[GAP Forum] algebra mapping problem

[R.N. Tsai]

Dear Willem and gap-forum,
   
  That's too bad. A simple "substitution rule" or "evaluation map"
  for algebras would be a useful feature to have in GAP. 
   
  BTW, this problem occurs outside the WeylAlg package.

  Here's another simple example with lie algebras:
   
   A:=SimpleLieAlgebra("A",1,Rationals);
 U:=UniversalEnvelopingAlgebra(A);
 g:=GeneratorsOfAlgebraWithOne(U);
 a:=g[1];
 hom:=AlgebraWithOneGeneralMappingByImages(U,U,g,g);
 # This gets stuck
 b:=Image(hom,a);
   
  Regards,
  R.N.

Willem de Graaf <degraaf at science.unitn.it> wrote:
  Dear Tsai,

You noted the following behaviour:

> Read("weylalg.gap");
> A:=WeylAlgebra(Rationals,1);
> g:=GeneratorsOfAlgebra(A);
> h:=[g[1],g[2]-g[3],g[2]+g[3]];
> hom:=AlgebraWithOneGeneralMappingByImages(A,A,g,h);
> a:=g[2];
> # This operation gets stuck
> b:=Image(hom,a);
> 
> Any ideas why this is happening?

As far as I can see the following happens.

Unfortunately, GAP doesn't know how to compute the image in this
case, i.e., there is no special method implemented. So GAP
falls back on a method that is based on computing a basis
of the algebra in question. But because the algebra here is
infinite dimensional, this never terminates. 


Best wishes,

Willem de Graaf



       
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