a:=1$

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

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

phase 1 de la resolution des equations$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

sa valeur doit etre WW:= - d(1,6)$

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

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

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

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

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

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

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

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

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

on resout l'equation {{0,6},3} 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},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 {{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) + d(0,2))$

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

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

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(1,0))$

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

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

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

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

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

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

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

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

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

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

il n'y a pas de phase 2$

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

derivation generique de gtildedelta:$

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

pour delta:=

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


pour shortformdelta:={1,

   ss,

   1,

   0,

   ss,

   0,

   ss,

   0,

   1,

   ss,

   0,

   0}

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

dim Der(gtildedelta):=12$

un element t1 d'un tore $

t1:=D(0,0)$

t1:=

[1  0  0  0  0  0  0]
[                   ]
[0  2  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  3  0  0  0]
[                   ]
[0  0  0  0  3  0  0]
[                   ]
[0  0  0  0  0  4  0]
[                   ]
[0  0  0  0  0  0  4]


MATD:=

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

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

    ( - d(0,2),0,d(0,0),0,0,0,0),

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

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

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

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

     ))


Unknowns: {d(0,2),d(0,0)}

Unknowns: {d(0,2),d(0,0)}

commutant de t1 dans der(gtildedelta):

[ d(0,0)       0      d(0,2)     0          0           0          0    ]
[                                                                       ]
[    0      2*d(0,0)    0        0          0           0          0    ]
[                                                                       ]
[ - d(0,2)     0      d(0,0)     0          0           0          0    ]
[                                                                       ]
[    0         0        0     3*d(0,0)   - d(0,2)       0          0    ]
[                                                                       ]
[    0         0        0      d(0,2)   3*d(0,0)        0          0    ]
[                                                                       ]
[    0         0        0        0          0       4*d(0,0)    2*d(0,2)]
[                                                                       ]
[    0         0        0        0          0       - 2*d(0,2)  4*d(0,0)]


Unknowns: {d(0,2),d(0,0)}

Unknowns: {d(0,2),d(0,0)}

t2:=D(0,2):=

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


{{x,1,

  [      0       ]
  [              ]
  [arbcomplex(61)]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]

  },

   2
 {x  + 1,

  2,


  [  arbcomplex(62)   ]
  [ ----------------  ]
  [        x          ]
  [                   ]
  [         0         ]
  [                   ]
  [  arbcomplex(62)   ]
  [                   ]
  [  - arbcomplex(63) ]
  [-------------------]
  [         x         ]
  [                   ]
  [  arbcomplex(63)   ]
  [                   ]
  [         0         ]
  [                   ]
  [         0         ]

  },

   2
 {x  + 4,

  1,


  [          0          ]
  [                     ]
  [          0          ]
  [                     ]
  [          0          ]
  [                     ]
  [          0          ]
  [                     ]
  [          0          ]
  [                     ]
  [  - arbcomplex(64)*x ]
  [---------------------]
  [          2          ]
  [                     ]
  [   arbcomplex(64)    ]

  }}

Unknowns: {d(0,2),d(0,0)}

Unknowns: {d(0,2),d(0,0)}

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

[ d(0,0)       0      d(0,2)     0          0           0          0    ]
[                                                                       ]
[    0      2*d(0,0)    0        0          0           0          0    ]
[                                                                       ]
[ - d(0,2)     0      d(0,0)     0          0           0          0    ]
[                                                                       ]
[    0         0        0     3*d(0,0)   - d(0,2)       0          0    ]
[                                                                       ]
[    0         0        0      d(0,2)   3*d(0,0)        0          0    ]
[                                                                       ]
[    0         0        0        0          0       4*d(0,0)    2*d(0,2)]
[                                                                       ]
[    0         0        0        0          0       - 2*d(0,2)  4*d(0,0)]


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


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

[1  0  0  0  0  0  0]
[                   ]
[0  2  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  3  0  0  0]
[                   ]
[0  0  0  0  3  0  0]
[                   ]
[0  0  0  0  0  4  0]
[                   ]
[0  0  0  0  0  0  4]


