For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta_R)$, such that $$ (G \otimes V)^{\text{co}H} = \lbrace g_{(1)} \otimes v_{(0)} \otimes \pi(g_{(2)})v_{(1)} = g \otimes v \otimes 1 ~ | ~ g \otimes v \in G \otimes V) \rbrace = \lbrace 0 \rbrace. $$


Edit: Ideally, I'd like $G$ to be the coordinate algebra of a Drinfeld--Jimbo quantum group. But I'm interested in other (non-trivial) examples also.


1 Answer 1


Zero comodule!!

If you want something more halloweeny, take $G$ to be a field, $H$ - a Hopf algebra, not a field, $V$ - any simple nontrivial $H$-comodule...

EDIT: as asked, $G$ is any Hopf algebra with a simple non-trivial comodoule, $H=G\otimes G$, the map is $x\mapsto x \otimes 1$, $V$ is a simple nontrivial $1\otimes G$-comodule (and hence $H$-comodule)... Bingo!

  • $\begingroup$ I was hoping for something more like $G$ the coordinate algebra of a Drinfeld--Jimbo quantum group . . . maybe I should have specified this in the question. $\endgroup$ Jul 23, 2012 at 17:12
  • 3
    $\begingroup$ Doc, I will do my best to read your mind next times! I can think of an example at a root of unity of the top of head, but if you want not root of unity, I'd better think again when I am sober:-)) $\endgroup$
    – Bugs Bunny
    Jul 23, 2012 at 20:56
  • $\begingroup$ Yes, I would prefer not at a root of unity. Thanks. $\endgroup$ Jul 23, 2012 at 23:03

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