generic derivation : delta:=
mat((xi(1,1),xi(1,2),0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),xi(2,2)
 + xi(1,1),0,0,0),(xi(4,1),xi(4,2),xi(3,2),xi(2,2) + 2*xi(1,1),xi(1,2),xi(4,6)),
(xi(5,1),xi(5,2), - xi(3,1),xi(2,1),2*xi(2,2) + xi(1,1),xi(5,6)),(xi(6,1),xi(6,2
),0,0,0,xi(6,6)))$
$


        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  1  0  0  0  0]
adx1 := [                ]
        [0  0  1  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
adx2 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  1  0  0  0]
        [                 ]
        [0   0  0  0  0  0]



        [0   0   0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]
adx3 := [                  ]
        [-1  0   0  0  0  0]
        [                  ]
        [0   -1  0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]


The generic nilpotent derivation : the eigenvalues are 0

xi(6,6):=0

And the matrix A:=(xi(1,1),xi(1,2)),(xi(2,1),xi(2,2)) is nilpotent

We hence get 2 cases acoording to whether A neq 0 or A=0.

We consider here the case 1 where A neq 0.

In that case, one may suppose A:=((0,1),(0,0)).

xi(1,1):=0,xi(1,2):=1,xi(2,1):=0,xi(2,2):=0

by subtracting adjoints one then may suppose:

xi(3,1):=0,xi(3,2):=0,xi(4,1):=0

delta:=

[   0        1     0  0  0     0   ]
[                                  ]
[   0        0     0  0  0     0   ]
[                                  ]
[   0        0     0  0  0     0   ]
[                                  ]
[   0     xi(4,2)  0  0  1  xi(4,6)]
[                                  ]
[xi(5,1)  xi(5,2)  0  0  0  xi(5,6)]
[                                  ]
[xi(6,1)  xi(6,2)  0  0  0     0   ]


We denote this delta by the shortform

shortformdelta:={xi(4,2),

                 xi(4,6),

                 ss,

                 xi(5,1),

                 xi(5,2),

                 xi(5,6),

                 ss,

                 xi(6,1),

                 xi(6,2)}

paramindexeslist:={{4,2},{4,6},{5,1},{5,2},{5,6},{6,1},{6,2}}

a neq {}$

a:=a$

delta:=
mat((0,1,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,1,0),(1,0,0,0,0,a),(1,0,0
,0,0,0))$
$

shortformdelta:={0,0,ss,1,0,a,ss,1,0}$

on resout l'equation {{0,1},0} qui est  maintenant AA:= - (d(0,6) + d(0,5))$

Unknowns: {d(0,6),d(0,5)}
Unknowns: {d(0,6),d(0,5)}
bonne inconnue W:=d(0,6)$

sa valeur doit etre WW:= - d(0,5)$

on resout l'equation {{0,1},1} qui est  maintenant AA:=d(2,1) - d(1,6) - d(1,5)$

Unknowns: {d(2,1),d(1,6),d(1,5)}
Unknowns: {d(2,1),d(1,6),d(1,5)}
bonne inconnue W:=d(2,1)$

sa valeur doit etre WW:=d(1,6) + d(1,5)$

on resout l'equation {{0,1},2} qui est  maintenant AA:= - (d(2,6) + d(2,5))$

Unknowns: {d(2,6),d(2,5)}
Unknowns: {d(2,6),d(2,5)}
bonne inconnue W:=d(2,6)$

sa valeur doit etre WW:= - d(2,5)$

on resout l'equation {{0,1},3} qui est  maintenant AA:= - (d(3,6) + d(3,5) + d(2
,0))$

Unknowns: {d(3,6),d(3,5),d(2,0)}
Unknowns: {d(3,6),d(3,5),d(2,0)}
bonne inconnue W:=d(3,6)$

sa valeur doit etre WW:= - (d(3,5) + d(2,0))$

on resout l'equation {{0,1},4} qui est  maintenant AA:=d(5,1) - d(4,6) - d(4,5) 
- d(3,0)$

