# Image of the Coproduct of a Hopf Algebra

Let $H$ ba a Hopf algebra with coaction $\Delta: H \to H \otimes H$. Denote the action of $\Delta$ by $\Delta (h) = h_{(1)} \otimes h_{(2)}$. I was wondering if every element of $H$ can arise as a $h_{(2)}$, for some $h$. To be more precise, for any $g \in H$, does there exist a $h \in H$, such that $\Delta(h) = f \otimes g + \sum_i h_i \otimes h'_i$, for some set $h'_i$ which is linearily indpt with $g$? Or, alternatively, does there exist $H' \subseteq H$, such that $\Delta(H) \subseteq H \otimes H'$?

• To be pedantic, I guess you need that $\{f,h_i\}$ forms a linearly independent set... – Matthew Daws Jun 20 '11 at 19:30
• Yes, I do. Thanks, that's not being pedantic, just clear. – Dyke Acland Jun 20 '11 at 20:20
• The $h_i$'s need not be independent with $f$ and the $h_i$'s, though. For well-posedness of the question one needs linear independence on one side of the $\otimes$. – Mariano Suárez-Álvarez Jun 20 '11 at 22:26

Counitarity implies that $$(\varepsilon\otimes\mathrm{id}_H\circ\Delta)(H)=H.$$ If $\Delta(H)\subseteq H\otimes H'$ for some subspace $H'\subseteq H$, then this tells us ---since $\varepsilon$ takes scalar values!--- that $H\subseteq H'$.