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7.7.9.0. findInvo
Procedure from library involut.lib (see involut_lib).
- Usage:
- findInvo();
- Return:
- a ring containing a list L of pairs, where
L[i][1] = ideal; a Groebner Basis of an i-th associated prime,
L[i][2] = matrix, defining a linear map, with entries, reduced with respect to L[i][1]
- Purpose:
- computed the ideal of linear involutions of the basering
- Assume:
- the relations on the algebra are of the form YX = XY + D, that is
the current ring is a G-algebra of Lie type.
- Note:
- for convenience, the full ideal of relations
idJ
and the initial matrix with indeterminates matD are exported in the output ring
Example:
| | LIB "involut.lib";
def a = makeWeyl(1);
setring a; // this algebra is a first Weyl algebra
a;
==> // characteristic : 0
==> // number of vars : 2
==> // block 1 : ordering dp
==> // : names x D
==> // block 2 : ordering C
==> // noncommutative relations:
==> // Dx=xD+1
def X = findInvo();
setring X; // ring with new variables, corr. to unknown coefficients
X;
==> // characteristic : 0
==> // number of vars : 4
==> // block 1 : ordering dp
==> // : names a11 a12 a21 a22
==> // block 2 : ordering C
L;
==> [1]:
==> [1]:
==> _[1]=a11+a22
==> _[2]=a12*a21+a22^2-1
==> [2]:
==> _[1,1]=-a22
==> _[1,2]=a12
==> _[2,1]=a21
==> _[2,2]=a22
// look at the matrix in the new variables, defining the linear involution
print(L[1][2]);
==> -a22,a12,
==> a21, a22
L[1][1]; // where new variables obey these relations
==> _[1]=a11+a22
==> _[2]=a12*a21+a22^2-1
idJ;
==> idJ[1]=-a12*a21+a11*a22+1
==> idJ[2]=a11^2+a12*a21-1
==> idJ[3]=a11*a12+a12*a22
==> idJ[4]=a11*a21+a21*a22
==> idJ[5]=a12*a21+a22^2-1
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See also:
findInvoDiag;
involution.
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