Unknowns: {d(5,1),d(4,6),d(4,5),d(3,0)}
Unknowns: {d(5,1),d(4,6),d(4,5),d(3,0)}
bonne inconnue W:=d(5,1)$

sa valeur doit etre WW:=d(4,6) + d(4,5) + d(3,0)$

on resout l'equation {{0,1},5} qui est  maintenant AA:=d(6,1)*a - d(5,6) - d(5,5
) + d(1,1) + d(0,0)$

Unknowns: {d(6,1),d(5,6),d(5,5),d(1,1),d(0,0),a}
Unknowns: {d(6,1),d(5,6),d(5,5),d(1,1),d(0,0),a}
bonne inconnue W:=d(5,6)$

sa valeur doit etre WW:=d(6,1)*a - d(5,5) + d(1,1) + d(0,0)$

on resout l'equation {{0,1},6} qui est  maintenant AA:= - d(6,6) - d(6,5) + d(1,
1) + d(0,0)$

Unknowns: {d(6,6),d(6,5),d(1,1),d(0,0)}
Unknowns: {d(6,6),d(6,5),d(1,1),d(0,0)}
bonne inconnue W:=d(6,6)$

sa valeur doit etre WW:= - d(6,5) + d(1,1) + d(0,0)$

on resout l'equation {{0,2},0} qui est  maintenant AA:= - d(0,1)$

Unknown: d(0,1)
Unknown: d(0,1)
bonne inconnue W:=d(0,1)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},1} qui est  maintenant AA:=d(2,2) - d(1,1) + d(0,0)$

Unknowns: {d(2,2),d(1,1),d(0,0)}
Unknowns: {d(2,2),d(1,1),d(0,0)}
bonne inconnue W:=d(2,2)$

sa valeur doit etre WW:=d(1,1) - d(0,0)$

on resout l'equation {{0,2},2} qui est  maintenant AA:= - (d(1,6) + d(1,5))$

Unknowns: {d(1,6),d(1,5)}
Unknowns: {d(1,6),d(1,5)}
bonne inconnue W:=d(1,6)$

sa valeur doit etre WW:= - d(1,5)$

on resout l'equation {{0,2},3} qui est  maintenant AA:= - d(3,1) + d(1,0)$

Unknowns: {d(3,1),d(1,0)}
Unknowns: {d(3,1),d(1,0)}
bonne inconnue W:=d(3,1)$

sa valeur doit etre WW:=d(1,0)$

on resout l'equation {{0,2},4} qui est  maintenant AA:=d(5,2) - d(4,1)$

Unknowns: {d(5,2),d(4,1)}
Unknowns: {d(5,2),d(4,1)}
bonne inconnue W:=d(5,2)$

sa valeur doit etre WW:=d(4,1)$

on resout l'equation {{0,2},5} qui est  maintenant AA:=d(6,2)*a - d(4,6) - d(4,5
) - 2*d(3,0) + d(1,2)$

Unknowns: {d(6,2),d(4,6),d(4,5),d(3,0),d(1,2),a}
Unknowns: {d(6,2),d(4,6),d(4,5),d(3,0),d(1,2),a}
bonne inconnue W:=d(4,6)$

sa valeur doit etre WW:=d(6,2)*a - d(4,5) - 2*d(3,0) + d(1,2)$

on resout l'equation {{0,2},6} qui est  maintenant AA:= - d(6,1) + d(1,2)$

Unknowns: {d(6,1),d(1,2)}
Unknowns: {d(6,1),d(1,2)}
bonne inconnue W:=d(6,1)$

sa valeur doit etre WW:=d(1,2)$

on resout l'equation {{0,3},1} qui est  maintenant AA:=d(2,3)$

Unknown: d(2,3)
Unknown: d(2,3)
bonne inconnue W:=d(2,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,3},4} qui est  maintenant AA:=d(5,3) + d(1,0)$

Unknowns: {d(5,3),d(1,0)}
Unknowns: {d(5,3),d(1,0)}
bonne inconnue W:=d(5,3)$

sa valeur doit etre WW:= - d(1,0)$

on resout l'equation {{0,3},5} qui est  maintenant AA:=d(6,3)*a + d(2,0) + d(1,3
)$

