[GAP Forum] Quotient of an Algebra by its Radical

[Max Horn]

Dear Marc,


On 14.08.2012, at 18:12, Mbg Nimda wrote:

> Dear Membres,
> 
> I have constructed an algebra A, starting by defining a set of matrices
> M,N, and P, with  coefficients in the rationals. I used following commands:
> *
> *
> *A:=Algebra(Rationals, [M,N,P]);*
> <algebra over Rationals, with 3 generators>
> *Dimension(A);inA,*
> 24
> *R:=RadicalOfAlgebra(A);*
> <algebra of dimension 18 over Rationals>
> *Q:=A/R;*
> <algebra of dimension 6 over Rationals>
> *gQ:=GeneratorsOfAlgebra(Q);*
> [ v.1, v.2, v.3, v.4, v.5, v.6 ]
> 
> I would like to know if it is possible to either express the generators as
> representatives or to construct some homomorphism of algebras from A:->Q.

Please try the following code:

gap> m:= [ [ 0, 1, 2 ], [ 0, 0, 3 ], [ 0, 0, 0 ] ];;
gap> A:= AlgebraWithOneByGenerators( Rationals, [ m ] );
<algebra-with-one over Rationals, with 1 generators>
gap> Dimension(A);
3
gap> R:=RadicalOfAlgebra(A);
<algebra of dimension 2 over Rationals>
gap> Dimension(R);
2
gap> hom:=NaturalHomomorphismByIdeal(A,R);;

# Now we can compute Q as image of hom, and also lift elements from Q back to A.

gap> Q:=Image(hom);
<algebra of dimension 1 over Rationals>
gap> gQ:=GeneratorsOfAlgebra(Q);
[ v.1 ]
gap> PreImagesRepresentative(hom, gQ[1]);
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]


Hope that helps,
Max


> 
> Thanks,
> 
> Marc Bogaerts
> _______________________________________________
> Forum mailing list
> Forum at mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>