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

        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
adx1 := [                ]
        [0  1  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]
adx2 := [                 ]
        [-1  0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  1  1  0  0]


        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
adx3 := [                 ]
        [0  0   0  0  0  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]
adx4 := [                  ]
        [0   0   0  0  0  0]
        [                  ]
        [-1  0   0  0  0  0]
        [                  ]
        [0   -1  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(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,2):=0

delta:=

[   0        0        0         0       0  0]
[                                           ]
[   0        0        0         0       0  0]
[                                           ]
[xi(3,1)  xi(3,2)     0         0       0  0]
[                                           ]
[   0        0        0         0       0  0]
[                                           ]
[   0     xi(5,2)  xi(5,3)      0       0  0]
[                                           ]
[xi(6,1)     0     xi(6,3)   - xi(3,1)  0  0]


We denote this delta by the shortform

shortformdelta:={xi(3,1),

                 xi(3,2),

                 ss,

                 xi(5,2),

                 xi(5,3),

                 ss,

                 xi(6,1),

                 xi(6,3)}

paramindexeslist:={{3,1},{3,2},{5,2},{5,3},{6,1},{6,3}}

In case 2 one has$

xi(3,1):=1$

xi(3,2):=0$

a neq {}$

a:=a$

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

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

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

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

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

sa valeur doit etre WW:= - d(4,0) + d(3,1)$

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

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

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

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

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

sa valeur doit etre WW:=0$

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

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:= - d(2,6)*a$

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

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

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

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

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

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

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

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

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

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

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

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

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

Unknown: d(0,6)
Unknown: d(0,6)
bonne inconnue W:=d(0,6)$

sa valeur doit etre WW:=0$

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

Unknown: d(1,6)
Unknown: d(1,6)
bonne inconnue W:=d(1,6)$

sa valeur doit etre WW:=0$

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

Unknown: d(2,6)
Unknown: d(2,6)
bonne inconnue W:=d(2,6)$

sa valeur doit etre WW:=0$

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

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

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

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

Unknown: d(4,6)
Unknown: d(4,6)
bonne inconnue W:=d(4,6)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:= - (d(4,1) + d(3,1))$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

a neq {-1}$

derivation generique de gtildedelta:$

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

pour delta:=

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


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

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

dim Der(gtildedelta):=11$

t1:=D(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  2  0  0  0]
[                   ]
[0  0  0  0  2  0  0]
[                   ]
[0  0  0  0  0  3  0]
[                   ]
[0  0  0  0  0  0  3]


MATD:=

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

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

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

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

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

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

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

     ))


Unknown: d(0,0)

Unknown: d(0,0)

commutant de t1 dans der(gtildedelta):

[d(0,0)    0       0        0         0         0         0    ]
[                                                              ]
[  0     d(0,0)    0        0         0         0         0    ]
[                                                              ]
[  0       0     d(0,0)     0         0         0         0    ]
[                                                              ]
[  0       0       0     2*d(0,0)     0         0         0    ]
[                                                              ]
[  0       0       0        0      2*d(0,0)     0         0    ]
[                                                              ]
[  0       0       0        0         0      3*d(0,0)     0    ]
[                                                              ]
[  0       0       0        0         0         0      3*d(0,0)]


rank 1 with maximal torus t1

1


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]


P**(-1)*t1*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  2  0  0  0]
[                   ]
[0  0  0  0  2  0  0]
[                   ]
[0  0  0  0  0  3  0]
[                   ]
[0  0  0  0  0  0  3]


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

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

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

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

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

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

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

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

     ))


on voit apparaitre les poids sur la diagonale

r(1) := d(0,0)

r(2) := d(0,0)

r(3) := d(0,0)

r(4) := 2*d(0,0)

r(5) := 2*d(0,0)

r(6) := 3*d(0,0)

r(7) := 3*d(0,0)

r(1) := gamma1

r(2) := gamma1

r(3) := gamma1

r(4) := 2*gamma1

r(5) := 2*gamma1

r(6) := 3*gamma1

r(7) := 3*gamma1

Le systeme de poids est le systeme 1.19

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

listcommutateursdesx := {{{0,1},x(3)},

   {{0,2},0},

   {{0,3},x(6)*a + x(5)},

   {{0,4}, - x(6)},

   {{0,5},0},

   {{0,6},0},

   {{1,2},x(4)},

   {{1,3},0},

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

   {{1,5},0},

   {{1,6},0},

   {{2,3},x(6)},

   {{2,4},x(6)},

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

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

Namely g_{7,1.19}$

and that for a neq{-1}$

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 isom computed by calculisom6_5case2I.red$

mat((1,0, - (a + 1)**(1/3),0,0,0,0),(1,0,0,0,0,0,0),(a + 2, - (a + 1)**(2/3), - 
(a + 1)**(1/3),0,0,0,0),(0,0,0,0,(a + 1)**(1/3),0,0),(0,0,0, - (a + 1)**(2/3), -
 (a + 1)**(1/3),0,0),(0,0,0,0,0,0, - (a + 1)**(2/3)),(0,0,0,0,0,(a + 1)**(1/3)*(
a + 1), - (a + 1)**(2/3)*(a + 1)))$

det(isom):= (a + 1)**4$

ZZ(1):=diay(2) + diay(1) + (a + 2)*diay(3)$

ZZ(2):= - (a + 1)**(2/3)*diay(3)$

ZZ(3):= - (a + 1)**(1/3)*(diay(3) + diay(1))$

ZZ(4):= - (a + 1)**(2/3)*diay(5)$

ZZ(5):= - (a + 1)**(1/3)*(diay(5) - diay(4))$

ZZ(6):=(a + 1)**(1/3)*(a + 1)*diay(7)$

ZZ(7):= - (a + 1)**(2/3)*((a + 1)*diay(7) + diay(6))$

listcommutateursdesZZ:=$

{{1,2},zz(4)}$

{{1,3},zz(5)}$

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

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

{{1,6},0}$

{{1,7},0}$

{{2,3},0}$

{{2,4},zz(6)}$

{{2,5},0}$

{{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_{7,1.19}$

Et cela pour a:=a, b:=b.$

and that for a neq {-1}$

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

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