Unknowns: {d(6,3),d(2,0),d(1,3),a}
Unknowns: {d(6,3),d(2,0),d(1,3),a}
bonne inconnue W:=d(2,0)$

sa valeur doit etre WW:= - (d(6,3)*a + d(1,3))$

on resout l'equation {{0,3},6} qui est  maintenant AA:=d(1,3)$

Unknown: d(1,3)
Unknown: d(1,3)
bonne inconnue W:=d(1,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},1} qui est  maintenant AA:=d(2,4)$

Unknown: d(2,4)
Unknown: d(2,4)
bonne inconnue W:=d(2,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},4} qui est  maintenant AA:=d(5,4)$

Unknown: d(5,4)
Unknown: d(5,4)
bonne inconnue W:=d(5,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},5} qui est  maintenant AA:=d(6,4)*a + d(1,4)$

Unknowns: {d(6,4),d(1,4),a}
Unknowns: {d(6,4),d(1,4),a}
bonne inconnue W:=d(1,4)$

sa valeur doit etre WW:= - d(6,4)*a$

on resout l'equation {{0,4},6} qui est  maintenant AA:= - d(6,4)*a$

Unknowns: {d(6,4),a}
Unknowns: {d(6,4),a}
pas de selection possible de variable a coefficient numerique dans  - d(6,4)*a
on resout l'equation {{0,5},0} qui est  maintenant AA:= - d(0,4)$

Unknown: d(0,4)
Unknown: d(0,4)
bonne inconnue W:=d(0,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},1} qui est  maintenant AA:=d(6,4)*a + d(2,5)$

Unknowns: {d(6,4),d(2,5),a}
Unknowns: {d(6,4),d(2,5),a}
bonne inconnue W:=d(2,5)$

sa valeur doit etre WW:= - d(6,4)*a$

on resout l'equation {{0,5},3} qui est  maintenant AA:= - d(3,4)$

Unknown: d(3,4)
Unknown: d(3,4)
bonne inconnue W:=d(3,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},4} qui est  maintenant AA:=d(5,5) - d(4,4) + d(0,0)$

Unknowns: {d(5,5),d(4,4),d(0,0)}
Unknowns: {d(5,5),d(4,4),d(0,0)}
bonne inconnue W:=d(5,5)$

sa valeur doit etre WW:=d(4,4) - d(0,0)$

on resout l'equation {{0,5},5} qui est  maintenant AA:=d(6,5)*a + d(1,5)$

Unknowns: {d(6,5),d(1,5),a}
Unknowns: {d(6,5),d(1,5),a}
bonne inconnue W:=d(1,5)$

sa valeur doit etre WW:= - d(6,5)*a$

on resout l'equation {{0,5},6} qui est  maintenant AA:= - (d(6,5)*a + d(6,4))$

Unknowns: {d(6,5),d(6,4),a}
Unknowns: {d(6,5),d(6,4),a}
bonne inconnue W:=d(6,4)$

sa valeur doit etre WW:= - d(6,5)*a$

on resout l'equation {{0,6},0} qui est  maintenant AA:= - d(0,5)*a$

Unknowns: {d(0,5),a}
Unknowns: {d(0,5),a}
pas de selection possible de variable a coefficient numerique dans  - d(0,5)*a
on resout l'equation {{0,6},2} qui est  maintenant AA:= - d(6,5)*a**3$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans  - d(6,5)*a**
3
on resout l'equation {{0,6},3} qui est  maintenant AA:= - d(3,5)*a$

Unknowns: {d(3,5),a}
Unknowns: {d(3,5),a}
pas de selection possible de variable a coefficient numerique dans  - d(3,5)*a
on resout l'equation {{0,6},4} qui est  maintenant AA:= - d(4,5)*a - d(4,4) + d(
1,2)*a + d(1,1) + 2*d(0,0)$

Unknowns: {d(4,5),d(4,4),d(1,2),d(1,1),d(0,0),a}
Unknowns: {d(4,5),d(4,4),d(1,2),d(1,1),d(0,0),a}
bonne inconnue W:=d(4,4)$

sa valeur doit etre WW:= - d(4,5)*a + d(1,2)*a + d(1,1) + 2*d(0,0)$

on resout l'equation {{0,6},5} qui est  maintenant AA:=a*(d(4,5)*a - d(1,2)*a + 
d(0,0))$

