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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 $$ (G \otimes V)^{\text{co}H} = \lbrace g_{(1)} \otimes v_{(0)} \otimes \pi(g_{(2)})v_{(1)} = g \otimes v \otimes 1 ~ | ~ g \otimes v \in G \otimes V) \rbrace = \lbrace 0 \rbrace. $$

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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.

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up vote 2 down vote accepted

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!

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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. –  Dyke Acland Jul 23 '12 at 17:12
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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:-)) –  Bugs Bunny Jul 23 '12 at 20:56
    
Yes, I would prefer not at a root of unity. Thanks. –  Dyke Acland Jul 23 '12 at 23:03
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