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About: About this document fullSerreRelations

Procedure from library fpalgebras.lib (see fpalgebras_lib).

fullSerreRelations(A,N,C,P,d); A an intmat, N,C,P ideals, d an int

ring (and ideal)

compute the inhomogeneous Serre's relations associated to A in given variable names

three ideals in the input are of the same sizes and contain merely variables which are interpreted as follows: N resp. P stand for negative resp. positive roots, C stand for Cartan elements. d is the degree bound for letterplace ring, which will be returned.
The matrix A is a generalized Cartan matrix with integer entries The result is the ideal called 'fsRel' in the returned ring.

LIB "fpalgebras.lib";
intmat A[2][2] =
2, -1,
-1, 2; // A_2 = sl_3 Cartan matrix
ring r = 0,(f1,f2,h1,h2,e1,e2),dp;
ideal negroots = f1,f2; ideal cartans = h1,h2; ideal posroots = e1,e2;
int uptodeg = 5;
def RS = fullSerreRelations(A,negroots,cartans,posroots,uptodeg);
setring RS; fsRel;
==> fsRel[1]=f2*f2*f1-2*f2*f1*f2+f1*f2*f2
==> fsRel[2]=f2*f1*f1-2*f1*f2*f1+f1*f1*f2
==> fsRel[3]=e2*e2*e1-2*e2*e1*e2+e1*e2*e2
==> fsRel[4]=e2*e1*e1-2*e1*e2*e1+e1*e1*e2
==> fsRel[5]=e1*f2-f2*e1
==> fsRel[6]=e2*f1-f1*e2
==> fsRel[7]=e1*f1-f1*e1-h1
==> fsRel[8]=e2*f2-f2*e2-h2
==> fsRel[9]=h2*h1-h1*h2
==> fsRel[10]=e1*h1-h1*e1+2*e1
==> fsRel[11]=h1*f1-f1*h1+2*f1
==> fsRel[12]=e2*h1-h1*e2-e2
==> fsRel[13]=h1*f2-f2*h1-f2
==> fsRel[14]=e1*h2-h2*e1-e1
==> fsRel[15]=h2*f1-f1*h2-f1
==> fsRel[16]=e2*h2-h2*e2+2*e2
==> fsRel[17]=h2*f2-f2*h2+2*f2