# Coideals in the grouplike colagebra are spanned by differences

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 $$s\neq s'$$ ($$\lambda_{s,s'}\in k$$).

• Does the definition of coideal you are using include $\epsilon(I) = 0$? – Joppy Apr 5 '20 at 1:18
• Yes, it does: $\epsilon(I)=0$ and $\Delta(I)\subseteq I\otimes H+H\otimes I$. – user213008 Apr 6 '20 at 3:15
• 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 $s - s'$. – Joppy Apr 6 '20 at 3:25
• You can but that doesn't mean it's spanned by differences just contained in such a span. – user213008 Apr 6 '20 at 9:41
• How can you know that every nonzero summand is as well contained in $I$? – user213008 Apr 6 '20 at 9:44