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7.5.1.0. linearMapKernel
Procedure from library central.lib (see central_lib).

Usage:
linearMapKernel( Images ); Images ideal

Purpose:
Computes the syzygy module of the linear map given by Images.

Return:
syzygy module, or int(0) if all images are zeroes

Example:
 
LIB "central.lib";
ring A = 0,(e,f,h),dp;
matrix D[3][3]=0;
D[1,2]=-h;  D[1,3]=2*e;  D[2,3]=-2*f;
nc_algebra(1,D); // this algebra is U(sl_2)
==> //   characteristic : 0
==> //   number of vars : 3
==> //        block   1 : ordering dp
==> //                  : names    e f h 
==> //        block   2 : ordering C
==> //   noncommutative relations: ...
// Let us consider the linear map Ad_{e} from A_2 into A.
// Compute the PBW basis of A_2:
ideal Basis = PBW_maxDeg( 2 ); Basis;
==> Basis[1]=e
==> Basis[2]=f
==> Basis[3]=h
==> Basis[4]=h2
==> Basis[5]=fh
==> Basis[6]=f2
==> Basis[7]=eh
==> Basis[8]=ef
==> Basis[9]=e2
// Compute images of basis elements under the linear map Ad_e:
ideal Image = applyAdF( Basis, e ); Image;
==> Image[1]=0
==> Image[2]=0
==> Image[3]=0
==> Image[4]=0
==> Image[5]=0
==> Image[6]=0
==> Image[7]=0
==> Image[8]=0
==> Image[9]=0
// Now we have a linear map given by: Basis_i --> Image_i
// Let's compute its kernel K:
// 1. compute syzygy module C:
module C = linearMapKernel( Image ); C;
==> C[1]=0
// 2. compute corresponding combinations of basis vectors:
ideal K = linearCombinations(Basis, C); K;
==> K[1]=0
// Let's check that Ad_e(K) is zero:
ideal Z = applyAdF( K, e ); Z;
==> Z[1]=0
// Now linearMapKernel will return a single integer 0:
def CC  = linearMapKernel(Z); typeof(CC); CC;
==> int
==> 0
applyAdF, linearMapKernel


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