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

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

conditionssura:={0}$

phase 1 de la resolution des equations$

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)$

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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},4} qui est  maintenant AA:=d(3,4)*a$

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

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

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

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

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

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

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

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

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

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

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

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

Unknowns: {d(5,6),a}
Unknowns: {d(5,6),a}
pas de selection possible de variable a coefficient independant des xi dans d(5,
6)*(a - 1)
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},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 {{1,2},4} qui est  maintenant AA:= - d(4,4) + 2*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:=2*d(1,1) + d(0,0)$

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

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

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:=0$

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

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

sa valeur doit etre WW:=0$

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

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

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

on resout l'equation {{2,3},6} 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)$

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

Il y a une phase 2$

conditionssura:={1,0}$

conditionssura:={1/2,1,0}$

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

derivation generique de gtildedelta:$

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

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

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

Unknowns: {d(6,3),
d(6,1),
d(6,0),
d(5,3),
d(5,1),
d(5,0),
d(4,1),
d(4,0),
d(3,1),
d(3,0),
d(2,1),
d(0,0),
a}
Unknowns: {d(6,3),
d(6,1),
d(6,0),
d(5,3),
d(5,1),
d(5,0),
d(4,1),
d(4,0),
d(3,1),
d(3,0),
d(2,1),
d(0,0),
a}
Unknowns: {d(6,3),
d(6,1),
d(6,0),
d(5,3),
d(5,1),
d(5,0),
d(4,1),
d(4,0),
d(3,1),
d(3,0),
d(2,1),
d(0,0),
a}
Unknowns: {d(6,3),
d(6,1),
d(6,0),
d(5,3),
d(5,1),
d(5,0),
d(4,1),
d(4,0),
d(3,1),
d(3,0),
d(2,1),
d(0,0),
a}
dim Der(gtildedelta):=12$

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

un element t1 d'un tore $

t1:=D(0,0)$

t1:=

[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  0]
[                   ]
[0  0  0  0  0  0  4]


 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:=

[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  0]
[                   ]
[0  0  0  0  0  0  4]


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),

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

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

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

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

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

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

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

     4*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),

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

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

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

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

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

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

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

     4*d(0,0)))


on voit apparaitre les poids sur la diagonale

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

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

r(3) := 2*d(0,0)

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

r(5) := 3*d(0,0)

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

r(7) := 4*d(0,0)

r(1) := gamma1

r(2) := gamma1

r(3) := 2*gamma1

r(4) := 2*gamma1

r(5) := 3*gamma1

r(6) := 3*gamma1

r(7) := 4*gamma1

Le systeme de poids est le systeme 1.2

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

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

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

   {{0,3},a*x(4)},

   {{0,4},0},

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

   {{0,6},0},

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

   {{1,3},x(5)},

   {{1,4},x(6)},

   {{1,5},0},

   {{1,6},0},

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

   {{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(6)},
{{1,4},diay(5)*a},
{{1,5},0},
{{1,6},diay(7)*(a - 1)},
{{1,7},0},
{{2,3},diay(5)},
{{2,4},diay(6)},
{{2,5},diay(7)},
{{2,6},0},
{{2,7},0},
{{3,4}, - diay(7)},
{{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_{7,1.2}$

(iL)$

and that for a neq{1/2,1,0}$

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/sqrt(a - 1),0,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,( - 1)/sqrt(a - 1),0,0,0)
,(0,0,( - 1)/(a - 1),0,0,0,0),(0,0,0,0,( - 1)/sqrt(a - 1),0,0),(0,0,0,0,0,( - 1)
/(a - 1),0),(0,0,0,0,0,0,( - 1)/sqrt(a - 1)))$

det(isom):= ( - 1)/(a - 1)**4$

ZZ(1):=diay(2)$

ZZ(2):=diay(1)/sqrt(a - 1)$

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

ZZ(4):=( - diay(3))/sqrt(a - 1)$

ZZ(5):=( - diay(5))/sqrt(a - 1)$

ZZ(6):=( - diay(6))/(a - 1)$

ZZ(7):=( - diay(7))/sqrt(a - 1)$

listcommutateursdesZZ:=$

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

{{1,3},zz(6)}$

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

{{1,5},zz(7)}$

{{1,6},0}$

{{1,7},0}$

{{2,3},(zz(5)*a)/(a - 1)}$

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

{{2,5},0}$

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

{{2,7},0}$

{{3,4},( - zz(7))/(a - 1)}$

{{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_{7,1.2}$

(iL)$

with L:=a/(a - 1)$

Et cela pour a:=a$

and that for a neq {1/2,1,0}$

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

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