[ALGTOP-L] self-dual Hopf algebras

Jack Morava jack at math.jhu.edu
Fri Dec 10 13:54:33 EST 2010


There's something funny about these dualities, which I don't
know how to understand: they interchange positive and negative
gradings.

The graded duality algebraic topologists use in the
theory of Hopf algebras corresponds to Cartier duality in
the language of groupschemes: that sends a (representable)
group-valued functor

  G : (k-algs) ---> (ab gps)

to the (representable) group-valued functor

  A |--> G^*(A) = Hom(G,\G_m)(A) := Hom_{ab gps}(G(A),A^x)

but I don't see how the pos/neg grading exchange fits into
this.

If I sound confused, it's only because I am. All these dualities
floating around in homotopy theory [Spanier-Whitehead, Brown-Comenetz, 
Anderson, Gross-Hopkins, Morita/Koszul (tip o'the hat to RC for recent 
conversations), maybe even Langlands?] make my head hurt...


> Peter's email reminded me of something Mahowald pointed out to me
> more than 30 years ago:
>
> The cohomology of BU(n) is isomorphic as graded algebras,
> to the homology of the loop space, \Omega SU(n+1).
>
> Question:  What kind of duality phenomenon (if any) does this represent?
>
> Clearly as n tends to infinity one obtains, with Bott periodicity, the self
> duality of the hopf algebra of H^*(BU) that Peter mentions. But does 
> the situation for finite n have some geometric meaning or explanation?
>
>    -Ralph
>
>> ... self-dual Hopf algebras, in general, may deserve more attention 
>> than they have received...




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