generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),2*xi(1,1),0,0 ,0),(xi(4,1),xi(4,2), - xi(2,1),xi(2,2) + xi(1,1),0,0),(xi(5,1),xi(5,2),xi(5,3), xi(4,2) - xi(3,1),xi(2,2) + 2*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),0,0,xi(6 ,6)))$ [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 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] adx2 := [ ] [-1 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 0 0 0 0] adx3 := [ ] [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 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=0 delta:= [ 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 - xi(2,1) 0 0 0 ] [ ] [ 0 0 xi(5,3) - xi(3,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,3), xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,3},{5,6},{6,1},{6,2},{6,3}} a neq {-2}$ a:=a$ delta:= mat((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,-1,0,a),(0,1, 0,0,0,0))$ shortformdelta:={0, ss, 1, 0, ss, 0, a, ss, 0, 1, 0}$ 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(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 {{0,1},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(6,1)*a - d(5,3) - d(4,1 ) - d(4,0)$ Unknowns: {d(6,1),d(5,3),d(4,1),d(4,0),a} Unknowns: {d(6,1),d(5,3),d(4,1),d(4,0),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(6,1)*a - d(4,1) - d(4,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,3) + d(2,1)$ Unknowns: {d(6,3),d(2,1)} Unknowns: {d(6,3),d(2,1)} bonne inconnue W:=d(6,3)$ 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) + d(1,2)$ Unknowns: {d(3,6),d(1,2)} Unknowns: {d(3,6),d(1,2)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(1,2)$ 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(6,2)*a - d(5,6) - d(4,2 ) - d(3,0)$ Unknowns: {d(6,2),d(5,6),d(4,2),d(3,0),a} Unknowns: {d(6,2),d(5,6),d(4,2),d(3,0),a} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=d(6,2)*a - d(4,2) - d(3,0)$ 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(2,1)*a + 2*d(2,0)$ Unknowns: {d(2,1),d(2,0),a} Unknowns: {d(2,1),d(2,0),a} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=( - d(2,1)*a)/2$ on resout l'equation {{0,4},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 {{0,4},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 {{0,4},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 {{0,4},3} qui est maintenant AA:=d(3,5) + d(1,4)$ Unknowns: {d(3,5),d(1,4)} Unknowns: {d(3,5),d(1,4)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,4},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 {{0,4},5} qui est maintenant AA:=d(6,4)*a + d(5,5) - d(4,4 ) + d(1,0) - d(0,0)$ Unknowns: {d(6,4),d(5,5),d(4,4),d(1,0),d(0,0),a} Unknowns: {d(6,4),d(5,5),d(4,4),d(1,0),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(6,4)*a + d(4,4) - d(1,0) + d(0,0)$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(6,5) + d(2,4)$ Unknowns: {d(6,5),d(2,4)} Unknowns: {d(6,5),d(2,4)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(2,4)$ on resout l'equation {{0,5},5} qui est maintenant AA:= - d(2,4)*a$ Unknowns: {d(2,4),a} Unknowns: {d(2,4),a} pas de selection possible de variable a coefficient numerique dans - d(2,4)*a on resout l'equation {{0,6},3} qui est maintenant AA:=d(1,4)*a$ Unknowns: {d(1,4),a} Unknowns: {d(1,4),a} pas de selection possible de variable a coefficient numerique dans d(1,4)*a on resout l'equation {{0,6},5} qui est maintenant AA:=d(6,4)*a**2 - d(4,4)*a + d(2,2)*a + d(1,0)*a - d(1,0) + d(0,0)*a$ Unknowns: {d(6,4),d(4,4),d(2,2),d(1,0),d(0,0),a} Unknowns: {d(6,4),d(4,4),d(2,2),d(1,0),d(0,0),a} pas de selection possible de variable a coefficient numerique dans d(6,4)*a**2 - d(4,4)*a + d(2,2)*a + d(1,0)*a - d(1,0) + d(0,0)*a on resout l'equation {{0,6},6} qui est maintenant AA:=d(2,4)*a$ Unknowns: {d(2,4),a} Unknowns: {d(2,4),a} pas de selection possible de variable a coefficient numerique dans d(2,4)*a 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},2} 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 {{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) + 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) - d(3, 1)$ Unknowns: {d(5,4),d(4,2),d(3,1)} Unknowns: {d(5,4),d(4,2),d(3,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,4) + d(0,1)$ Unknowns: {d(6,4),d(0,1)} Unknowns: {d(6,4),d(0,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,3},5} qui est maintenant AA:=(d(2,1)*(a + 2))/2$ Unknowns: {d(2,1),a} Unknowns: {d(2,1),a} pas de selection possible de variable a coefficient numerique dans (d(2,1)*(a + 2))/2 on resout l'equation {{1,4},5} qui est maintenant AA:=d(1,1) + d(1,0) + d(0,1)* a - d(0,1) - d(0,0)$ Unknowns: {d(1,1),d(1,0),d(0,1),d(0,0),a} Unknowns: {d(1,1),d(1,0),d(0,1),d(0,0),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:= - d(1,1) - d(0,1)*a + d(0,1) + d(0,0)$ on resout l'equation {{1,6},5} 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)$ on resout l'equation {{2,4},5} qui est maintenant AA:=d(1,2) - 2*d(0,2)$ Unknowns: {d(1,2),d(0,2)} Unknowns: {d(1,2),d(0,2)} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=2*d(0,2)$ on resout l'equation {{2,6},5} qui est maintenant AA:=d(0,2)*(a + 2)$ Unknowns: {d(0,2),a} Unknowns: {d(0,2),a} pas de selection possible de variable a coefficient numerique dans d(0,2)*(a + 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},3},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},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},0},0}, {{{0,6},1},0}, {{{0,6},2},0}, {{{0,6},3},0}, {{{0,6},4},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},(d(2,1)*(a + 2))/2}, {{{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},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,6},3},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},d(0,2)*(a + 2)}, {{{2,6},6},0}, {{{3,4},5},0}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}, {{{5,6},5},0}}$ Il y a une phase 2$ a neq {-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},3},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},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},0},0}, {{{0,6},1},0}, {{{0,6},2},0}, {{{0,6},3},0}, {{{0,6},4},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}, {{{1,4},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},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,6},3},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}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}, {{{5,6},5},0}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(1,1) - d(0,0),0,0,0,0,0),(a*( - d(1,1) + d(0,0)),d(1,1),0,0,0,0,0) ,(0,0,d(2,2),0,0,0,0),(d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0),0,0,0),(d(4,0),d(4,1 ),d(4,2),0,d(2,2) + d(1,1),0,a*( - d(1,1) + d(0,0))),(d(5,0),d(5,1),d(5,2),d(6,1 )*a - d(4,1) - d(4,0),d(4,2) - d(3,1),d(2,2) + d(1,1) + d(0,0),d(6,2)*a - d(4,2) - d(3,0)),(d(6,0),d(6,1),d(6,2),0,d(1,1) - d(0,0),0,d(2,2) + d(0,0)))$ pour delta:= [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 -1 0 a] [ ] [0 1 0 0 0 0] pour shortformdelta:={0, ss, 1, 0, ss, 0, a, ss, 0, 1, 0} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,2), d(1,1), d(0,0), a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,2), d(1,1), d(0,0), a} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,2), d(1,1), d(0,0)}$ dim Der(gtildedelta):=15$ t1:=D(0,0):= [1 -1 0 0 0 0 0] [ ] [a 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 a] [ ] [0 0 0 0 0 1 0] [ ] [0 0 0 0 -1 0 1] {{x,1, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(139)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 1, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(140)] [ ] [ 0 ] [ ] [arbcomplex(141)] [ ] [ 0 ] }, 2 {a + x - x, 2, [ - arbcomplex(142) ] [ -------------------- ] [ x - 1 ] [ ] [ arbcomplex(142) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(143)*(a + x - 1) ] [-----------------------------] [ x - 1 ] [ ] [ 0 ] [ ] [ arbcomplex(143) ] }} Unknowns: {d(6,0),d(5,0),d(4,2),d(4,1),d(4,0),d(3,0),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(6,0),d(5,0),d(4,2),d(4,1),d(4,0),d(3,0),d(2,2),d(1,1),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),d(1,1) - d(0,0),0,0,0,0,0), (a*( - d(1,1) + d(0,0)),d(1,1),0,0,0,0,0), (0,0,d(2,2),0,0,0,0), (d(3,0), - d(3,0),0,d(1,1) + d(0,0),0,0,0), (d(4,0),d(4,1),d(4,2),0,d(2,2) + d(1,1),0,a*( - d(1,1) + d(0,0))), (d(5,0), - d(5,0),0, - d(4,1) - 2*d(4,0),d(4,2) + d(3,0), d(2,2) + d(1,1) + d(0,0), - (d(4,2) + d(3,0)*a + d(3,0))), (d(6,0), - d(6,0) + d(4,1),d(4,2),0,d(1,1) - d(0,0),0,d(2,2) + d(0,0))) Unknowns: {d(6,0),d(5,0),d(4,2),d(4,1),d(4,0),d(3,0),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(6,0),d(5,0),d(4,2),d(4,1),d(4,0),d(3,0),d(2,2),d(1,1),d(0,0),a} t2:=D(1,1):= [ 0 1 0 0 0 0 0 ] [ ] [ - a 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 - a] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 0 0 0 0 1 0 0 ] {{x,1, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(148)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 1, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(149)] [ ] [ 0 ] [ ] [arbcomplex(150)] [ ] [ 0 ] }, 2 {a + x - x, 2, [ arbcomplex(151)*(a + x - 1) ] [-----------------------------] [ a*(x - 1) ] [ ] [ arbcomplex(151) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(152)*a ] [ ---------------------- ] [ x - 1 ] [ ] [ 0 ] [ ] [ arbcomplex(152) ] }} Unknowns: {d(6,0),d(4,1),d(4,0),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(6,0),d(4,1),d(4,0),d(2,2),d(1,1),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),d(1,1) - d(0,0),0,0,0,0,0),(a*( - d(1,1) + d(0,0)),d(1,1),0,0,0,0,0) ,(0,0,d(2,2),0,0,0,0),(0,0,0,d(1,1) + d(0,0),0,0,0),(d(4,0),d(4,1),0,0,d(2,2) + d(1,1),0,a*( - d(1,1) + d(0,0))),(0,0,0, - d(4,1) - 