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Idea section for a new entry cohomological induction and a new stub induced comodule. I have separated corepresentation from comodule&coaction. Sometimes corepresentation is the same as coaction, sometimes there are small differences (defined on dense subspaces etc.) but more important, there is a different notion of corepresentation in Leibniz algebra theory, which will be explained in a separate section later.
A remark at induction.
Nice to have this. I have put a link to cohomological induction from induced representation.
I have now expanded the above mentioned entry corepresentation by a material from Leibniz algebra. Namely, corepresentation in the Leibniz context has another meaning than coaction.
Well, it is still dual in some sense. What is your proposal for an alternative name ?
So from Urs’ remark here, there should ’cohomological coinduction’ too, though not a single Google hit?
What cohomological coinduction has to do with a left-right dual notion of certain notion which is in Leibniz algebras called representation and involves both left and right actions of a sort ?? Or this is an independent remark not intended to be related with the discussion on the corepresentation terminology in the Leihniz algebra context ?
Zoran,
the question is: if “cohomological induction” is another term (maybe at least in some situations, as you seem to say in the $n$Lab entry) for “derived functor of the induction functor on representations”, then: what is the corresponding term for the derived functor of the coinduction functor on representations?
Yes that’s the sort of question I meant.
There is
which sounds like it might help, but maybe it doesn’t.
Rosenberg worked much on the derived representation theory, in quite categorical language but the main part of this work is not yet publically available yet, hopefully the main part will be out next year. Cohomological induction is not quite precisely another term for the derived functor of induction, but morally it is so.
morally it is so.
I browsed through some GoogleBooks book on cohomological induction, trying to find that statement. Can you point me to a specific page with the relevant information?
I do not know off hand a reference for your purpose, but will think of it. I'll try to ask Pavle at some point. I don't believe you will find it in Knapp-Vogan. Look instead maybe into
already the first page has a statement of that form with details later. Of course, there are so many specifics about the representations of real reductive groups in place there. Look at Theorem 1.13 as well.
Thanks, Zoran, thant helps!
In fact, that also answers David’s question in #6, in a way: as that article says on its page 1, the Zuckermann functor is indeed the derived co-induction functor, instead of the derived induction functor!
(Hm, only that these authors don’t seem to use the term “cohomological induction”, do they?)
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