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


        [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  1  0  0]
        [                ]
        [0  0  0  0  1  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  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]
adx3 := [                 ]
        [-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  0  0]
        [                 ]
        [0   0  0  0  0  0]
adx4 := [                 ]
        [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  0  0]
adx5 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]


The generic nilpotent derivation : the eigenvalues are 0

xi(1,1):=0

xi(2,2):=0

by subtracting adjoints one then may suppose:

xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0,xi(6,1):=0.

delta:=

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


We denote this delta by the shortform

shortformdelta:={xi(2,1),

                 ss,

                 xi(4,2),

                 ss,

                 xi(5,2),

                 ss,

                 xi(6,2)}

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

a:=1$

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

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

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

Unknown: d(1,2)
Unknown: d(1,2)
bonne inconnue W:=d(1,2)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:= - d(3,5) + d(1,0)$

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

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

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

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

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

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

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

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

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

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

Unknown: d(1,5)
Unknown: d(1,5)
bonne inconnue W:=d(1,5)$

sa valeur doit etre WW:=0$

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

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

sa valeur doit etre WW:= - d(1,3)$

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:= - d(3,3) - d(2,3) - d(2,0) + d(1,1) + d(0,0)$

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

Unknown: d(1,4)
Unknown: d(1,4)
bonne inconnue W:=d(1,4)$

sa valeur doit etre WW:=0$

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

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

sa valeur doit etre WW:= - d(2,4)$

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

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

sa valeur doit etre WW:= - d(3,4)$

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

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

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

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

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

sa valeur doit etre WW:= - d(3,0) + d(0,1)$

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

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

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

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

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

sa valeur doit etre WW:=3*d(1,1) + d(0,0)$

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

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

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

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

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

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

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

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

sa valeur doit etre WW:=d(0,0)/3$

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

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

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},0},0},
{{{0,3},1},0},
{{{0,3},2},0},
{{{0,3},3},0},
{{{0,3},4},0},
{{{0,3},5},0},
{{{0,3},6},0},
{{{0,4},2},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},2},0},
{{{0,5},5},0},
{{{0,5},6},0},
{{{0,6},2},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},0},0},
{{{1,4},1},0},
{{{1,4},2},0},
{{{1,4},3},0},
{{{1,4},4},0},
{{{1,4},5},0},
{{{1,4},6},0},
{{{1,5},0},0},
{{{1,5},1},0},
{{{1,5},2},0},
{{{1,5},3},0},
{{{1,5},4},0},
{{{1,5},5},0},
{{{1,5},6},0},
{{{1,6},2},0},
{{{1,6},3},0},
{{{1,6},4},0},
{{{1,6},5},0},
{{{1,6},6},0},
{{{2,3},3},0},
{{{2,3},4},0},
{{{2,3},5},0},
{{{2,3},6},0},
{{{2,4},3},0},
{{{2,4},5},0},
{{{2,4},6},0},
{{{2,5},3},0},
{{{2,5},5},0},
{{{2,5},6},0},
{{{2,6},3},0},
{{{2,6},5},0},
{{{2,6},6},0},
{{{3,4},4},0},
{{{3,4},5},0},
{{{3,4},6},0},
{{{3,5},4},0},
{{{3,5},6},0},
{{{3,6},4},0},
{{{3,6},6},0},
{{{4,5},5},0},
{{{4,5},6},0},
{{{4,6},5},0},
{{{5,6},6},0}}$

Il n'y a pas de phase 2$

derivation generique de gtildedelta:$

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

pour delta:=

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


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

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

dim Der(gtildedelta):=11$

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

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

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

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

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

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

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

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

***********   gtildedelta est caracteristiquement nilpotente d'ordre7$

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

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

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

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

    (d(4,0),d(4,1), - d(3,0), - d(2,0),0,0,0),

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

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

      - d(3,0) + 2*d(0,1), - d(2,0),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 0.3

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

listcommutateursdesx := {{{0,1},x(2)},

   {{0,2},x(6) + x(5)},

   {{0,3},x(6)},

   {{0,4},0},

   {{0,5},0},

   {{0,6},0},

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

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

   {{1,4},x(5)},

   {{1,5},x(6)},

   {{1,6},0},

   {{2,3},0},

   {{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(3)},
{{1,3},diay(7) + diay(6)},
{{1,4},diay(7)},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3},diay(4)},
{{2,4},diay(5)},
{{2,5},diay(6)},
{{2,6},diay(7)},
{{2,7},0},
{{3,4},0},
{{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_{7,0.3}$

and that for a neqconditionssura$

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 :$


mat((0,1,0,0,0,0,0),(1,0,0,0,0,0,0),(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))$
$

det(isom):= 1$

ZZ(1):=diay(2)$

ZZ(2):=diay(1)$

ZZ(3):= - diay(3)$

ZZ(4):= - diay(4)$

ZZ(5):= - diay(5)$

ZZ(6):= - diay(6)$

ZZ(7):= - diay(7)$

listcommutateursdesZZ:=$

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

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

{{1,4},zz(5)}$

{{1,5},zz(6)}$

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

{{1,7},0}$

{{2,3},zz(7) + zz(6)}$

{{2,4},zz(7)}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4},0}$

{{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}$

We get the commutation relations of$

g_{7,0.3}$

Et cela pour a:=1$

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

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

The isomorphism from g_{7,0.3} to gtildedelta$

was constructed in 2 steps and is given by$

the product matrix P*isom:=
mat((0,1,0,0,0,0,0),(1,0,0,0,0,0,0),(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))$
$

which we record here under the name PSI$

PSI_III2:=
mat((0,1,0,0,0,0,0),(1,0,0,0,0,0,0),(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))$
$