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 $\Delta(H) \subseteq H \otimes H'$?

  • $\begingroup$ To be pedantic, I guess you need that $\{f,h_i\}$ forms a linearly independent set... $\endgroup$ – Matthew Daws Jun 20 '11 at 19:30
  • $\begingroup$ Yes, I do. Thanks, that's not being pedantic, just clear. $\endgroup$ – Dyke Acland Jun 20 '11 at 20:20
  • 1
    $\begingroup$ 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 $\otimes$. $\endgroup$ – 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 $H\subseteq H'$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.