Re: Identities in an algebra

From: Jyrki Lahtonen (lahtonen_at_utu.fi)
Date: 02/14/05

  • Next message: Robert Israel: "Re: C^infinity function c-to-1" 
    Date: Mon, 14 Feb 2005 17:07:49 +0200

    Kevin Buzzard wrote:

    [At the end of the day the real reason I don't want
    > to push this approach through is that the identities above
    > are in some sense a theorem (hopefully) about Sp_4, and I would one day
    > like to prove analogous ones for an arbitrary reductive group, when
    > brute force calculations would be doomed.]

    I was about to suggest that your Lie algebra looks a lot
    like the linear span of the positive (or negative) root
    algebra of type C_2 (or B_2). It's been a while since I
    worked on Lie algebras, but I have a feeling that it
    might be sufficient to verify the identities in question
    in the symmetric powers of the natural representation.
    I.e if the 4-dimensional representation has a basis
    {x,y,u,v}, then the symmetric powers are the homogeneous
    parts of the polynomial algebra k[x,y,u,v], and the
    generators of your Lie algebra act on it via the operators
    D = x*(d/dy)- v*(d/du) and
    E = y*(d/dv)
    (As all the homogeneous parts are finite-dimensional,
    no single such representation will suffice, but together
    they might. All the finite-dimensional simple
    representations appear in the tensor products of these
    guys as direct summands, but at the moment I cannot
    complete this argument/hunch, and prove that the enveloping
    algebra would be faithfully represented by these operators)

    OTOH I dunno, if this helps at all. The remaining identities
    don't appear to be much simpler :(

    I really feel tempted to just fix an ordering of your
    basis, say D>E>F>G, and simply crank out a sufficient number
    of identities to prove your claim:) That won't be pretty,
    but expressing everything with respect to a PBW-basis
    should do the trick.

    If I think of something, I will get back to you.

    Cheers,

    Jyrki Lahtonen, Turku, Finland

  • Next message: Robert Israel: "Re: C^infinity function c-to-1"

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