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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.

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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.

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    $\begingroup$ Note that any finite dimensional Hopf algebra over a field has the space of left integrals as two-sided 1-dimensional ideal. So, if your criterium is true, this would answer the question positively. $\endgroup$ – tj_ Oct 17 at 16:30
  • $\begingroup$ @tj_ I made my answer more definitive, and slightly different. One is left with: does every finite dimensional Hopf algebra over a field have a one dimensional two sided ideal not in the kernel of the augmentation? $\endgroup$ – Nicholas Kuhn Oct 18 at 16:11
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    $\begingroup$ I believe I worked this out, although using a different method. Let $T \in H^*$ be an integral for $H$, and choose a basis $\{ h_1, \dots, h_n \}$ for $H$, where $h_1 = 1$ (possible since $H$ is semisimple) and $T(h_i) = 0$ for $i \neq 1$ (which can be achieved by subtracting a scalar multiple of $1$ from each basis element, if necessary). Then, writing $\Delta(h_i)$ as in my question and calculating $\delta_{h_i} * T$ using the fact that $T$ is an integral, you find that this basis has the desired properties. $\endgroup$ – ren Oct 18 at 16:19
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    $\begingroup$ @Nicholas Kuhn: Cute. I don't know the answer in general, but under the assumption of the OP's question, $H$ is semisimple, which is equivalent to $\epsilon(L)\neq 0$ where $\epsilon$ is the augmentation and $L$ the 1-dim. space of left integrals. So $L$ works. $\endgroup$ – tj_ Oct 18 at 21:39

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