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

[i  0   0     0    0   0     0   ]
[                                ]
[0  0   0     0    0   0     0   ]
[                                ]
[0  0   - i   0    0   0     0   ]
[                                ]
[0  0   0     - i  0   0     0   ]
[                                ]
[0  0   0     0    i   0     0   ]
[                                ]
[0  0   0     0    0  2*i    0   ]
[                                ]
[0  0   0     0    0   0    - 2*i]


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

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

PP:=
mat((1,0,i,0,0,0,0),(0,1,0,0,0,0,0),(i,0,1,0,0,0,0),(0,0,0,1,i,0,0),(0,0,0,i,1,0
,0),(0,0,0,0,0,1,i),(0,0,0,0,0,i,1))$

avec PP:=P*Q:=
mat((1,0,i,0,0,0,0),(0,1,0,0,0,0,0),(i,0,1,0,0,0,0),(0,0,0,1,i,0,0),(0,0,0,i,1,0
,0),(0,0,0,0,0,1,i),(0,0,0,0,0,i,1))$

on voit apparaitre les poids sur la diagonale$

r(1) := i*d(0,2) + d(0,0)$

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

r(3) :=  - (i*d(0,2) - d(0,0))$

r(4) :=  - i*(d(0,2) + 3*i*d(0,0))$

r(5) := i*(d(0,2) - 3*i*d(0,0))$

r(6) := 2*i*(d(0,2) - 2*i*d(0,0))$

r(7) :=  - 2*i*(d(0,2) + 2*i*d(0,0))$

r(1) := gamma1$

r(2) := gamma1 + gamma2$

r(3) := gamma2$

r(4) := gamma1 + 2*gamma2$

r(5) := 2*gamma1 + gamma2$

r(6) := 3*gamma1 + gamma2$

r(7) := gamma1 + 3*gamma2$

Le systeme de poids est le systeme 2.9$

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

listcommutateursdesx := {{{0,1},x(3)},
{{0,2},x(1)},
{{0,3},x(5)},
{{0,4}, - x(6)},
{{0,5},0},
{{0,6},0},
{{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):=i*x(2) + x(0)$

diaY(2):=x(1)$

diaY(3):=x(2) + i*x(0)$

diaY(4):=i*x(4) + x(3)$

diaY(5):=x(4) + i*x(3)$

diaY(6):=i*x(6) + x(5)$

diaY(7):=x(6) + i*x(5)$

liste des commutateurs des diaY(i) :$

listcommutateurdesdiaY:={{{1,2}, - i*diay(5)},
{{1,3},2*diay(2)},
{{1,4},0},
{{1,5},2*i*diay(6)},
{{1,6},0},
{{1,7},0},
{{2,3}, - i*diay(4)},
{{2,4},0},
{{2,5},0},
{{2,6},0},
{{2,7},0},
{{3,4},2*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_{7,2.9}$

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 :$

[1  0  0    0      0   0   0 ]
[                            ]
[0  0  2    0      0   0   0 ]
[                            ]
[0  1  0    0      0   0   0 ]
[                            ]
[0  0  0    0     2*i  0   0 ]
[                            ]
[0  0  0   - 2*i   0   0   0 ]
[                            ]
[0  0  0    0      0   4   0 ]
[                            ]
[0  0  0    0      0   0  4*i]


det(isom):= 128*i

ZZ(1):=diay(1)

ZZ(2):=diay(3)

ZZ(3):=2*diay(2)

ZZ(4):= - 2*i*diay(5)

ZZ(5):=2*i*diay(4)

ZZ(6):=4*diay(6)

ZZ(7):=4*i*diay(7)

listcommutateursdesZZ:=

{{1,2},zz(3)}

{{1,3},zz(4)}

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

{{1,5},0}

{{1,6},0}

{{1,7},0}

{{2,3},zz(5)}

{{2,4},0}

{{2,5},zz(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 obtient donc les relations de commutations de  g_{7,2.9}

Et cela pour a:=1.

shortformdelta:={1,

                 ss,

                 1,

                 0,

                 ss,

                 0,

                 ss,

                 0,

                 1,

                 ss,

                 0,

                 0}

delta:=

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