a:=0$

b:=0$

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

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

phase 1 de la resolution des equations$

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

Unknown: d(3,1)
Unknown: d(3,1)
bonne inconnue W:=d(3,1)$

sa valeur doit etre WW:=0$

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},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))$

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

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

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

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

sa valeur doit etre WW:=d(3,2) - 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(2,3) - d(1,6)$

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

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

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)$

Unknown: d(4,6)
Unknown: d(4,6)
bonne inconnue W:=d(4,6)$

sa valeur doit etre WW:=0$

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

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

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

on resout l'equation {{0,4},1} 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:=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 {{0,5},1} 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 {{0,5},6} 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 {{1,2},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},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)$

Unknown: d(6,4)
Unknown: d(6,4)
bonne inconnue W:=d(6,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,3},4} 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 {{2,3},1} 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 {{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(4,3)$

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

on resout l'equation {{2,4},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$

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

il n'y a pas de phase 2$

collect_eq:={{{{0,1},1},0},
{{{0,1},4},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},1},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},1},0},
{{{0,5},6},0},
{{{0,6},1},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},6},0},
{{{1,4},4},0},
{{{1,4},5},0},
{{{1,5},4},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},6},0},
{{{3,6},6},0},
{{{4,5},5},0},
{{{4,6},5},0}}$

derivation generique de gtildedelta:$

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

pour delta:=

[0  1  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]


pour shortformdelta:={1,

   ss,

   0,

   0,

   ss,

   0,

   ss,

   0,

   0,

   ss,

   0,

   1}

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

dim Der(gtildedelta):=15$

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


MATD:=

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

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

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

    (d(3,0),0,d(3,2),d(3,3),0,0,0),

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

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

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


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

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

commutant de t1 dans der(gtildedelta):

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

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

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

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

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

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

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


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

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

t2:=D(0,2):=

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


{{x + 1,

  2,


  [ - arbcomplex(34)]
  [                 ]
  [ arbcomplex(33)  ]
  [                 ]
  [ arbcomplex(34)  ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]

  },

 {x + 2,1,

  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(35)]
  [              ]
  [      0       ]
  [              ]
  [      0       ]

  },

 {x + 3,1,

  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(36)]
  [              ]
  [      0       ]

  },

 {x,

  3,


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

  }}

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

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

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

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

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

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

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

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

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

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


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

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

t3:=D(3,3):=

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


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

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

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

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

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

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

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

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

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

    (0,0,0,0,0,0,d(3,3) + 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  -1  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  2  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  0  0  0  0]
[                   ]
[0  0  0  0  3  0  0]
[                   ]
[0  0  0  0  0  4  0]
[                   ]
[0  0  0  0  0  0  1]


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


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

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

on voit apparaitre les poids sur la diagonale$

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

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

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

r(4) := d(3,3)$

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

r(6) :=  - (3*d(0,2) - 4*d(0,0))$

r(7) := d(3,3) + d(0,0)$

r(1) := gamma2$

r(2) := gamma1 + gamma2$

r(3) := gamma1$

r(4) := gamma3$

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

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

r(7) := gamma2 + gamma3$

Le systeme de poids est le systeme 3.3$

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

listcommutateursdesx := {{{0,1},0},
{{0,2},x(1)},
{{0,3},x(6)},
{{0,4},0},
{{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):=x(0)$

diaY(2):=x(1)$

diaY(3):=x(2) - x(0)$

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},0},
{{1,3},diay(2)},
{{1,4},diay(7)},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3},diay(5)},
{{2,4},0},
{{2,5},0},
{{2,6},0},
{{2,7},0},
{{3,4},0},
{{3,5},diay(6)},
{{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_{7,3.3}$

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 :$

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


det(isom):= 1

ZZ(1):=diay(3)

ZZ(2):=diay(1)

ZZ(3):=diay(4)

ZZ(4):= - diay(2)

ZZ(5):=diay(7)

ZZ(6):=diay(5)

ZZ(7):=diay(6)

listcommutateursdesZZ:=

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

{{1,3},0}

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

{{1,5},0}

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

{{1,7},0}

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

{{2,4},0}

{{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 commutations de  g_{7,3.3}

Et cela pour a:=0, b:=0.

shortformdelta:={1,

                 ss,

                 0,

                 0,

                 ss,

                 0,

                 ss,

                 0,

                 0,

                 ss,

                 0,

                 1}

delta:=

[0  1  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]