!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 neq {}$ a:=a$ 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,a,0,0,0,0),(0,0,1 ,a,0,0))$ shortformdelta:={0,ss,0,1,ss,a,ss,1}$ 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) + d(4,1)*a + d( 3,1)$ Unknowns: {d(5,0),d(4,1),d(3,1),a} Unknowns: {d(5,0),d(4,1),d(3,1),a} bonne inconnue W:=d(5,0)$ sa valeur doit etre WW:=d(4,1)*a + d(3,1)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,5)*a + d(0,3))$ Unknowns: {d(0,5),d(0,3),a} Unknowns: {d(0,5),d(0,3),a} bonne inconnue W:=d(0,3)$ sa valeur doit etre WW:= - d(0,5)*a$ on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,5)*a + d(1,3))$ Unknowns: {d(1,5),d(1,3),a} Unknowns: {d(1,5),d(1,3),a} bonne inconnue W:=d(1,3)$ sa valeur doit etre WW:= - d(1,5)*a$ on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,5)*a + d(2,3))$ Unknowns: {d(2,5),d(2,3),a} Unknowns: {d(2,5),d(2,3),a} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:= - d(2,5)*a$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,5)*a - d(3,3) + d( 2,2) + d(0,0)$ Unknowns: {d(3,5),d(3,3),d(2,2),d(0,0),a} Unknowns: {d(3,5),d(3,3),d(2,2),d(0,0),a} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:= - d(3,5)*a + d(2,2) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,5)*a - d(4,3) + d( 1,0)$ Unknowns: {d(4,5),d(4,3),d(1,0),a} Unknowns: {d(4,5),d(4,3),d(1,0),a} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(4,5)*a + d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,5)*a - d(5,3) + d( 2,2)*a + d(0,0)*a$ Unknowns: {d(5,5),d(5,3),d(2,2),d(0,0),a} Unknowns: {d(5,5),d(5,3),d(2,2),d(0,0),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=a*( - d(5,5) + d(2,2) + d(0,0))$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,5)*a - d(6,3) + d( 4,2)*a + d(3,2) - d(3,0)$ Unknowns: {d(6,5),d(6,3),d(4,2),d(3,2),d(3,0),a} Unknowns: {d(6,5),d(6,3),d(4,2),d(3,2),d(3,0),a} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - d(6,5)*a + d(4,2)*a + 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(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) + d(2,5)*a)$ Unknowns: {d(3,6),d(2,5),a} Unknowns: {d(3,6),d(2,5),a} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(2,5)*a$ 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(2,5)*a**2) $ Unknowns: {d(5,6),d(2,5),a} Unknowns: {d(5,6),d(2,5),a} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - d(2,5)*a**2$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6) - d(4,5)*a**2 - d(3,5)*a + d(2,2) + d(1,0)*a + 2*d(0,0)$ Unknowns: {d(6,6),d(4,5),d(3,5),d(2,2),d(1,0),d(0,0),a} Unknowns: {d(6,6),d(4,5),d(3,5),d(2,2),d(1,0),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(4,5)*a**2 - d(3,5)*a + d(2,2) + d(1,0)*a + 2*d(0,0) $ on resout l'equation {{0,4},3} qui est maintenant AA:=d(2,5)*a**2 + d(2,4)$ Unknowns: {d(2,5),d(2,4),a} Unknowns: {d(2,5),d(2,4),a} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:= - d(2,5)*a**2$ 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 {{0,4},6} qui est maintenant AA:=d(4,5)*a**3 + d(4,4)*a + d(3,5)*a**2 + d(3,4) - d(2,2)*a - d(0,0)*a$ Unknowns: {d(4,5),d(4,4),d(3,5),d(3,4),d(2,2),d(0,0),a} Unknowns: {d(4,5),d(4,4),d(3,5),d(3,4),d(2,2),d(0,0),a} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=a*( - d(4,5)*a**2 - d(4,4) - d(3,5)*a + d(2,2) + d(0,0)) $ 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 {{0,5},6} qui est maintenant AA:=d(4,5)*a + d(3,5)$ Unknowns: {d(4,5),d(3,5),a} Unknowns: {d(4,5),d(3,5),a} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(4,5)*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},3} qui est maintenant AA:=d(4,4)*a - d(2,2)*a + d(0 ,1) - d(0,0)*a$ Unknowns: {d(4,4),d(2,2),d(0,1),d(0,0),a} Unknowns: {d(4,4),d(2,2),d(0,1),d(0,0),a} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=a*( - d(4,4) + d(2,2) + d(0,0))$ 