Unknowns: {d(4,5),d(1,2),d(0,0),a}
Unknowns: {d(4,5),d(1,2),d(0,0),a}
pas de selection possible de variable a coefficient numerique dans a*(d(4,5)*a -
 d(1,2)*a + d(0,0))
on resout l'equation {{1,2},0} qui est  maintenant AA:= - d(0,3)$

Unknown: d(0,3)
Unknown: d(0,3)
bonne inconnue W:=d(0,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},3} qui est  maintenant AA:= - d(3,3) + 2*d(1,1) - d(
0,0)$

Unknowns: {d(3,3),d(1,1),d(0,0)}
Unknowns: {d(3,3),d(1,1),d(0,0)}
bonne inconnue W:=d(3,3)$

sa valeur doit etre WW:=2*d(1,1) - d(0,0)$

on resout l'equation {{1,2},4} qui est  maintenant AA:= - d(4,3) + d(3,2)$

Unknowns: {d(4,3),d(3,2)}
Unknowns: {d(4,3),d(3,2)}
bonne inconnue W:=d(4,3)$

sa valeur doit etre WW:=d(3,2)$

on resout l'equation {{1,2},5} qui est  maintenant AA:= - d(0,2)$

Unknown: d(0,2)
Unknown: d(0,2)
bonne inconnue W:=d(0,2)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},6} qui est  maintenant AA:= - d(6,3)$

Unknown: d(6,3)
Unknown: d(6,3)
bonne inconnue W:=d(6,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,3},1} qui est  maintenant AA:= - d(6,5)*a**2$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans  - d(6,5)*a**
2
on resout l'equation {{1,3},4} qui est  maintenant AA:=d(4,5)*a - d(1,2)*a + 2*d
(1,1) - 3*d(0,0)$

Unknowns: {d(4,5),d(1,2),d(1,1),d(0,0),a}
Unknowns: {d(4,5),d(1,2),d(1,1),d(0,0),a}
bonne inconnue W:=d(1,1)$

sa valeur doit etre WW:=( - d(4,5)*a + d(1,2)*a + 3*d(0,0))/2$

on resout l'equation {{1,3},6} qui est  maintenant AA:=d(6,5)*a$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans d(6,5)*a
on resout l'equation {{1,5},3} qui est  maintenant AA:=d(6,5)*a**2$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans d(6,5)*a**2
on resout l'equation {{1,5},4} qui est  maintenant AA:=d(3,5)$

Unknown: d(3,5)
Unknown: d(3,5)
bonne inconnue W:=d(3,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,5},5} qui est  maintenant AA:= - d(0,5)$

Unknown: d(0,5)
Unknown: d(0,5)
bonne inconnue W:=d(0,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,6},3} qui est  maintenant AA:= - d(6,5)*a**2$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans  - d(6,5)*a**
2
on resout l'equation {{2,3},1} qui est  maintenant AA:=d(6,5)*a$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans d(6,5)*a
on resout l'equation {{2,3},2} qui est  maintenant AA:= - d(6,5)*a**2$

Unknowns: {d(6,5),a}
Unknowns: {d(6,5),a}
pas de selection possible de variable a coefficient numerique dans  - d(6,5)*a**
2
on resout l'equation {{2,3},4} qui est  maintenant AA:= - d(4,5) + d(1,2)$

