In the present case 2 we suppose A=((0,0),(0,0)).$

a neq {}$

a:=a$

b:=0$

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

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

phase 1 de la resolution des equations$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Unknowns: {d(0,3),a}
Unknowns: {d(0,3),a}
pas de selection possible de variable a coefficient numerique dans d(0,3)*a
Derivation equations to cancel (Reduce output) : \\{{{{0,1},4},d(2,1)*a},
{{{0,1},5},0},
{{{0,1},6},0},
{{{0,2},0},0},
{{{0,2},1},0},
{{{0,2},2},0},
{{{0,2},3},2*d(5,4)*a**2},
{{{0,2},4},
 - (d(3,3) - d(1,1) + d(0,2) - d(0,0))*a},
{{{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},4},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},4},0},
{{{0,6},6},0},
{{{1,2},0},0},
{{{1,2},1},0},
{{{1,2},2},0},
{{{1,2},3},
 - (d(0,2) - d(0,0) - d(1,1) + d(3,3))*a},
{{{1,2},4},0},
{{{1,2},5},0},
{{{1,2},6},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},5},0},
{{{1,5},6},2*d(0,1)*a},
{{{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}, - d(0,3)*a},
{{{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},
(d(0,2) - d(0,0) - d(1,1) + d(3,3) + d(0,2))*a},
{{{2,6},4},0},
{{{2,6},5},0},
{{{2,6},6},0},
{{{3,4},6},0},
{{{3,5},6},d(0,3)*a},
{{{3,6},6},0},
{{{4,5},6},0},
{{{4,6},6},0},
{{{5,6},6},0}}$

Il y a une phase 2$

a neq {0}$

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

pour delta:=

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


pour shortformdelta:={0,

   0,

   ss,

   0,

   a,

   ss,

   0,

   0,

   ss,

   1,

   0}

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

dim Der(gtildedelta):=15$

t1:=D(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  2  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  2]


MATD:=

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

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

    (0,0,d(5,3)*a + d(0,0),0,0,0,0),

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

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

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

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

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


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

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

commutant de t1 dans der(gtildedelta):

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

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

    (0,0,d(5,3)*a + d(0,0),0,0,0,0),

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

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

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

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


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

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

t2:=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  0  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


Unknowns: {d(5,3),d(1,1),d(0,0),a}

Unknowns: {d(5,3),d(1,1),d(0,0),a}

commutant simultane de t1,t2 dans der(gtildedelta):

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

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

    (0,0,d(5,3)*a + d(0,0),0,0,0,0),

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

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

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

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


Unknowns: {d(5,3),d(1,1),d(0,0),a}

Unknowns: {d(5,3),d(1,1),d(0,0),a}

t3:=D(5,3):=

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


{{x,

  3,


  [  arbcomplex(215)   ]
  [                    ]
  [  arbcomplex(216)   ]
  [                    ]
  [         0          ]
  [                    ]
  [ - arbcomplex(217)*a]
  [                    ]
  [         0          ]
  [                    ]
  [  arbcomplex(217)   ]
  [                    ]
  [         0          ]

  },

 { - (a - x),

  4,


  [       0       ]
  [               ]
  [       0       ]
  [               ]
  [arbcomplex(218)]
  [               ]
  [       0       ]
  [               ]
  [arbcomplex(219)]
  [               ]
  [arbcomplex(220)]
  [               ]
  [arbcomplex(221)]

  }}

Unknowns: {d(5,3),d(1,1),d(0,0),a}

Unknowns: {d(5,3),d(1,1),d(0,0),a}

commutant simultane de t1,t2,t3 dans der(gtildedelta):

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

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

    (0,0,d(5,3)*a + d(0,0),0,0,0,0),

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

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

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

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


rank 3 with maximal torus t1,t2,t3

3


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


P**(-1)*t1*P:=

[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  2  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  2]

P**(-1)*t2*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  0  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]

P**(-1)*t3*P:=

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


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

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


MATDDIAGONALISE:=

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

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

    (0,0,d(5,3)*a + d(0,0),0,0,0,0),

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

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

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

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

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

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


on voit apparaitre les poids sur la diagonale

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

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

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

r(4) := d(1,1) + d(0,0)

r(5) := d(5,3)*a + 2*d(0,0)

r(6) := d(1,1) + d(0,0) + d(5,3)*a

r(7) := d(1,1) + 2*d(0,0) + d(5,3)*a

r(1) := gamma2

r(2) := gamma3

r(3) := gamma1

r(4) := gamma2 + gamma3

r(5) := gamma1 + gamma2

r(6) := gamma1 + gamma3

r(7) := gamma2 + gamma3 + gamma1

Le systeme de poids est le systeme 3.1

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

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

   {{0,2},a*x(4)},

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

   {{0,4},0},

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

   {{0,6},0},

   {{1,2},x(5)},

   {{1,3},0},

   {{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)

           - x(5) + x(3)*a
diaY(4):=------------------
                 a

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(5)*a},
{{1,4},0},
{{1,5},0},
{{1,6},diay(7)*a},
{{1,7},0},
{{2,3},diay(6)},
{{2,4},0},
{{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_{7,3.1}$

(iii)$

pour a neq{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((0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,0,0, - a,0),(0,0,0, -
 a,0,0,0),(0,0,0,0,-1,0,0),(0,0,0,0,0,0, - a))$

det(isom):= a**3$

ZZ(1):=diay(3)$

ZZ(2):=diay(1)$

ZZ(3):=diay(2)$

ZZ(4):= - diay(5)*a$

ZZ(5):= - diay(6)$

ZZ(6):= - diay(4)*a$

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

listcommutateursdesZZ:=$

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

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

{{1,4},0}$

{{1,5},0}$

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

{{1,7},0}$

{{2,3},0}$

{{2,4},0}$

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

{{2,6},0}$

{{2,7},0}$

{{3,4},zz(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}$

On obtient donc les relations de commutations de $

g_{7,3.1}$

(iii)$

Et cela pour a:=a.$

Et cela pour b:=0.$

Et cela pour a different de {0}.$

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

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