"Nice" bases for finite dimensional semisimple Hopf algebras

Question

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.

Answer 1

Given a basis for $H$ of the sort you are looking for, let $V$ be the linear span of $h_2, \dots, h_n$. Then $V \cap \mathbb C1 = 0$ and $\Delta(V) \subset V \otimes V$. Conversely, given such a $V$, any basis for $V$, together with 1, will do what you ask.

Carefully dualizing, we are asking if the dual Hopf algebra $H^*$ must have a 1-dimensional two sided ideal that is not contained in the kernel of the dual of the unit $\eta^*: H^* \rightarrow \mathbb C$.

Let try this out when $H$ is the dual of a group algebra, so $H^* = \mathbb C G$. Is there such a 1-dimensional ideal? Yes: let $\bar G = \sum_{g \in G}g \in H^* = \mathbb C G$. The span of this element is a one dimensional ideal, and the element has augmentation $|G|$, not 0.