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 neq {}$

a:=a$

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

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

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

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Unknown: d(2,5)
Unknown: d(2,5)
bonne inconnue W:=d(2,5)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Il y a une phase 2$

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

a neq {2/sqrt(5),( - 2)/sqrt(5)}$

derivation generique de gtildedelta:$

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

pour delta:=

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


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

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

dim Der(gtildedelta):=11$

t1:=D(2,2):=

[0  0  0  0  0  0  0]
[                   ]
[0  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]


Unknowns: {d(6,2),d(5,2),d(4,2),d(3,2),d(2,2)}

Unknowns: {d(6,2),d(5,2),d(4,2),d(3,2),d(2,2)}

commutant de t1 dans der(gtildedelta):


[0  0    0       0       0       0       0   ]
[                                            ]
[0  0    0       0       0       0       0   ]
[                                            ]
[0  0  d(2,2)    0       0       0       0   ]
[                                            ]
[0  0  d(3,2)  d(2,2)    0       0       0   ]
[                                            ]
[0  0  d(4,2)  d(3,2)  d(2,2)    0       0   ]
[                                            ]
[0  0  d(5,2)  d(4,2)  d(3,2)  d(2,2)    0   ]
[                                            ]
[0  0  d(6,2)  d(5,2)  d(4,2)  d(3,2)  d(2,2)]


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:=

[0  0  0  0  0  0  0]
[                   ]
[0  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]


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

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

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:=

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


on voit apparaitre les poids sur la diagonale

r(1) := 0

r(2) := 0

r(3) := d(2,2)

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

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

r(6) := d(2,2)

r(7) := d(2,2)

r(1) := 0

r(2) := 0

r(3) := gamma

r(4) := gamma

r(5) := gamma

r(6) := gamma

r(7) := gamma

Le systeme de poids est le systeme 1.{0,point,1}

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

listcommutateursdesx := {{{0,1},0},

   {{0,2},x(6) + x(5)*a + x(4)},

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

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

   {{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},0},
{{1,3},diay(7) + diay(6)*a + diay(5)},
{{1,4},diay(7)*a + diay(6)},
{{1,5},diay(7)},
{{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_{7,1.{0,point,1}}$

(i)$

and that for a neq{2/sqrt(5),( - 2)/sqrt(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 the program calculisom6_16V.red$


mat(((sqrt( - 5*a**2 + 4)*a)/4,0,(5*a**2 - 4)/4,0,0,0,0),(sqrt( - 5*a**2 + 4)/2,
0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,0,sqrt( - 5*a**2 + 4)/2,0,0,0),(0,0,0,(sqrt( -
 5*a**2 + 4)*a)/4,( - 5*a**2 + 4)/4,0,0),(0,0,0,(sqrt( - 5*a**2 + 4)*a**2)/4,(a*
( - 5*a**2 + 4))/4,(sqrt( - 5*a**2 + 4)*( - 5*a**2 + 4))/8,0),(0,0,0,(sqrt( - 5*
a**2 + 4)*a)/4,(5*a**2*( - 5*a**2 + 4))/16,(3*sqrt( - 5*a**2 + 4)*a*( - 5*a**2 +
 4))/16,(25*a**4 - 40*a**2 + 16)/16))$
$

det(isom):= ( - sqrt( - 5*a**2 + 4)*(5*a**2 - 4)**6)/8192$

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

ZZ(2):=diay(3)$

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

ZZ(4):=(sqrt( - 5*a**2 + 4)*(diay(5)*a + 2*diay(4) + diay(6)*a**2 + diay(7)*a))/
4$

ZZ(5):=( - (4*(diay(6)*a + diay(5)) + 5*diay(7)*a**2)*(5*a**2 - 4))/16$

ZZ(6):=( - sqrt( - 5*a**2 + 4)*(3*diay(7)*a + 2*diay(6))*(5*a**2 - 4))/16$

ZZ(7):=((5*a**2 - 4)**2*diay(7))/16$

listcommutateursdesZZ:=$

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

{{1,3},0}$

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

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

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

{{1,7},0}$

{{2,3},zz(7) + zz(5)}$

{{2,4},0}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4}, - zz(6)}$

{{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.{0,point,1}}$

(i)$

Et cela pour a:=a$

and that for a neq {2/sqrt(5),( - 2)/sqrt(5)}$

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

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

The isomorphism from g_{7,1.{0,point,1}}$

(i)$

to gtildedelta$

was constructed in 2 steps and is given by$

the product matrix P*isom:=
mat(((sqrt( - 5*a**2 + 4)*a)/4,0,(5*a**2 - 4)/4,0,0,0,0),(sqrt( - 5*a**2 + 4)/2,
0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,0,sqrt( - 5*a**2 + 4)/2,0,0,0),(0,0,0,(sqrt( -
 5*a**2 + 4)*a)/4,( - (5*a**2 - 4))/4,0,0),(0,0,0,(sqrt( - 5*a**2 + 4)*a**2)/4,(
 - (5*a**2 - 4)*a)/4,( - sqrt( - 5*a**2 + 4)*(5*a**2 - 4))/8,0),(0,0,0,(sqrt( - 
5*a**2 + 4)*a)/4,( - 5*(5*a**2 - 4)*a**2)/16,( - 3*sqrt( - 5*a**2 + 4)*(5*a**2 -
 4)*a)/16,(5*a**2 - 4)**2/16))$
$

which we record here under the name PSI$

PSI_V:=
mat(((sqrt( - 5*a**2 + 4)*a)/4,0,(5*a**2 - 4)/4,0,0,0,0),(sqrt( - 5*a**2 + 4)/2,
0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,0,sqrt( - 5*a**2 + 4)/2,0,0,0),(0,0,0,(sqrt( -
 5*a**2 + 4)*a)/4,( - (5*a**2 - 4))/4,0,0),(0,0,0,(sqrt( - 5*a**2 + 4)*a**2)/4,(
 - (5*a**2 - 4)*a)/4,( - sqrt( - 5*a**2 + 4)*(5*a**2 - 4))/8,0),(0,0,0,(sqrt( - 
5*a**2 + 4)*a)/4,( - 5*(5*a**2 - 4)*a**2)/16,( - 3*sqrt( - 5*a**2 + 4)*(5*a**2 -
 4)*a)/16,(5*a**2 - 4)**2/16))$
$