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7.5.1.0. centerVS
Procedure from library central.lib (see central_lib).
- Usage:
- centerVS( D ); D int
- Return:
- ideal, generated by computed elements
- Purpose:
- computes a vector space basis of the center up to degree D
- Note:
- D must be non-negative
Example:
| LIB "central.lib";
ring A = 0,(x,y,z),dp;
matrix D[3][3]=0;
D[1,2]=-z; D[1,3]=2*x; D[2,3]=-2*y;
nc_algebra(1,D); // this algebra is U(sl_2)
==> // characteristic : 0
==> // number of vars : 3
==> // block 1 : ordering dp
==> // : names x y z
==> // block 2 : ordering C
==> // noncommutative relations: ...
// find a basis of the vector space of all
// central elements of degree <= 4:
ideal Z = centerVS(4);
Z;
==> Z[1]=x
==> Z[2]=y
==> Z[3]=z
==> Z[4]=z2
==> Z[5]=yz
==> Z[6]=y2
==> Z[7]=xz
==> Z[8]=xy
==> Z[9]=x2
==> Z[10]=z3
==> Z[11]=yz2
==> Z[12]=y2z
==> Z[13]=y3
==> Z[14]=xz2
==> Z[15]=xyz
==> Z[16]=xy2
==> Z[17]=x2z
==> Z[18]=x2y
==> Z[19]=x3
==> Z[20]=z4
==> Z[21]=yz3
==> Z[22]=y2z2
==> Z[23]=y3z
==> Z[24]=y4
==> Z[25]=xz3
==> Z[26]=xyz2
==> Z[27]=xy2z
==> Z[28]=xy3
==> Z[29]=x2z2
==> Z[30]=x2yz
==> Z[31]=x2y2
==> Z[32]=x3z
==> Z[33]=x3y
==> Z[34]=x4
// note that the second element is the square of the first
// plus a multiple of the first:
Z[2] - Z[1]^2 + 8*Z[1];
==> -x2+8x+y
inCenter(Z); // check the result
==> 1
| centralizerVS, center, inCenter
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