##### 7.5.8.0. 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;
    ==> // coefficients: QQ considered as a field
    ==> // 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;
    ==> // coefficients: QQ considered as a field
    ==> // 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

See also: findInvoDiag; involution.

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