Singular

7.7.3.0. center

Usage:

center(D[, N]); D int, N optional int

Return:

ideal, generated by computed elements

Purpose:

computes subalgebra generators of the center up to degree D

Note:

In general, one cannot compute the whole center.
Hence, one has to specify a termination condition via arguments D and/or N.
If D is positive, only central elements up to degree D will be found.
If D is negative, the termination is determined by N only.
If N is given, the computation stops if at least N elements have been found.
Warning: if N is given and bigger than the actual number of generators,
the procedure may not terminate.
Current ordering must be a degree compatible well-ordering.

LIB "central.lib";
ring AA = 0,(x,y,z,t),dp;
matrix D[4][4]=0;
D[1,2]=-z;  D[1,3]=2*x;  D[2,3]=-2*y;
def A = nc_algebra(1,D); setring A; // this algebra is U(sl_2) tensored with K[t]
// find generators of the center of degree <= 3:
ideal Z = center(3);
Z;
==> Z[1]=t
==> Z[2]=4xy+z2-2z
inCenter(Z); // check the result
==> 1
// find at least one generator of the center:
ideal Z2 = center(-1, 1);
Z2;
==> Z2[1]=t
inCenter(Z2); // check the result
==> 1