2*d(4,0),0,d(2,2) + d(1,1) + d(0,0),0),(d(6,0), - d(6,0) + d(4,1),0,0,d(1,1) - d(0,0),0,d(2,2) + d(0,0)))$ Unknowns: {d(6,0),d(4,1),d(4,0),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(6,0),d(4,1),d(4,0),d(2,2),d(1,1),d(0,0),a} t3:=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 1 0] [ ] [0 0 0 0 0 0 1] Unknowns: {d(2,2),d(1,1),d(0,0),a} Unknowns: {d(2,2),d(1,1),d(0,0),a} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),d(1,1) - d(0,0),0,0,0,0,0), (a*( - d(1,1) + d(0,0)),d(1,1),0,0,0,0,0), (0,0,d(2,2),0,0,0,0), (0,0,0,d(1,1) + d(0,0),0,0,0), (0,0,0,0,d(2,2) + d(1,1),0,a*( - d(1,1) + d(0,0))), (0,0,0,0,0,d(2,2) + d(1,1) + d(0,0),0), (0,0,0,0,d(1,1) - d(0,0),0,d(2,2) + d(0,0))) 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 We suppose a neq 1/4 and a neq 0. P:= sqrt( - 4*a + 1) + 1 mat((1,----------------------,0,0,0,0,0), 2*a sqrt( - 4*a + 1) + 1 (----------------------,1,0,0,0,0,0), 2 (0,0,1,0,0,0,0), (0,0,0,1,0,0,0), sqrt( - 4*a + 1) + 1 (0,0,0,0,1,0,----------------------), 2 (0,0,0,0,0,1,0), sqrt( - 4*a + 1) + 1 (0,0,0,0,----------------------,0,1)) 2*a P**(-1)*t1*P:= 2*sqrt( - 4*a + 1)*a mat((----------------------------,0,0,0,0,0,0), sqrt( - 4*a + 1) - 4*a + 1 2 (0,( - 4*sqrt( - 4*a + 1)*a + 5*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) 2 2 - 12*a + 7*a - 1)/(4*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) - 8*a + 6*a - 1),0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,1,0,0,0), 2 (0,0,0,0,( - 4*sqrt( - 4*a + 1)*a + 5*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) 2 - 12*a + 7*a - 1)/(4*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) 2 - 8*a + 6*a - 1),0,0), (0,0,0,0,0,1,0), 2*sqrt( - 4*a + 1)*a (0,0,0,0,0,0,----------------------------)) sqrt( - 4*a + 1) - 4*a + 1 P**(-1)*t2*P:= 2 2 mat((( - 4*sqrt( - 4*a + 1)*a + 5*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) - 12*a 2 + 7*a - 1)/(4*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) - 8*a + 6*a - 1),0,0 ,0,0,0,0), 2*sqrt( - 4*a + 1)*a (0,----------------------------,0,0,0,0,0), sqrt( - 4*a + 1) - 4*a + 1 (0,0,0,0,0,0,0), (0,0,0,1,0,0,0), 2*sqrt( - 4*a + 1)*a (0,0,0,0,----------------------------,0,0), sqrt( - 4*a + 1) - 4*a + 1 (0,0,0,0,0,1,0), 2 (0,0,0,0,0,0,( - 4*sqrt( - 4*a + 1)*a + 5*sqrt( - 4*a + 1)*a 2 - sqrt( - 4*a + 1) - 12*a + 7*a - 1)/(4*sqrt( - 4*a + 1)*a 2 - sqrt( - 4*a + 1) - 8*a + 6*a - 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 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] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((( - 2*sqrt( - 4*a + 1)*d(1,1)*a + sqrt( - 4*a + 1)*d(1,1) - 4*d(1,1)*a + d( 1,1) + 2*sqrt( - 4*a + 1)*d(0,0)*a)/(sqrt( - 4*a + 1) - 4*a + 1),0,0,0,0,0,0),(0 ,(2*sqrt( - 4*a + 1)*d(1,1)*a - 2*sqrt( - 4*a + 1)*d(0,0)*a + sqrt( - 4*a + 1)*d (0,0) - 4*d(0,0)*a + d(0,0))/(sqrt( - 4*a + 1) - 4*a + 1),0,0,0,0,0),(0,0,d(2,2) ,0,0,0,0),((sqrt( - 4*a + 1)*d(3,1) + d(3,1) + 2*d(3,0))/2,(d(3,0)*(sqrt( - 4*a + 1) + 1))/(2*a),d(3,2),d(1,1) + d(0,0),0,0,0),(0,(sqrt( - 4*a + 1)*d(6,0) + d(6 ,0) - 4*d(4,1)*a - 2*sqrt( - 4*a + 1)*d(4,0) - 4*d(4,0))/(sqrt( - 4*a + 1) - 4*a + 1),0,0,(sqrt( - 4*a + 1)*d(2,2) - 4*d(2,2)*a + d(2,2) + 2*sqrt( - 4*a + 1)*d( 1,1)*a - 2*sqrt( - 4*a + 1)*d(0,0)*a + sqrt( - 4*a + 1)*d(0,0) - 