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Let $k$ be a field, and let $S$ be a nonempty set. Let $k[S]$ be the grouplike coalgebra of $S$ over $k$, i.e. the free vector space with basis $S$ equipped with the coproduct $\Delta(s)=s\otimes s$ for all $s\in S$. (If $S$ is a group, then $k[S]$ is the usual group algebra of $S$ - which is also a Hopf algebra.) Let $I$ be a coideal in $k[S]$. Then, the coideal $I$ is spanned by differences $s-s'$ for some $s,s'$ in $S$.

This is Exercise 2.1.26(b) in Radford's Hopf algebras. It is clear that differences $s-s'$ for some $s,s'$ in $S$ always span a coideal in $k[S]$. But how to prove the other direction, that every coideal in $k[S]$ is of this form?

It should be simple because the other exercises in this book are not too hard. I was thinking that for a coideal $I$ in $k[S]$, we have $I\subseteq\bigoplus_{\text{for some }s\neq s'}k(s-s')$, where each summand $k(s-s')$ is a simple coideal, and conclude from this...

Another thought: As in other exercises in this book, I think the rank of $\Delta(c)$ where $c\in I$ plays a role, possibly of $c$ written as a sum with the least possible amount of nonzero summands $\lambda_{s,s'}(s-s')$ for some $s\neq s'$ ($\lambda_{s,s'}\in k$).

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    $\begingroup$ Does the definition of coideal you are using include $\epsilon(I) = 0$? $\endgroup$ – Joppy Apr 5 '20 at 1:18
  • $\begingroup$ Yes, it does: $\epsilon(I)=0$ and $\Delta(I)\subseteq I\otimes H+H\otimes I$. $\endgroup$ – user213008 Apr 6 '20 at 3:15
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    $\begingroup$ Doesn't that give you the answer then? By the $\epsilon(I) = 0$ condition, the coideal $I$ must be spanned by elements of the form $\sum_{s \in S} a_s s$ where $\sum_s a_s = 0$. You can always rewrite a sum of that form into a linear combination of differences $s - s'$. $\endgroup$ – Joppy Apr 6 '20 at 3:25
  • $\begingroup$ You can but that doesn't mean it's spanned by differences just contained in such a span. $\endgroup$ – user213008 Apr 6 '20 at 9:41
  • $\begingroup$ How can you know that every nonzero summand is as well contained in $I$? $\endgroup$ – user213008 Apr 6 '20 at 9:44

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