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nLab > Latest Changes: cohomological induction

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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeNov 11th 2012

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2012

    Nice to have this. I have put a link to cohomological induction from induced representation.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeNov 23rd 2012

    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.

    • CommentRowNumber4.
    • CommentAuthorjim_stasheff
    • CommentTimeNov 23rd 2012
    Namely, corepresentation in the Leibniz context has another meaning than coaction. That seems to have been an unfortunate choice of terms - too late to establish an alternate?
    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeNov 23rd 2012

    Well, it is still dual in some sense. What is your proposal for an alternative name ?

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2012

    So from Urs’ remark here, there should ’cohomological coinduction’ too, though not a single Google hit?

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeNov 23rd 2012
    • (edited Nov 23rd 2012)

    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 ?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2012
    • (edited Nov 23rd 2012)

    Zoran,

    the question is: if “cohomological induction” is another term (maybe at least in some situations, as you seem to say in the nnLab 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?

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2012

    Yes that’s the sort of question I meant.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2012

    There is

    • Gunnar Carlsson, Derived Representation Theory and the Algebraic K-theory of Fields (pdf)

    which sounds like it might help, but maybe it doesn’t.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeNov 23rd 2012

    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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2012
    • (edited Nov 23rd 2012)

    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?

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeNov 24th 2012
    • (edited Nov 24th 2012)

    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

    • Dragan Miličić, Pavle Pandžić, Equivariant derived categories, Zuckerman functors and localization, from Geometry and Representation Theory of real and p-adic Lie Groups , J. Tirao, D. Vogan, J.A. Wolf, editors, Progress in Mathematics 158, Birkhäuser, Boston, 1997, 209-242, pdf

    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.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2012
    • (edited Nov 25th 2012)

    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?)

1 to 14 of 14

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