generic derivation : delta:=
mat((xi(1,1),xi(1,2),xi(1,3),0,0,0),(xi(2,1),xi(2,2),xi(2,3),0,0,0),(xi(3,1),xi(
3,2),xi(3,3),0,0,0),(xi(4,1),xi(4,2),xi(4,3),xi(2,2) + xi(1,1),xi(2,3), - xi(1,3
)),(xi(5,1),xi(5,2),xi(5,3),xi(3,2),xi(3,3) + xi(1,1),xi(1,2)),(xi(6,1),xi(6,2),
xi(6,3), - xi(3,1),xi(2,1),xi(3,3) + xi(2,2)))$
$


        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
adx1 := [                ]
        [0  1  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  0  0  0  0]
adx2 := [                 ]
        [-1  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]
adx3 := [                  ]
        [0   0   0  0  0  0]
        [                  ]
        [-1  0   0  0  0  0]
        [                  ]
        [0   -1  0  0  0  0]


by subtracting adjoints one then may suppose:

xi(4,1):=0,xi(4,2):=0,xi(5,1):=0.

In the case 3, supposes :

xi(1,1):=0

xi(1,2):=0

xi(1,3):=0

xi(2,1):=0

xi(2,2):=0

xi(2,3):=0

xi(3,1):=0

xi(3,2):=0

xi(3,3):=0

delta:=

[   0        0        0     0  0  0]
[                                  ]
[   0        0        0     0  0  0]
[                                  ]
[   0        0        0     0  0  0]
[                                  ]
[   0        0     xi(4,3)  0  0  0]
[                                  ]
[   0     xi(5,2)  xi(5,3)  0  0  0]
[                                  ]
[xi(6,1)  xi(6,2)  xi(6,3)  0  0  0]


We denote this delta by the shortform

shortformdelta:={xi(4,3),

                 ss,

                 xi(5,2),

                 xi(5,3),

                 ss,

                 xi(6,1),

                 xi(6,2),

                 xi(6,3)}

paramindexeslist:={{4,3},{5,2},{5,3},{6,1},{6,2},{6,3}}

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

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

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) + d(2,1)$

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

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

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(0,5)$

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

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(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 {{0,3},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) + d(2,3) + d(1,
0)$

Unknowns: {d(5,6),d(2,3),d(1,0)}
Unknowns: {d(5,6),d(2,3),d(1,0)}
bonne inconnue W:=d(5,6)$

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

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},5} 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 {{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) + 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) + d(0,
1)$

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

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

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

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

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

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

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

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

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

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

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

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:= - 2*d(2,3) + d(1,2) - d(
0,3)$

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

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

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

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

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


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

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

dim Der(gtildedelta):=19$

t1:=D(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  2  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  2  0]
[                   ]
[0  0  0  0  0  0  3]


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

Unknowns: {d(5,3),d(4,2),d(4,0),d(1,1),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(1,1),0,0,0,0,0),

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

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

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

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

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


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

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

t2:=D(1,1):=

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


Unknowns: {d(1,1),d(0,2),d(0,0)}
Unknowns: {d(1,1),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(1,1),0,0,0,0,0),(0,0, - (d(0,2) - d(0,0)),0,0
,0,0),(0,0,0, - (d(0,2) - 2*d(0,0) + d(1,1)),0,0,0),(0,0,0,0, - (d(0,2) - d(0,0)
 - d(1,1)),0,0),(0,0,0,0,0, - (d(0,2) - 2*d(0,0)),0),(0,0,0,0,0,0, - (d(0,2) - 3
*d(0,0) + d(1,1))))$
$

Unknowns: {d(1,1),d(0,2),d(0,0)}

Unknowns: {d(1,1),d(0,2),d(0,0)}

t3:=D(0,2):=

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


{{x + 1,

  5,


  [ - arbcomplex(49)]
  [                 ]
  [        0        ]
  [                 ]
  [ arbcomplex(49)  ]
  [                 ]
  [ arbcomplex(50)  ]
  [                 ]
  [ arbcomplex(51)  ]
  [                 ]
  [ arbcomplex(52)  ]
  [                 ]
  [ arbcomplex(53)  ]

  },

 {x,

  2,


  [arbcomplex(54)]
  [              ]
  [arbcomplex(55)]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]

  }}

Unknowns: {d(1,1),d(0,2),d(0,0)}

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

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

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

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

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

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


rank 3 with maximal torus t1,t2,t3

3


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

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

[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  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  -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]


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

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

PP:=

[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:=

[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]


MATDDIAGONALISE:=

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

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

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

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

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

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

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

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

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

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


on voit apparaitre les poids sur la diagonale

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

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

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

r(4) :=  - (d(0,2) - 2*d(0,0) + d(1,1))

r(5) :=  - (d(0,2) - d(0,0) - d(1,1))

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

r(7) :=  - (d(0,2) - 3*d(0,0) + d(1,1))

r(1) := gamma2

r(2) := gamma3

r(3) :=  - (gamma2 - gamma3 - gamma1)

r(4) := gamma1

r(5) :=  - (gamma2 - gamma3 - gamma1 - gamma3)

r(6) := gamma1 + gamma3

r(7) := gamma1 + gamma2

Le systeme de poids est le systeme 3.12

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

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

   {{0,2},x(5)},

   {{0,3},x(6)},

   {{0,4},0},

   {{0,5},0},

   {{0,6},0},

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

   {{1,3},x(5)},

   {{1,4},0},

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

Now we make explicit the isomorphism with an algebra of the book:$

Namely g_{7,3.12}$

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

det(isom):= -1$

ZZ(1):=diay(4)$

ZZ(2):=diay(1)$

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

ZZ(4):=diay(3)$

ZZ(5):= - diay(7)$

ZZ(6):=diay(6)$

ZZ(7):= - diay(5)$

listcommutateursdesZZ:=$

{{1,2},zz(5)}$

{{1,3},zz(6)}$

{{1,4},0}$

{{1,5},0}$

{{1,6},0}$

{{1,7},0}$

{{2,3},0}$

{{2,4},zz(6)}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4},zz(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}$

We get the commutation relations of$

g_{7,3.12}$

Et cela pour a:=a$

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

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

The isomorphism from g_{7,rkgtildedelta.1} to gtildedelta$

was constructed in 2 steps and is given by$

the product matrix P*isom:=
mat((0,1,0,-1,0,0,0),(0,0,-1,0,0,0,0),(0,0,0,1,0,0,0),(1,0,0,0,0,0,0),(0,0,0,0,0
,0,-1),(0,0,0,0,0,1,0),(0,0,0,0,-1,0,0))$
$

which we record here under the name PSI$

PSIcase3V:=
mat((0,1,0,-1,0,0,0),(0,0,-1,0,0,0,0),(0,0,0,1,0,0,0),(1,0,0,0,0,0,0),(0,0,0,0,0
,0,-1),(0,0,0,0,0,1,0),(0,0,0,0,-1,0,0))$
$