[GAP Forum] Algebras

[R.N. Tsai]

Dear Max and forum,



>I have used GBNP successfully in the past with e.g. BMW algebras. But of course anything >dealing with Gröbner bases in free associative algebras is bound to hit some snags at some >point... But in the examples you list below, GBNP should easily work... If you tell us what >exactly you tried, perhaps we can give some advice.

That's encouraging. I was expecting this to work too but even with the simplest example it fails. Here's what I tried :

LoadPackage("gbnp");
A := FreeAssociativeAlgebraWithOne(Rationals,"a","b");
a := A.a;
b := A.b;

# all three rels below produce the same error
rels := [a*b-b*a];
rels := [a*b-b*a-One(A)];
rels := [a*b+b*a-One(A)];

K := GP2NPList(rels);                           
G := SGrobner(K);
Display(DimQA(G,2));
PrintNPList(BaseQA(G, 2, 0));

I get the same error for all three cases : 

Error, recursion depth trap (5000)
 in
  if GBNP.LookUpOccurTreePTSLRPos( pol, ROT, false, 1 ) = 0  then
    count := count + countfun( pol, (lvl + 1) ) + 1;
    Unbind( pol[lvl + 1] );
fi; called from 
countfun( pol, lvl + 1 ) called from
countfun( pol, lvl + 1 ) called from
countfun( pol, lvl + 1 ) called from
countfun( pol, lvl + 1 ) called from
countfun( pol, lvl + 1 ) called from

You'd expect at least in the first case ([a*b-b*a]) things would work; but maybe there's
something subtle I missed that breaks things. Hopefully if it is fixed for one it will apply to all 3.

>Note that I'd prefer if this was doable via the "naive" (or "logical"?) approach of taking a >FreeAssociateAlgebra and factoring out an ideal of relations. Indeed, one way to implement that >would be to take GBNP, and writing some wrappers around it. This would be a win for both GAP >(gains functionality) and GBNP (gains a nicer user interface ;-). It might be a task for a >bachelor student, or Google Summer of Code or so...

I know the clifford algebra is finite dimensional and can be defined through structure constants. The Weyl algebra can also defined as the universal enveloping algebra of a finite dimensional (Heisenberg) algebra with a further mapping of the center to 1. But I prefer the
"naive" approach since it deals with clifford and weyl on the same footing.

>Finally, here is my code for clifford algebras (this mailing list doesn't allow attachments, so >I am putting it inline). The line you need to change is marked with a comment.


Thanks for the code. I will use this if it turns out the naive approach is a dead end.


R.N.