Re: Identities in an algebra

From: Bill Dubuque (wgd_at_nestle.csail.mit.edu)
Date: 02/13/05 

Date: 13 Feb 2005 11:02:29 -0500

Kevin Buzzard <buzzard@imperREMial.aOVEc.uk> wrote:
>
> A student of mine is finishing up writing his thesis; if he can
> prove that a certain diagram commutes then he will be done!
>
> We spent some time yesterday trying to work out what needs to be done.
> The question below isn't equivalent (the original question is about
> p-adic Siegel modular forms), but it suffices to finish the argument.
> The question below is basically equivalent to a messy combinatorial
> identity that nowadays could be checked by computer, but there is
> probably a neater way of finishing the job. Can anyone help?
>
> Let L be the 4-dimensional Lie algebra with basis D,E,F,G, and
> satisfying
>
> [D,E]=F, [D,F]=G, [D,G]=0,
> [E,F]=[E,G]=[F,G]=0
>
> (this is a Lie algebra; for example D could be d/dX
> and E could be X^2 d/dY acting on R[X,Y])
>
> Is it true that for all integers a,b>=0 the following identity
> is true in the universal enveloping algebra of L:
>
> a a+b a+2b b b a+2b a+b a
> D E D E = E D E D

With D = d/dX it suffices to verify the following

      a 2a+2b a+2b 2b 2b a+2b 2a+2b a
     D X D X = X D X D

This is easily verified by evaluating at X^n, i.e.

      (n+a+2b)..(n+2b+1) (n+2b)..(n-a+1) X^(n+2b)
    = (n+a+2b) ... (n+1) n(n-1)..(n-a+1) X^(n+2b)

--Bill Dubuque