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7.7.10.0. findInvoDiag
Procedure from library involut.lib (see involut_lib).
- Usage:
- findInvoDiag();
- Return:
- a ring together with a list of pairs L, 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:
- compute homothetic (diagonal) 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
==> // number of vars : 2
==> // block 1 : ordering dp
==> // : names x D
==> // block 2 : ordering C
==> // noncommutative relations:
==> // Dx=xD+1
def X = findInvoDiag();
setring X; // ring with new variables, corresponding to unknown coefficients
X;
==> // coefficients: QQ
==> // number of vars : 2
==> // block 1 : ordering dp
==> // : names a11 a22
==> // block 2 : ordering C
// print matrices, defining linear involutions
print(L[1][2]); // a first matrix: we see it is constant
==> -1,0,
==> 0, 1
print(L[2][2]); // and a second possible matrix; it is constant too
==> 1,0,
==> 0,-1
L; // let us take a look on the whole list
==> [1]:
==> [1]:
==> _[1]=a22-1
==> _[2]=a11+1
==> [2]:
==> _[1,1]=-1
==> _[1,2]=0
==> _[2,1]=0
==> _[2,2]=1
==> [2]:
==> [1]:
==> _[1]=a22+1
==> _[2]=a11-1
==> [2]:
==> _[1,1]=1
==> _[1,2]=0
==> _[2,1]=0
==> _[2,2]=-1
idJ;
==> idJ[1]=a11*a22+1
==> idJ[2]=a11^2-1
==> idJ[3]=a22^2-1
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See also:
findInvo;
involution.
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