Is it true that if $W$ is a submodule of $V$ then by duality $W^*$ is a sub-co-module of $V^*$? Can anybody confirm that or give an example when this doesn't happen?

I think it might be necessary to the spaces to be semisimple in order to have correspondence, but I didn't really get why.

Finally, would it change something if $W$ was a co-module of $V$ instead of a module?

Thanks in advance!

  • $\begingroup$ You used the tag 'hopf-algebras', so do you mean that $V$ is an $H$-module for some Hopf algebra $H$? For what coalgebra do you want $V^*$ to be a comodule? For $H^*$ to be a coalgebra, $H$ needs to be f.d., I believe. Of course there is also a dual Hopf algebra for infinite dimension, but this depends on the situation you consider and you should consider explaining this context in your question a bit more. In any case, note that an embedding $W \to V$ induces an surjection $V^* \to W^*$, so I suppose, if you should get anything you will most likely get a quotient comodule, not a subcomdule. $\endgroup$ – Matthias Klupsch Aug 3 '18 at 8:28
  • $\begingroup$ I had in mind Hopf Algebras that are duals to each other, such as $U_q(sl(2)$ and $Sl_q(2)$. If the case is semisimple I guess you have a complete correlation between sub-modules and sub-comodules but I didn't get exactly why it works only in the semisimple case $\endgroup$ – Dac0 Aug 3 '18 at 8:33
  • $\begingroup$ Would it change something if $W$ was a co-module of $V$ instead of a module? $\endgroup$ – Dac0 Aug 3 '18 at 8:51
  • $\begingroup$ As I said before if $V \mapsto V^*$ would induce a contravariant equivalence between modules and comodules, then submodules would correspond to quotient comodules. Of course, if every module and comodule is completely reducible (if your Hopf algebras are semisimple and cosemisimple, respectively), then quotients and submodules are the same and you would get your correspondence. I would encourage you to edit your question and explain the context you are considering: What is the definition of "Hopf algebras that are duals to each other"? What further things do you assume of your Hopf algebras? $\endgroup$ – Matthias Klupsch Aug 3 '18 at 9:10

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