Unknowns: {d(4,5),d(1,2)}
Unknowns: {d(4,5),d(1,2)}
bonne inconnue W:=d(1,2)$

sa valeur doit etre WW:=d(4,5)$

on resout l'equation {{2,3},6} qui est  maintenant AA:= - d(6,5)$

Unknown: d(6,5)
Unknown: d(6,5)
bonne inconnue W:=d(6,5)$

sa valeur doit etre WW:=0$

Derivation equations to cancel (Reduce output) : \\{{{{0,1},0},0},
{{{0,1},1},0},
{{{0,1},2},0},
{{{0,1},3},0},
{{{0,1},4},0},
{{{0,1},5},0},
{{{0,1},6},0},
{{{0,2},0},0},
{{{0,2},1},0},
{{{0,2},2},0},
{{{0,2},3},0},
{{{0,2},4},0},
{{{0,2},5},0},
{{{0,2},6},0},
{{{0,3},1},0},
{{{0,3},4},0},
{{{0,3},5},0},
{{{0,3},6},0},
{{{0,4},1},0},
{{{0,4},4},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},0},0},
{{{0,5},1},0},
{{{0,5},2},0},
{{{0,5},3},0},
{{{0,5},4},0},
{{{0,5},5},0},
{{{0,5},6},0},
{{{0,6},0},0},
{{{0,6},1},0},
{{{0,6},2},0},
{{{0,6},3},0},
{{{0,6},4},0},
{{{0,6},5},d(0,0)*a},
{{{0,6},6},0},
{{{1,2},0},0},
{{{1,2},1},0},
{{{1,2},2},0},
{{{1,2},3},0},
{{{1,2},4},0},
{{{1,2},5},0},
{{{1,2},6},0},
{{{1,3},0},0},
{{{1,3},1},0},
{{{1,3},2},0},
{{{1,3},3},0},
{{{1,3},4},0},
{{{1,3},5},0},
{{{1,3},6},0},
{{{1,4},3},0},
{{{1,4},4},0},
{{{1,4},5},0},
{{{1,4},6},0},
{{{1,5},3},0},
{{{1,5},4},0},
{{{1,5},5},0},
{{{1,5},6},0},
{{{1,6},3},0},
{{{1,6},4},0},
{{{1,6},5},0},
{{{1,6},6},0},
{{{2,3},0},0},
{{{2,3},1},0},
{{{2,3},2},0},
{{{2,3},3},0},
{{{2,3},4},0},
{{{2,3},5},0},
{{{2,3},6},0},
{{{2,4},1},0},
{{{2,4},3},0},
{{{2,4},5},0},
{{{2,5},1},0},
{{{2,5},3},0},
{{{2,5},4},0},
{{{2,5},5},0},
{{{2,6},1},0},
{{{2,6},3},0},
{{{2,6},5},0},
{{{3,4},4},0},
{{{3,4},5},0},
{{{3,5},4},0},
{{{3,5},5},0},
{{{3,6},4},0},
{{{3,6},5},0},
{{{4,5},4},0},
{{{4,6},5},0},
{{{5,6},4},0},
{{{5,6},5},0}}$

Il y a une phase 2$

collect_eq:={{{{0,1},0},0},
{{{0,1},1},0},
{{{0,1},2},0},
{{{0,1},3},0},
{{{0,1},4},0},
{{{0,1},5},0},
{{{0,1},6},0},
{{{0,2},0},0},
{{{0,2},1},0},
{{{0,2},2},0},
{{{0,2},3},0},
{{{0,2},4},0},
{{{0,2},5},0},
{{{0,2},6},0},
{{{0,3},1},0},
{{{0,3},4},0},
{{{0,3},5},0},
{{{0,3},6},0},
{{{0,4},1},0},
{{{0,4},4},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},0},0},
{{{0,5},1},0},
{{{0,5},2},0},
{{{0,5},3},0},
{{{0,5},4},0},
{{{0,5},5},0},
{{{0,5},6},0},
{{{0,6},0},0},
{{{0,6},1},0},
{{{0,6},2},0},
{{{0,6},3},0},
{{{0,6},4},0},
{{{0,6},5},0},
{{{0,6},6},0},
{{{1,2},0},0},
{{{1,2},1},0},
{{{1,2},2},0},
{{{1,2},3},0},
{{{1,2},4},0},
{{{1,2},5},0},
{{{1,2},6},0},
{{{1,3},0},0},
{{{1,3},1},0},
{{{1,3},2},0},
{{{1,3},3},0},
{{{1,3},4},0},
{{{1,3},5},0},
{{{1,3},6},0},
{{{1,4},3},0},
{{{1,4},4},0},
{{{1,4},5},0},
{{{1,4},6},0},
{{{1,5},3},0},
{{{1,5},4},0},
{{{1,5},5},0},
{{{1,5},6},0},
{{{1,6},3},0},
{{{1,6},4},0},
{{{1,6},5},0},
{{{1,6},6},0},
{{{2,3},0},0},
{{{2,3},1},0},
{{{2,3},2},0},
{{{2,3},3},0},
{{{2,3},4},0},
{{{2,3},5},0},
{{{2,3},6},0},
{{{2,4},1},0},
{{{2,4},3},0},
{{{2,4},5},0},
{{{2,5},1},0},
{{{2,5},3},0},
{{{2,5},4},0},
{{{2,5},5},0},
{{{2,6},1},0},
{{{2,6},3},0},
{{{2,6},5},0},
{{{3,4},4},0},
{{{3,4},5},0},
{{{3,5},4},0},
{{{3,5},5},0},
{{{3,6},4},0},
{{{3,6},5},0},
{{{4,5},4},0},
{{{4,6},5},0},
{{{5,6},4},0},
{{{5,6},5},0}}$