4*d(0,0)*a + d( 0,0))/(sqrt( - 4*a + 1) - 4*a + 1),0,0),((sqrt( - 4*a + 1)*d(5,1) + d(5,1) + 2*d (5,0))/2,(d(5,0)*(sqrt( - 4*a + 1) + 1))/(2*a),d(5,2), - d(4,1) - 2*d(4,0),( - ( sqrt( - 4*a + 1)*d(4,2) + d(4,2) + sqrt( - 4*a + 1)*d(3,0) + d(3,0)))/(2*a),d(2, 2) + d(1,1) + d(0,0),(sqrt( - 4*a + 1)*d(4,2) - d(4,2) - sqrt( - 4*a + 1)*d(3,1) - d(3,1) - 2*d(3,0))/2),((sqrt( - 4*a + 1)*d(4,1) - 4*d(4,1)*a + d(4,1) + 2* sqrt( - 4*a + 1)*d(4,0))/(sqrt( - 4*a + 1) - 4*a + 1),0,(d(4,2)*(sqrt( - 4*a + 1 ) + 1))/(sqrt( - 4*a + 1) - 4*a + 1),0,0,0,(sqrt( - 4*a + 1)*d(2,2) - 4*d(2,2)*a + d(2,2) - 2*sqrt( - 4*a + 1)*d(1,1)*a + sqrt( - 4*a + 1)*d(1,1) - 4*d(1,1)*a + d(1,1) + 2*sqrt( - 4*a + 1)*d(0,0)*a)/(sqrt( - 4*a + 1) - 4*a + 1)))$ PP:= sqrt( - 4*a + 1) + 1 mat((1,----------------------,0,0,0,0,0), 2*a sqrt( - 4*a + 1) + 1 (----------------------,1,0,0,0,0,0), 2 (0,0,1,0,0,0,0), (0,0,0,1,0,0,0), sqrt( - 4*a + 1) + 1 (0,0,0,0,1,0,----------------------), 2 (0,0,0,0,0,1,0), sqrt( - 4*a + 1) + 1 (0,0,0,0,----------------------,0,1)) 2*a avec PP:=P*Q:= sqrt( - 4*a + 1) + 1 mat((1,----------------------,0,0,0,0,0), 2*a sqrt( - 4*a + 1) + 1 (----------------------,1,0,0,0,0,0), 2 (0,0,1,0,0,0,0), (0,0,0,1,0,0,0), sqrt( - 4*a + 1) + 1 (0,0,0,0,1,0,----------------------), 2 (0,0,0,0,0,1,0), sqrt( - 4*a + 1) + 1 (0,0,0,0,----------------------,0,1)) 2*a MATDDIAGONALISE:= mat((( - 2*sqrt( - 4*a + 1)*d(1,1)*a + sqrt( - 4*a + 1)*d(1,1) - 4*d(1,1)*a + d(1,1) + 2*sqrt( - 4*a + 1)*d(0,0)*a)/(sqrt( - 4*a + 1) - 4*a + 1),0,0, 0,0,0,0), (0,(2*sqrt( - 4*a + 1)*d(1,1)*a - 2*sqrt( - 4*a + 1)*d(0,0)*a + sqrt( - 4*a + 1)*d(0,0) - 4*d(0,0)*a + d(0,0))/(sqrt( - 4*a + 1) - 4*a + 1),0,0,0,0,0), (0,0,d(2,2),0,0,0,0), sqrt( - 4*a + 1)*d(3,1) + d(3,1) + 2*d(3,0) (---------------------------------------------, 2 d(3,0)*(sqrt( - 4*a + 1) + 1) -------------------------------,d(3,2),d(1,1) + d(0,0),0,0,0), 2*a (0,(sqrt( - 4*a + 1)*d(6,0) + d(6,0) - 4*d(4,1)*a - 2*sqrt( - 4*a + 1)*d(4,0) - 4*d(4,0))/(sqrt( - 4*a + 1) - 4*a + 1),0, 0,(sqrt( - 4*a + 1)*d(2,2) - 4*d(2,2)*a + d(2,2) + 2*sqrt( - 4*a + 1)*d(1,1)*a - 2*sqrt( - 4*a + 1)*d(0,0)*a + sqrt( - 4*a + 1)*d(0,0) - 4*d(0,0)*a + d(0,0))/(sqrt( - 4*a + 1) - 4*a + 1),0,0), sqrt( - 4*a + 1)*d(5,1) + d(5,1) + 2*d(5,0) (---------------------------------------------, 2 d(5,0)*(sqrt( - 4*a + 1) + 1) -------------------------------,d(5,2), - d(4,1) - 2*d(4,0), 2*a - (sqrt( - 4*a + 1)*d(4,2) + d(4,2) + sqrt( - 4*a + 1)*d(3,0) + d(3,0)) --------------------------------------------------------------------------, 2*a d(2,2) + d(1,1) + d(0,0),(sqrt( - 4*a + 1)*d(4,2) - d(4,2) - sqrt( - 4*a + 1)*d(3,1) - d(3,1) - 2*d(3,0))/2), sqrt( - 4*a + 1)*d(4,1) - 4*d(4,1)*a + d(4,1) + 2*sqrt( - 4*a + 1)*d(4,0) (--------------------------------------------------------------------------- sqrt( - 4*a + 1) - 4*a + 1 d(4,2)*(sqrt( - 4*a + 1) + 1) ,0,-------------------------------,0,0,0,(sqrt( - 4*a + 1)*d(2,2) sqrt( - 4*a + 1) - 4*a + 1 - 4*d(2,2)*a + d(2,2) - 2*sqrt( - 4*a + 1)*d(1,1)*a + sqrt( - 4*a + 1)*d(1,1) - 4*d(1,1)*a + d(1,1) + 2*sqrt( - 4*a + 1)*d(0,0)*a)/(sqrt( - 4*a + 1) - 4*a + 1))) on voit apparaitre les poids sur la diagonale r(1) := ( - 2*sqrt( - 4*a + 1)*d(1,1)*a + sqrt( - 4*a + 1)*d(1,1) - 