\Delta(h_i)$ as the sum of $n$ tensors where the second tensorand is a basis vector, is $h_1$ the only basis vector with a nonzero summand of the form $- \otimes 1$?

If $H = \mathbb{C}G$ is a group algebra, then the basis consisting of grouplike elements obviously has this property. If $G = \mathbb{Z}_p$ is cyclic of order $p$, then $(\mathbb{C} \mathbb{Z}_p)^*$ has this property (and I think this is true for any finite abelian group). Many examples I know of have this property (e.g. the Kac-Palyutkin Hopf algebra). However, I can't tell if this is true for $(\mathbb{C} S_3)^*$, for example.

In general, if we let $G(H)$ denote the set of grouplikes, then it might be sensible to consider a basis consisting of unions of cosets of $G(H)$. However, I haven't been able to get this to work.