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))$

shortformdelta:={0,
ss,
0,
0,
ss,
0,
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(2,0)$

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},5} qui est  maintenant AA:= - d(3,0)$

Unknown: d(3,0)
Unknown: d(3,0)
bonne inconnue W:=d(3,0)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

on resout l'equation {{1,4},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)/2$

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

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

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

il n'y a pas de phase 2$

derivation generique de gtildedelta:$

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

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]


pour shortformdelta:={0,

   ss,

   0,

   0,

   ss,

   0,

   ss,

   0,

   0,

   ss,

   1,

   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(4,3),
d(4,2),
d(4,1),
d(3,2),
d(3,1),
d(2,2),
d(2,1),
d(0,3),
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(4,3),
d(4,2),
d(4,1),
d(3,2),
d(3,1),
d(2,2),
d(2,1),
d(0,3),
d(0,2),
d(0,1),
d(0,0)}
listeparametresMATD{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(4,3),
d(4,2),
d(4,1),
d(3,2),
d(3,1),
d(2,2),
d(2,1),
d(0,3),
d(0,2),
d(0,1),
d(0,0)}$

dim Der(gtildedelta):=19$

un element t1 d'un tore $

t1:=D(0,0)$

t1:=

[1   0   0  0   0    0   0]
[                         ]
[    1                    ]
[0  ---  0  0   0    0   0]
[    2                    ]
[                         ]
[0   0   0  0   0    0   0]
[                         ]
[0   0   0  1   0    0   0]
[                         ]
[               1         ]
[0   0   0  0  ---   0   0]
[               2         ]
[                         ]
[                    3    ]
[0   0   0  0   0   ---  0]
[                    2    ]
[                         ]
[0   0   0  0   0    0   1]


MATD:=

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

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

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

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

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

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

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

     d(2,2) + d(0,0)))


             2                  3
  - (2*x - 1) *(2*x - 3)*(x - 1) *x
------------------------------------
                 8

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

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

commutant de t1 dans der(gtildedelta):

mat((d(0,0),0,0,d(0,3),0,0,0),

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

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

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

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

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

    (d(6,0),0,0,d(6,3),0,0,d(2,2) + d(0,0)))


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

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

t2:=D(2,2):=

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


Unknowns: {d(2,2),d(0,3),d(0,0)}

Unknowns: {d(2,2),d(0,3),d(0,0)}

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

mat((d(0,0),0,0,d(0,3),0,0,0),

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

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

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

              2*d(2,2) + d(0,0)
    (0,0,0,0,-------------------,0,0),
                      2

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

    (0,0,0,0,0,0,d(2,2) + d(0,0)))


Unknowns: {d(2,2),d(0,3),d(0,0)}

Unknowns: {d(2,2),d(0,3),d(0,0)}

t3:=D(0,3):=

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


{{x + 1,

  2,


  [ - arbcomplex(35)]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [ arbcomplex(35)  ]
  [                 ]
  [        0        ]
  [                 ]
  [ arbcomplex(36)  ]
  [                 ]
  [        0        ]

  },

 {x,

  5,


  [arbcomplex(37)]
  [              ]
  [arbcomplex(38)]
  [              ]
  [arbcomplex(39)]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(40)]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(41)]

  }}

Unknowns: {d(2,2),d(0,3),d(0,0)}

Unknowns: {d(2,2),d(0,3),d(0,0)}

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

mat((d(0,0),0,0,d(0,3),0,0,0),

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

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

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

              2*d(2,2) + d(0,0)
    (0,0,0,0,-------------------,0,0),
                      2

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

    (0,0,0,0,0,0,d(2,2) + d(0,0)))



le calcul du commutant de t1 et t2 et t3 dans der(gtildedelta) montre  que t1,\
t2,t3 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  -1  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]
[                         ]
[    1                    ]
[0  ---  0  0   0    0   0]
[    2                    ]
[                         ]
[0   0   0  0   0    0   0]
[                         ]
[0   0   0  1   0    0   0]
[                         ]
[               1         ]
[0   0   0  0  ---   0   0]
[               2         ]
[                         ]
[                    3    ]
[0   0   0  0   0   ---  0]
[                    2    ]
[                         ]
[0   0   0  0   0    0   1]


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

[0  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  0  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  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]


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

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

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

ladiag := {{1,d(0,0)},
{2,d(0,0)/2},
{3,d(2,2)},
{4, - (d(0,3) - d(0,0))},
{5,(2*d(2,2) + d(0,0))/2},
{6,( - (2*d(0,3) - 3*d(0,0)))/2},
{7,d(2,2) + d(0,0)}}$

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},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) - x(0)$

diaY(5):=x(4)$

diaY(6):=x(5)$

diaY(7):=x(6)$

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

{{{1,2},0},
{{1,3},diadiay(7)},
{{1,4},0},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3},diadiay(5)},
{{2,4},diadiay(6)},
{{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 :$

avec comme matrice de changement de base :$

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


det(isom):= 1

*** zz declared operator 

ZZ(1):=diay(2)

ZZ(2):=diay(3)

ZZ(3):=diay(4)

ZZ(4):= - diay(1)

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(6)},
{{1,4},0},
{{1,5},zzz(7)},
{{1,6},0},
{{1,7},0},
{{2,3},0},
{{2,4},zzz(7)},
{{2,5},0},
{{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 commutation de g_{7,3.8}.$