a neq {0}$

derivation generique de gtildedelta:$

MATD:=
mat((0,0,0,0,0,0,0),(d(1,0),0,d(4,5),0,0,0,0),(0,0,0,0,0,0,0),(d(3,0),d(1,0),d(3
,2),0,0,0,0),(d(4,0),d(4,1),d(4,2),d(3,2),0,d(4,5),d(6,2)*a - 2*d(3,0)),(d(5,0),
d(4,5) - d(3,0) + d(6,2)*a,d(4,1), - d(1,0),0,0,d(4,5)*a),(d(6,0),d(4,5),d(6,2),
0,0,0,0))$
$

pour delta:=

[0  1  0  0  0  0]
[                ]
[0  0  0  0  0  0]
[                ]
[0  0  0  0  0  0]
[                ]
[0  0  0  0  1  0]
[                ]
[1  0  0  0  0  a]
[                ]
[1  0  0  0  0  0]


pour shortformdelta:={0,0,ss,1,0,a,ss,1,0}

Unknowns: {d(6,2),
d(6,0),
d(5,0),
d(4,5),
d(4,2),
d(4,1),
d(4,0),
d(3,2),
d(3,0),
d(1,0),
a}
Unknowns: {d(6,2),
d(6,0),
d(5,0),
d(4,5),
d(4,2),
d(4,1),
d(4,0),
d(3,2),
d(3,0),
d(1,0),
a}
listeparametresMATD{d(6,2),
d(6,0),
d(5,0),
d(4,5),
d(4,2),
d(4,1),
d(4,0),
d(3,2),
d(3,0),
d(1,0)}$

dim Der(gtildedelta):=10$

MATD**1:=
mat((0,0,0,0,0,0,0),(d(1,0),0,d(4,5),0,0,0,0),(0,0,0,0,0,0,0),(d(3,0),d(1,0),d(3
,2),0,0,0,0),(d(4,0),d(4,1),d(4,2),d(3,2),0,d(4,5),d(6,2)*a - 2*d(3,0)),(d(5,0),
d(4,5) - d(3,0) + d(6,2)*a,d(4,1), - d(1,0),0,0,d(4,5)*a),(d(6,0),d(4,5),d(6,2),
0,0,0,0))$
$

MATD**2:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(d(1,0)**2,0,d(4,5)*d(1,0),0
,0,0,0),(d(4,1)*d(1,0) + d(3,2)*d(3,0) + d(5,0)*d(4,5) + (d(6,2)*a - 2*d(3,0))*d
(6,0),2*d(6,2)*d(4,5)*a + d(4,5)**2 - 3*d(4,5)*d(3,0) + d(3,2)*d(1,0),(d(6,2)*a 
- 2*d(3,0))*d(6,2) + 2*d(4,5)*d(4,1) + d(3,2)**2, - d(4,5)*d(1,0),0,0,d(4,5)**2*
a),((d(4,5) - d(3,0) + d(6,2)*a)*d(1,0) + d(6,0)*d(4,5)*a - d(3,0)*d(1,0),d(4,5)
**2*a - d(1,0)**2,(d(4,5) - d(3,0) + d(6,2)*a)*d(4,5) + d(6,2)*d(4,5)*a - d(3,2)
*d(1,0),0,0,0,0),(d(4,5)*d(1,0),0,d(4,5)**2,0,0,0,0))$
$

