[GAP Forum] universal enveloping algebras

[R.N. Tsai]

Dear gap forum,
   
  I'm having difficulty in manipulating elements of the universal enveloping algebra
of a Lie algebra. This simple example brings out the problem :
   
   Test:=function()local L,UL,g,x,d,t,e,f,h,a1,a2,a3,M,emb;
  L:=SimpleLieAlgebra("A",1,Rationals);
  UL:=UniversalEnvelopingAlgebra(L);
  g:=GeneratorsOfAlgebraWithOne(UL);
  d:=g[1];x:=g[2];t:=g[3];
  a1:=d*x-x*d;Print(" [d,x] ",a1,"\n");
  e:=x+d;f:=x-d;h:=f*e;
  a2:=h*e-e*h;Print(" [h,f] ",a2,"\n");
  a3:=h*f-f*h;Print(" [h,e] ",a3,"\n");
  M:=FreeMagmaRing(Rationals,UL);
  emb:=Embedding(UL,M);
 end;
   
  when executed this gives :
   gap> Read("Test.g");
 gap> Test();
  [d,x] [(1)*x.3]
  [h,f] [(-4)*x.1+(-2)*x.1*x.3+(4)*x.2+(-2)*x.2*x.3]
  [h,e] [(-2)*x.1*x.3+(2)*x.2*x.3]
 Error, this case will eventually be handled by the Vector Enumerator
 which is not available yet called from
 OperationAlgebraHomomorphism( A, [ [ Zero( A ) ] ], OnRight ) called from
 IsomorphismMatrixFLMLOR( A ) called from
 IsFiniteDimensional( V ) called from
 IsFinite( V ) called from
 GeneratorsOfMagma( M ) called from
 ...
 Entering break read-eval-print loop ...
 you can 'quit;' to quit to outer loop, or
 you can 'return;' to continue
 brk> 
   
  If there's another way to map the generators of UL such that :
  x -> x
  d -> d
  t -> 1 
  so that [d,x]=1 and not [d,x]=t, then this would suffice for what I'm doing.
I am running "GAP4, Version 4.4.6 of 02-Sep-2005, i686-pc-cygwin-gcc"
  Thanks for you help.
  R.N.

		
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