Re: Identities in an algebra

From: edgar (edgar_at_dizzy.math.ohio-state.edu)
Date: 02/16/05 

Date: 16 Feb 2005 11:08:13 -0500

Kevin Buzzard wrote:

>
> 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^2d/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:
>
> D^a E^(a+b) D^(a+2b) E^b = E^b D^(a+2*b) E^(a+b) D^a
>
> ?

As already noted this Lie algebra is isomorphic to the "negative" part
of the Lie algebra of type B2. If I am not mistaken the identity can be
proved by using the theory of Verma modules.
Let V(a,b) be the Verma module with highest weight (a,b) and
set \lambda = (a,b) - (2a+2b)\alpha - (a+2b)\beta. Here \alpha and
\beta are the simple roots and D, E correspond to \alpha and \beta
respectively.
Then both elements above give the embedding V(\lambda) --> V(a,b) (in
both cases the embedding is the composition of four embeddings). By a
theorem of Bernstein-Gelfand-Gelfand, and Verma this implies that both
elements differ by a scalar multiple. However, if one writes the
elements on a PBW-basis then the monomials that only contain D and E
in both cases will have coefficient 1. Therefore the sclalar multiple
has to be 1.
If I remember correctly, this is essentially the same as an argument
contained in

@article {MR0218417,
     AUTHOR = {Verma, Daya-Nand},
      TITLE = {Structure of certain induced representations of complex
               semisimple {L}ie algebras},
    JOURNAL = {Bull. Amer. Math. Soc.},
     VOLUME = {74},
       YEAR = {1968},
      PAGES = {160--166},
    MRCLASS = {17.30},
   MRNUMBER = {MR0218417 (36 \#1503)},
MRREVIEWER = {D. J. Winter},
}

I don't have the paper here to check, but I think that the above
identity is also stated there.

I hope this makes sense.

Best wishes,

Willem de Graaf