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

phase 1 de la resolution des equations$

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

Unknown: d(4,0)
Unknown: d(4,0)
bonne inconnue W:=d(4,0)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Unknown: d(4,3)
Unknown: d(4,3)
bonne inconnue W:=d(4,3)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

on resoud l'equation {{2,4},6} 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)$

Derivation equations to cancel (Reduce output) : \\{{{{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},5},0},
{{{0,3},6},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},6},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},4},0},
{{{1,3},5},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},4},0},
{{{1,5},5},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},5},0},
{{{2,5},6},0},
{{{2,6},4},0},
{{{2,6},5},0},
{{{2,6},6},0},
{{{3,4},5},0},
{{{3,4},6},0},
{{{3,5},5},0},
{{{3,6},5},0},
{{{4,5},5},0},
{{{4,5},6},0},
{{{4,6},5},0},
{{{4,6},6},0}}$

phase 2$

pas de phase 2$

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

derivation generique de gtildedelta:

MATD:=

mat((d(0,0),d(0,1),d(0,2),0,0,0,0),

    (0,d(1,1),d(1,2),0,0,0,0),

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

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

    (0,d(4,1),d(4,2),0,d(0,0),0,0),

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

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


pour delta:=

[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  1  0  0  0  0]


Unknowns: {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(3,2),
d(3,1),
d(1,2),
d(1,1),
d(0,2),
d(0,1),
d(0,0)}
Unknowns: {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(3,2),
d(3,1),
d(1,2),
d(1,1),
d(0,2),
d(0,1),
d(0,0)}
dim Der(gtildedelta):=17$

un element t1 d'un tore$

t1:=D(0,0)$

t1:=

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



on peut prendre comme element semi simple du commutant de t1 dans der(gtildede\
lta):

t2:=

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


 t1,t2 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  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  2]


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

[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  0  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((d(0,0),d(0,1),d(0,2),0,0,0,0),

    (0,d(1,1),d(1,2),0,0,0,0),

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

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

    (0,d(4,1),d(4,2),0,d(0,0),0,0),

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

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



 A la lecture de cette matrice, on constate qu'elle n'est pas triangulaire; po\
ur la rendre triangulaire, il faut mettre Y(3) en 1, Y(1) en 3, et permuter Y6\
 et Y7 i.e. faire le changement de base de matrice de passage Q:

PP**(-1)*MATD*PP:=

mat((d(0,0) - d(1,1),0,0,0,0,0,0),

    (d(1,2),d(1,1),0,0,0,0,0),

    (d(0,2),d(0,1),d(0,0),0,0,0,0),

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

    (d(4,2),d(4,1),0,0,d(0,0),0,0),

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

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


avec PP:=P*Q:=

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


MATDDIAGONALISE:=

mat((d(0,0) - d(1,1),0,0,0,0,0,0),

    (d(1,2),d(1,1),0,0,0,0,0),

    (d(0,2),d(0,1),d(0,0),0,0,0,0),

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

    (d(4,2),d(4,1),0,0,d(0,0),0,0),

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

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


on voit apparaitre les poids sur la diagonale

ladiag := {{1,d(0,0) - d(1,1)},

           {2,d(1,1)},

           {3,d(0,0)},

           {4,2*d(1,1)},

           {5,d(0,0)},

           {6,2*d(0,0) - d(1,1)},

           {7,d(0,0) + d(1,1)}}

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

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

   {{0,2},x(6)},

   {{0,3},0},

   {{0,4},0},

   {{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(5)},

   {{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 declared operator 

diaY(1):=x(2)

diaY(2):=x(1)

diaY(3):=x(0)

diaY(4):=x(3)

diaY(5):=x(4)

diaY(6):=x(6)

diaY(7):=x(5)

listcommutateursdesdiay := {{{1,2}, - x(4)},

   {{1,3}, - x(6)},

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

   {{1,5},x(6)},

   {{1,6},0},

   {{1,7},0},

   {{2,3},0},

   {{2,4},0},

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

   {{2,6},0},

   {{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}}

*** yy declared operator 

*** diadiay declared operator 

liste des commutateurs des diaY(i) := (diadiaY=diaY

{{{1,2}, - diadiay(5)},

 {{1,3}, - diadiay(6)},

 {{1,4},diadiay(7)},

 {{1,5},diadiay(6)},

 {{1,6},0},

 {{1,7},0},

 {{2,3},0},

 {{2,4},0},

 {{2,5},diadiay(7)},

 {{2,6},0},

 {{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}}

on pose :

*** zz declared operator 

ZZ(1):=diay(1)

ZZ(2):= - diay(2)

ZZ(3):= - diay(4)

ZZ(4):=diay(3) + diay(5)

ZZ(5):=diay(5)

ZZ(6):=diay(6)

ZZ(7):= - diay(7)

*** zzz declared operator 

liste des commutateurs des ZZ(i) (ZZZ=ZZ:=

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

 {{1,3},zzz(7)},

 {{1,4},0},

 {{1,5},zzz(6)},

 {{1,6},0},

 {{1,7},0},

 {{2,3},0},

 {{2,4},zzz(7)},

 {{2,5},zzz(7)},

 {{2,6},0},

 {{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}}

C'est les relations de commutations de g_{7,2.27} page 599