[GAP Forum] Algebras

[Mohamed Barakat]

Dear Tsai,

the Weyl algebra and the exterior algebra (including GB computations) are supported through the homalg project’s interface to Singular/Plural/SCA. The clifford algebra can also be supported easily:

gap> LoadPackage( "RingsForHomalg" );
true

If this returns fail then probably because the package io fails to load; in this case please compile io, using ./configure and make

gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := PolynomialRing( Q, "x,y,z" );
Q[x,y,z]
gap> A := RingOfDerivations( R, "Dx,Dy,Dz" );
Q[x,y,z]<Dx,Dy,Dz>

All this and more is documented:

gap> ?RingsForHomalg:Singular

AssignGeneratorVariables does not yet work properly for Weyl algebras. This will be fixed in the next GAP release. For the moment:

gap> x := "x" / A;
x
gap> Dx := "Dx" / A;
Dx
gap> Dx * x;
x*Dx+1

The homalg project has at the moment its own matrix interface:

gap> ?HomalgMatrix

We are planning to comply with the new matrix interface in GAP once it becomes widely used.

gap> mat := HomalgMatrix( "[x*Dy, Dx*x, z*Dx, 1+Dy]", 2, 2, A );
<A 2 x 2 matrix over an external ring>
gap> Display( mat );
x*Dy,x*Dx+1,
z*Dx,Dy+1

See also:

gap> ?MatricesForHomalg:Matrices

Best wishes,

Mohamed