MATD**3:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(2*d(6,2)*d(
4,5)*d(1,0)*a + d(6,0)*d(4,5)**2*a + d(4,5)**2*d(1,0) - 4*d(4,5)*d(3,0)*d(1,0) +
 d(3,2)*d(1,0)**2,(d(4,5)**2*a - d(1,0)**2)*d(4,5),(3*d(6,2)*a + d(4,5) - 3*d(3,
0))*d(4,5)**2,0,0,0,0),((d(4,5)**2*a - d(1,0)**2)*d(1,0),0,(d(4,5)**2*a - d(1,0)
**2)*d(4,5),0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**4:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),((d(4,5)**2*
a - d(1,0)**2)*d(4,5)*d(1,0),0,(d(4,5)**2*a - d(1,0)**2)*d(4,5)**2,0,0,0,0),(0,0
,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**5:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0
,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**6:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0
,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**7:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0
,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**2:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(d(1,0)**2,0,d(4,5)*d(1,0),0
,0,0,0),(d(4,1)*d(1,0) + d(3,2)*d(3,0) + d(5,0)*d(4,5) + (d(6,2)*a - 2*d(3,0))*d
(6,0),2*d(6,2)*d(4,5)*a + d(4,5)**2 - 3*d(4,5)*d(3,0) + d(3,2)*d(1,0),(d(6,2)*a 
- 2*d(3,0))*d(6,2) + 2*d(4,5)*d(4,1) + d(3,2)**2, - d(4,5)*d(1,0),0,0,d(4,5)**2*
a),((d(4,5) - d(3,0) + d(6,2)*a)*d(1,0) + d(6,0)*d(4,5)*a - d(3,0)*d(1,0),d(4,5)
**2*a - d(1,0)**2,(d(4,5) - d(3,0) + d(6,2)*a)*d(4,5) + d(6,2)*d(4,5)*a - d(3,2)
*d(1,0),0,0,0,0),(d(4,5)*d(1,0),0,d(4,5)**2,0,0,0,0))$
$

MATD**3:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(2*d(6,2)*d(
4,5)*d(1,0)*a + d(6,0)*d(4,5)**2*a + d(4,5)**2*d(1,0) - 4*d(4,5)*d(3,0)*d(1,0) +
 d(3,2)*d(1,0)**2,(d(4,5)**2*a - d(1,0)**2)*d(4,5),(3*d(6,2)*a + d(4,5) - 3*d(3,
0))*d(4,5)**2,0,0,0,0),((d(4,5)**2*a - d(1,0)**2)*d(1,0),0,(d(4,5)**2*a - d(1,0)
**2)*d(4,5),0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**4:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),((d(4,5)**2*
a - d(1,0)**2)*d(4,5)*d(1,0),0,(d(4,5)**2*a - d(1,0)**2)*d(4,5)**2,0,0,0,0),(0,0
,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

MATD**5:=
mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0
,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

***********   gtildedelta est caracteristiquement nilpotente d'ordre5$

rank 0 :gtildedelta is characteristically nilpotent

rkgtildedelta


matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base  diagonali\
sant le tore maximal: on peut prendre

P:=

[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


matrice des derivations dans cette base diagonalisante Y(1),...,Y(7):

P**(-1)*MATD*P:=
mat((0,0,0,0,0,0,0),(d(1,0),0,d(4,5),0,0,0,0),(0,0,0,0,0,0,0),(d(3,0),d(1,0),d(3
,2),0,0,0,0),(d(4,0),d(4,1),d(4,2),d(3,2),0,d(4,5),d(6,2)*a - 2*d(3,0)),(d(5,0),
d(4,5) - d(3,0) + d(6,2)*a,d(4,1), - d(1,0),0,0,d(4,5)*a),(d(6,0),d(4,5),d(6,2),
0,0,0,0))$
$

PP:=

[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


avec PP:=P*Q:=

[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


MATDDIAGONALISE:=

mat((0,0,0,0,0,0,0),

    (d(1,0),0,d(4,5),0,0,0,0),

    (0,0,0,0,0,0,0),

    (d(3,0),d(1,0),d(3,2),0,0,0,0),

    (d(4,0),d(4,1),d(4,2),d(3,2),0,d(4,5),d(6,2)*a - 2*d(3,0)),

    (d(5,0),d(4,5) - d(3,0) + d(6,2)*a,d(4,1), - d(1,0),0,0,d(4,5)*a),

    (d(6,0),d(4,5),d(6,2),0,0,0,0))


