Identities in an algebra

From: Kevin Buzzard (buzzard_at_imperREMial.aOVEc.uk)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 21:06:51 +0000 (UTC)

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^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

?

It's true if one of a,b is zero and it's true for a,b<=3 (computer
calculation). I tried proving it by hand for a=b=1 and got very bogged
down; I still don't "know why" it's true. On the other hand, for
experts in this sort of associative algebra it might all be easy.
The question is about identites in the free algebra generated
by D,E,F,G, modulo the bi-ideal generated by the 6 relations above.
A computer algebra package like magma can check instances of the
equality e.g.

> P<D,E,F,G>:=FreeAlgebra(Rationals(),4);
> I:=ideal<P|D*E-E*D-F,D*F-F*D-G,D*G-G*D,E*F-F*E,E*G-G*E,F*G-G*F>;
> D*E^2*D^3*E-E*D^3*E^2*D in I;
true

but this isn't giving me any feeling as to what's going on.

Cheers,

Kevin Buzzard

Relevant Pages

  • This Weeks Finds in Mathematical Physics (Week 239)
    ... This Week's Finds in Mathematical Physics ... Urs Schreiber and I have started up a new blog! ... graded-commutative algebra is the same as making L into a Lie algebra. ...
    (sci.math.research)
  • Re: boson calculus and half inter spin? huh?
    ... >> get half integer spin out of combining integer spin components. ... several harmonic oscillators. ... relations as the standard generators of the Lie algebra gl, ... X_ij to generate the Lie algebra of suor many other Lie groups. ...
    (sci.physics.research)
  • Re: Lie groups and Lie alegbras -- some simple questions
    ... >> body rotations in a dynamic system. ... >> disturbed by the mixing of group elements with algebra elements. ... > Then there is the action of G on the Lie algebra LG by ...
    (sci.math)
  • Re: Pontryagin duality (again)
    ... Calling my comments "rude" is rude. ... whether one can speak of Banach algebra or Lie algebra ... "The Campbell-Hausdorff-Baker theorem ...
    (sci.math)
  • Doubly Affine Extension of GL(n) & Its Gauge Theory
    ... product of the Lie algebra for GLwith the Heisenberg algebra H ... on n pairs of generators? ...
    (sci.physics.research)