4*d(1,1)*a + d(1,1) + 2*sqrt( - 4*a + 1)*d(0,0)*a)/(sqrt( - 4*a + 1) - 4*a + 1) r(2) := (2*sqrt( - 4*a + 1)*d(1,1)*a - 2*sqrt( - 4*a + 1)*d(0,0)*a + sqrt( - 4*a + 1)*d(0,0) - 4*d(0,0)*a + d(0,0))/(sqrt( - 4*a + 1) - 4*a + 1) r(3) := d(2,2) r(4) := d(1,1) + d(0,0) r(5) := (sqrt( - 4*a + 1)*d(2,2) - 4*d(2,2)*a + d(2,2) + 2*sqrt( - 4*a + 1)*d(1,1)*a - 2*sqrt( - 4*a + 1)*d(0,0)*a + sqrt( - 4*a + 1)*d(0,0) - 4*d(0,0)*a + d(0,0))/(sqrt( - 4*a + 1) - 4*a + 1) r(6) := d(2,2) + d(1,1) + d(0,0) r(7) := (sqrt( - 4*a + 1)*d(2,2) - 4*d(2,2)*a + d(2,2) - 2*sqrt( - 4*a + 1)*d(1,1)*a + sqrt( - 4*a + 1)*d(1,1) - 4*d(1,1)*a + d(1,1) + 2*sqrt( - 4*a + 1)*d(0,0)*a)/(sqrt( - 4*a + 1) - 4*a + 1) r(4)-(r(1)+ r(2)) :=0 r(7)-(r(1)+ r(3)) :=0 r(5)-(r(2)+ r(3)) :=0 r(6)-(r(1)+r(2)+ r(3)) :=0 Le systeme de poids est le systeme 3.1 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(3)}, {{0,2},x(6)}, {{0,3},0}, {{0,4}, - x(5)}, {{0,5},0}, {{0,6},a*x(5)}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},0}, {{1,6},0}, {{2,3},x(5)}, {{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}} sqrt( - 4*a + 1)*x(1) + x(1) + 2*x(0) diaY(1):=--------------------------------------- 2 2*x(1)*a + sqrt( - 4*a + 1)*x(0) + x(0) diaY(2):=----------------------------------------- 2*a diaY(3):=x(2) diaY(4):=x(3) sqrt( - 4*a + 1)*x(6) + x(6) + 2*x(4)*a diaY(5):=----------------------------------------- 2*a diaY(6):=x(5) 2*x(6) + sqrt( - 4*a + 1)*x(4) + x(4) diaY(7):=--------------------------------------- 2 liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},(diay(4)*( - sqrt( - 4*a + 1) + 4*a - 1))/(2*a)} , {{1,3},diay(7)}, {{1,4},0}, {{1,5},sqrt( - 4*a + 1)*diay(6)}, {{1,6},0}, {{1,7},0}, {{2,3},diay(5)}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{2,7}, (diay(6)*(2*sqrt( - 4*a + 1)*a - sqrt( - 4*a + 1) + 4*a - 1))/(2*a)}, {{3,4},diay(6)}, {{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_{7,3.1}$ (iL)$ 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((1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,( - sqrt( - 4*a + 1) + 4*a - 1)/(2*a),0,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,sqrt( - 4*a + 1)),(0,0,0,0, 1,0,0))$ det(isom):= ((sqrt( - 4*a + 1) + 1)*(4*a - 1))/(2*a)$ ZZ(1):=diay(1)$ ZZ(2):=diay(2)$ ZZ(3):=diay(3)$ ZZ(4):=((4*a - 1 - sqrt( - 4*a + 1))*diay(4))/(2*a)$ ZZ(5):=diay(7)$ ZZ(6):=diay(5)$ ZZ(7):=sqrt( - 4*a + 1)*diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(5)}$ {{1,4},0}$ {{1,5},0}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(6)}$ {{2,4},0}$ {{2,5}, (zz(7)*(4*a - 1) + sqrt( - 4*a + 1)*zz(7)*(2*a - 1))/(2*sqrt( - 4*a + 1)*a)}$ {{2,6},0}$ {{2,7},0}$ {{3,4}, (zz(7)*(4*a - 1) - sqrt( - 4*a + 1)*zz(7))/(2*sqrt( - 4*a + 1)*a)}$ {{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_{7,3.1}$ (iL)$ with L:=(2*a - 1 - sqrt( - 4*a + 1))/(2*a)$ Note that ((4*a - 1) + sqrt( - 4*a + 1)*(2*a - 1))/(2*sqrt( - 4*a + 1)*a) -L := 0$ and that ((4*a - 1) - sqrt( - 4*a + 1))/(2*sqrt( - 4*a + 1)*a) -(L-1):=0$ Note also that L-1:=( - (sqrt( - 4*a + 1) + 1))/(2*a)$ Et cela pour a:=a$ and that for a neq {1/4,0,-2}$ shortformdelta:={0, ss, 1, 0, ss, 0, a, ss, 0, 1, 0}$ delta:= mat((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,-1,0,a),(0,1, 0,0,0,0))$