on voit apparaitre les poids sur la diagonale

r(1) := 0

r(2) := 0

r(3) := 0

r(4) := 0

r(5) := 0

r(6) := 0

r(7) := 0

r(1) := 0

r(2) := 0

r(3) := 0

r(4) := 0

r(5) := 0

r(6) := 0

r(7) := 0

Le systeme de poids est le systeme 0.5

calcul de relations de commutation de la base diaY(j) diagonalisant le tore

listcommutateursdesx := {{{0,1},x(6) + x(5)},

   {{0,2},x(1)},

   {{0,3},0},

   {{0,4},0},

   {{0,5},x(4)},

   {{0,6},a*x(5)},

   {{1,2},x(3)},

   {{1,3},x(4)},

   {{1,4},0},

   {{1,5},0},

   {{1,6},0},

   {{2,3},x(5)},

   {{2,4},0},

   {{2,5},0},

   {{2,6},0},

   {{3,4},0},

   {{3,5},0},

   {{3,6},0},

   {{4,5},0},

   {{4,6},0},

   {{5,6},0}}

diaY(1):=x(0)

diaY(2):=x(1)

diaY(3):=x(2)

diaY(4):=x(3)

diaY(5):=x(4)

diaY(6):=x(5)

diaY(7):=x(6)

liste des commutateurs des diaY(i) :$

listcommutateurdesdiaY:={{{1,2},diay(7) + diay(6)},
{{1,3},diay(2)},
{{1,4},0},
{{1,5},0},
{{1,6},diay(5)},
{{1,7},diay(6)*a},
{{2,3},diay(4)},
{{2,4},diay(5)},
{{2,5},0},
{{2,6},0},
{{2,7},0},
{{3,4},diay(6)},
{{3,5},0},
{{3,6},0},
{{3,7},0},
{{4,5},0},
{{4,6},0},
{{4,7},0},
{{5,6},0},
{{5,7},0},
{{6,7},0}}$

Now we make explicit the isomorphism with an algebra of the book:$

Namely g_{7,0.5}$

i.e. we go from the basis diaY(i) to the new basis ZZ(i) defined by the matrix:$

on pose :$

avec comme matrice de changement de base :$

This isomorphism computed by calculisom6_54xCI.red$


mat((1/a,0,0,0,0,0,0),(0,0,(sqrt(a)*i)/a**2,0,0,0,0),(0,(sqrt(a)*i)/a,0,0,0,0,0)
,(0,0,0,0,1/a**2,0,0),(0,0,0,0,0,0,(sqrt(a)*i)/a**4),(0,0,0,(sqrt(a)*i)/a**3,0,(
sqrt(a)*i)/a**3,0),(0,0,0,(sqrt(a)*i)/a**3,0,0,0))$
$

det(isom):= ( - sqrt(a)*i)/a**14$

ZZ(1):=diay(1)/a$

ZZ(2):=(sqrt(a)*diay(3)*i)/a$

ZZ(3):=(sqrt(a)*diay(2)*i)/a**2$

ZZ(4):=(sqrt(a)*(diay(7) + diay(6))*i)/a**3$

ZZ(5):=diay(4)/a**2$

ZZ(6):=(sqrt(a)*diay(6)*i)/a**3$

ZZ(7):=(sqrt(a)*diay(5)*i)/a**4$

listcommutateursdesZZ:=$

{{1,2},zz(3)}$

{{1,3},zz(4)}$

{{1,4},zz(7) + zz(6)}$

{{1,5},0}$

{{1,6},zz(7)}$

{{1,7},0}$

{{2,3},zz(5)}$

{{2,4},0}$

{{2,5},zz(6)}$

{{2,6},0}$

{{2,7},0}$

{{3,4},0}$

{{3,5},zz(7)}$

{{3,6},0}$

{{3,7},0}$

{{4,5},0}$

{{4,6},0}$

{{4,7},0}$

{{5,6},0}$

{{5,7},0}$

{{6,7},0}$

We get the commutation relations of$

g_{7,0.5}$

Et cela pour a:=a$

and that for a neq {0}$

shortformdelta:={0,0,ss,1,0,a,ss,1,0}$

delta:=
mat((0,1,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,1,0),(1,0,0,0,0,a),(1,0,0
,0,0,0))$
$