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From: Johannes Huebschmann <huebschm@math.univ-lille1.fr>
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Subject: Principal bundles
Date: Sun, 21 Jul 2013 11:47:59 +0200 (CEST)
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Dear Colleagues

Here is a question.

An answer results perhaps from combining Gelfand and Tannaka duality.

Many thanks in advance

Best wishes

Johannes

Let $X$ be an affine variety
with coordinate ring $B$
endowed with an action
$X\times G \to X$ of an affine group $G$,? and let $Y$
denote the quotient, with coordinate ring $A=B^G$.
By a theorem of Oberst \cite{MR0444680},
the projection $p\colon X \to Y$ is an affine principal fiber bundle
if and only if the functor
from $(G,B)$-modules to $A$-modules
which assigns to a? $(G,B)$-module $M$ the invariant subspace $M^G$
is an equivalence of categories.

Is there a similar characterization of a topological
or smooth principal bundle with structure group
a (presumably compact?)? Lie group
in the literature?
?


@article {MR0444680,
      AUTHOR = {Oberst, Ulrich},
       TITLE = {Affine {Q}uotientenschemata nach affinen, algebraischen
                {G}ruppen und induzierte {D}arstellungen},
     JOURNAL = {J. Algebra},
    FJOURNAL = {Journal of Algebra},
      VOLUME = {44},
        YEAR = {1977},
      NUMBER = {2},
       PAGES = {503--538},
        ISSN = {0021-8693},
     MRCLASS = {14L99},
    MRNUMBER = {0444680 (56 \#3030)},
MRREVIEWER = {T. Kambayashi},
}

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