!The! generic! nilpotent! derivation! as! in! (!Cohomology! tables! page! 50)! :
! the! eigen\
values
are! 0$



by! subtracting! adjoints! one! then! may! suppose! xi(4,1)=xi(4,2)=xi(5,1)=xi(6
,1)=x\
i(6,2)=0$

delta:=
mat((0,0,0,0,0,0),(xi(2,1),0,0,0,0,0),(xi(3,1),xi(3,2),0,0,0,0),(0,0,0,0,0,0),(0
,xi(5,2), - xi(2,1),0,0,0),(0,0,xi(6,3),xi(5,2) - xi(3,1),xi(2,1),0))$

We denote this delta by the shortform$

shortformdelta:={xi(2,1),
ss,
xi(3,1),
xi(3,2),
ss,
xi(5,2),
ss,
xi(6,3)}$

paramindexeslist:={{2,1},{3,1},{3,2},{5,2},{6,3}}$

a:=0$

delta:=
mat((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))$

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

phase 1 de la resolution des equations$

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

Unknown: d(5,0)
Unknown: d(5,0)
bonne inconnue W:=d(5,0)$

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

Unknown: d(5,3)
Unknown: d(5,3)
bonne inconnue W:=d(5,3)$

sa valeur doit etre WW:=0$

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

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

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

on resout l'equation {{0,4},3} 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},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 {{0,5},3} 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 {{0,6},3} 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 {{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) + d(0,1)$

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

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

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(4,2)$

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

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

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

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

sa valeur doit etre WW:=d(5,2) - d(4,1) - d(3,1)$

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

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

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

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

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

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

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

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

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

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

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

il n'y a pas de phase 2$

derivation generique de gtildedelta:$

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

pour delta:=

[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]


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

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

dim Der(gtildedelta):=13$

t1:=D(0,0):=

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


MATD:=

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

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

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

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

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

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

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

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


{{x + 1,

  4,


  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [ arbcomplex(41)  ]
  [                 ]
  [ - arbcomplex(42)]
  [                 ]
  [ arbcomplex(42)  ]
  [                 ]
  [ arbcomplex(43)  ]
  [                 ]
  [ arbcomplex(44)  ]

  },

 {x,

  2,


  [ - arbcomplex(45)]
  [                 ]
  [ arbcomplex(45)  ]
  [                 ]
  [        0        ]
  [                 ]
  [ arbcomplex(46)  ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]

  },

 {x - 1,1,

  [arbcomplex(47)]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]

  }}

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

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

commutant de t1 dans der(gtildedelta):

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

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

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

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

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

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

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


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

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

t2:=D(1,1):=

[0  -1  0  0  0   0  0]
[                     ]
[0  1   0  0  0   0  0]
[                     ]
[0  0   3  0  0   0  0]
[                     ]
[0  0   0  3  -1  0  0]
[                     ]
[0  0   0  0  4   0  0]
[                     ]
[0  0   0  0  0   5  0]
[                     ]
[0  0   0  0  0   0  6]


{{x - 1,

  1,


  [ - arbcomplex(48)]
  [                 ]
  [ arbcomplex(48)  ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]

  },

 {x - 4,

  1,


  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]
  [                 ]
  [ - arbcomplex(49)]
  [                 ]
  [ arbcomplex(49)  ]
  [                 ]
  [        0        ]
  [                 ]
  [        0        ]

  },

 {x - 5,1,

  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(50)]
  [              ]
  [      0       ]

  },

 {x - 6,1,

  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(51)]

  },

 {x,1,

  [arbcomplex(52)]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]

  },

 {x - 3,

  2,


  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(53)]
  [              ]
  [arbcomplex(54)]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]

  }}

Unknowns: {d(1,1),d(0,0)}

Unknowns: {d(1,1),d(0,0)}

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

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

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

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

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

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

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

    (0,0,0,0,0,0,6*d(1,1) - d(0,0)))


rank 2 with maximal torus t1,t2

2


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

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

[0  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  5  0]
[                   ]
[0  0  0  0  0  0  6]


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

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

PP:=

[1  -1  0  0  0   0  0]
[                     ]
[0  1   0  0  0   0  0]
[                     ]
[0  0   1  0  0   0  0]
[                     ]
[0  0   0  1  -1  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:=

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


MATDDIAGONALISE:=

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

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

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

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

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

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

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

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


on voit apparaitre les poids sur la diagonale

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

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

r(3) := 3*d(1,1) - d(0,0)

r(4) := 3*d(1,1)

r(5) := 4*d(1,1) - d(0,0)

r(6) := 5*d(1,1) - d(0,0)

r(7) := 6*d(1,1) - d(0,0)

r(1) := 3*gamma1 - gamma2

r(2) := gamma1

r(3) := gamma2

r(4) := 3*gamma1

r(5) := gamma1 + gamma2

r(6) := 2*gamma1 + gamma2

r(7) := 3*gamma1 + gamma2

Le systeme de poids est le systeme 2.31

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

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

   {{0,2},x(3)},

   {{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},x(6)},

   {{1,6},0},

   {{2,3},x(6)},

   {{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(1):=x(0)

diaY(2):=x(1) - x(0)

diaY(3):=x(2)

diaY(4):=x(3)

diaY(5):=x(4) - x(3)

diaY(6):=x(5)

diaY(7):=x(6)

liste des commutateurs des diaY(i) :$

listcommutateurdesdiaY:={{{1,2},0},
{{1,3},diay(4)},
{{1,4},0},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3},diay(5)},
{{2,4},0},
{{2,5},diay(6)},
{{2,6},diay(7)},
{{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_{7,2.31}$

pour a neqconditionssura$

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((0,0,-1,0,0,0,0),(-1,0,0,0,0,0,0),(0,-1,0,0,0,0,0),(0,0,0,0,-1,0,0),(0,0,0,1
,0,0,0),(0,0,0,0,0,-1,0),(0,0,0,0,0,0,1))$

det(isom):= 1$

ZZ(1):= - diay(2)$

ZZ(2):= - diay(3)$

ZZ(3):= - diay(1)$

ZZ(4):=diay(5)$

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

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

ZZ(7):=diay(7)$

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},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}$

We get the commutation relations of$

g_{7,2.31}$

Et cela pour a:=0.$

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

delta:=
mat((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))$