a:=a$

b:=1$

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

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

phase 1 de la resolution des equations$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

on resout l'equation {{0,5},3} 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 {{0,5},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 {{0,5},5} 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$

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

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:=0$

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

Unknowns: {d(5,6),d(3,1),a}
Unknowns: {d(5,6),d(3,1),a}
pas de selection possible de variable a coefficient numerique dans (d(5,6)*a**2 
- 4*d(5,6) - d(3,1)*a**2 + 4*d(3,1))/6
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},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},
( - (d(5,6) - d(3,1))*(a + 2)*(a - 2))/6},
{{{1,3},3},0},
{{{1,3},4},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},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},
((d(5,6) - d(3,1))*(a + 2)*(a - 2))/6},
{{{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},5},0},
{{{2,5},6},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},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},5},0}}$

Il y a une phase 2$

On suppose a neq {-2,2}. Alors $

d(5,6):=d(3,1)$

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

derivation generique de gtildedelta:$

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

pour delta:=

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


pour shortformdelta:={0,

   ss,

   1,

   0,

   ss,

   1,

   ss,

   0,

   0,

   ss,

   1,

   a}

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

dim Der(gtildedelta):=11$

un element t1 d'un tore $

t1:=D(1,1):=

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


MATD:=

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

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

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

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

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

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

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


Unknowns: {d(5,3),d(5,1),d(4,1),d(3,1),d(1,1),a}

Unknowns: {d(5,3),d(5,1),d(4,1),d(3,1),d(1,1),a}

commutant de t1 dans der(gtildedelta):

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


 t1 est un tore maximal.


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  1  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]


P**(-1)*t2*P:=

[t2  0   0   0   0   0   0 ]
[                          ]
[0   t2  0   0   0   0   0 ]
[                          ]
[0   0   t2  0   0   0   0 ]
[                          ]
[0   0   0   t2  0   0   0 ]
[                          ]
[0   0   0   0   t2  0   0 ]
[                          ]
[0   0   0   0   0   t2  0 ]
[                          ]
[0   0   0   0   0   0   t2]


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

PP:=
mat((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:=
mat((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))$

on voit apparaitre les poids sur la diagonale$

r(1) := 0$

r(2) := d(1,1)$

r(3) := 0$

r(4) := d(1,1)$

r(5) := d(1,1)$

r(6) := d(1,1)$

r(7) := d(1,1)$

r(1) := 0$

r(2) := gamma1$

r(3) := 0$

r(4) := gamma1$

r(5) := gamma1$

r(6) := gamma1$

r(7) := gamma1$

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

               2
      sqrt( - a  + 4)
mat((-----------------,0,0,0,0,0,0),
             2

    (0,1,0,0,0,0,0),

                  2
       - sqrt( - a  + 4)*a      - (a + 2)*(a - 2)
    (----------------------,0,--------------------,0,0,0,0),
               4                       4

                     2
            sqrt( - a  + 4)
    (0,0,0,-----------------,0,0,0),
                   2

                     2
            sqrt( - a  + 4)*a    - (a + 2)*(a - 2)
    (0,0,0,-------------------,--------------------,0,0),
                    4                   4

                     2        2            2
              (a + 2) *(a - 2)    sqrt( - a  + 4)*(a + 2)*(a - 2)*a
    (0,0,0,0,-------------------,-----------------------------------,
                     16                          16

                2        2
       - (a + 2) *(a - 2)
     ----------------------),
               16

                     2                   2
            sqrt( - a  + 4)     sqrt( - a  + 4)*(a + 2)*(a - 2)
    (0,0,0,-----------------,0,---------------------------------,0))
                   2                           8


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

                    2
         - sqrt( - a  + 4)*(diay(3)*a - 2*diay(1))
ZZ(1):=--------------------------------------------
                            4

ZZ(2):=diay(2)

         - (a + 2)*(a - 2)*diay(3)
ZZ(3):=----------------------------
                    4

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

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

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

                  2        2
         - (a + 2) *(a - 2) *diay(6)
ZZ(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}

On obtient donc les relations de commutations de 

g_{7,1.{0,point,1}}

(i).

Et cela pour b:=1.

Et cela pour a different de {-2,2}.

shortformdelta:={0,

                 ss,

                 1,

                 0,

                 ss,

                 1,

                 ss,

                 0,

                 0,

                 ss,

                 1,

                 a}

delta:=

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