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(4,2) - d(1, 1)*a**2 + d(0,0)*a**2$ Unknowns: {d(5,4),d(4,2),d(1,1),d(0,0),a} Unknowns: {d(5,4),d(4,2),d(1,1),d(0,0),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) - d(1,1)*a**2 + d(0,0)*a**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 {{1,3},5} qui est maintenant AA:= - d(4,5)*a$ Unknowns: {d(4,5),a} Unknowns: {d(4,5),a} pas de selection possible de variable a coefficient numerique dans - d(4,5)*a on resout l'equation {{1,3},6} qui est maintenant AA:=a*( - d(5,5) + d(2,2) - d (1,1) + 2*d(0,0))$ Unknowns: {d(5,5),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(5,5),d(2,2),d(1,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(5,5) + d(2,2) - d(1,1) + 2*d(0,0)) 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(4,5)*a$ Unknowns: {d(4,5),a} Unknowns: {d(4,5),a} pas de selection possible de variable a coefficient numerique dans d(4,5)*a 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) - 2*d( 1,1)*a**2 + 2*d(0,0)*a**2$ Unknowns: {d(6,5),d(4,2),d(1,1),d(0,0),a} Unknowns: {d(6,5),d(4,2),d(1,1),d(0,0),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(4,2) - 2*d(1,1)*a**2 + 2*d(0,0)*a**2$ on resout l'equation {{1,5},6} qui est maintenant AA:=3*d(1,1) - 2*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:=(2*d(0,0))/3$ 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)$ 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:=(3*d(0,2)*a - 3*d(0,2) + d(0,0)*a - d(0,0))/3$ Unknowns: {d(0,2),d(0,0),a} Unknowns: {d(0,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (3*d(0,2)*a - 3*d(0,2) + d(0,0)*a - d(0,0))/3 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},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},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},3},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},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},((3*d(0,2) + d(0,0))*(a - 1))/3}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},3},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}, {{{4,5},6},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ Il y a une phase 2$ a neq {1}$ collect_eq:={{{{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},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},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},3},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},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},5},0}, {{{2,5},6},0}, {{{2,6},3},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}, {{{4,5},6},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ derivation generique de gtildedelta:$ MATD:= mat(( - 3*d(0,2), - d(0,2)*a,d(0,2),0,0,0,0),(0, - 2*d(0,2),0,0,0,0,0),(0,0, - 4 *d(0,2),0,0,0,0),(d(3,0),d(3,1),d(3,2), - 7*d(0,2), - d(0,2)*a,0,0),(0,d(4,1),d( 4,2),0, - 6*d(0,2),0,0),(d(4,1)*a + d(3,1),d(5,1),d(5,2),d(0,2)*a,d(4,2) - d(0,2 )*a**2, - 8*d(0,2),0),(d(6,0),d(6,1),d(6,2), - (d(3,0) - 2*d(0,2)*a**3 - d(3,2)) , - (d(4,1) + d(3,1) - d(5,2)),d(4,2) - 2*d(0,2)*a**2, - 10*d(0,2)))$ 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 a 0 0 0 0] [ ] [0 0 1 a 0 0] pour shortformdelta:={0,ss,0,1,ss,a,ss,1} 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(0,2), a} 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(0,2), a} 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(0,2)}$ dim Der(gtildedelta):=11$ t1:=D(0,2):= [-3 - a 1 0 0 0 0 ] [ ] [0 -2 0 0 0 0 0 ] [ ] [0 0 -4 0 0 0 0 ] [ ] [0 0 0 -7 - a 0 0 ] [ ] [0 0 0 0 -6 0 0 ] [ ] [ 2 ] [0 0 0 a - a -8 0 ] [ ] [ 3 2 ] [0 0 0 2*a 0 - 2*a -10] MATD:= mat(( - 3*d(0,2), - d(0,2)*a,d(0,2),0,0,0,0), (0, - 2*d(0,2),0,0,0,0,0), (0,0, - 4*d(0,2),0,0,0,0), (d(3,0),d(3,1),d(3,2), - 7*d(0,2), - d(0,2)*a,0,0), (0,d(4,1),d(4,2),0, - 6*d(0,2),0,0), 2 (d(4,1)*a + d(3,1),d(5,1),d(5,2),d(0,2)*a,d(4,2) - d(0,2)*a , - 8*d(0,2),0), 3 (d(6,0),d(6,1),d(6,2), - (d(3,0) - 2*d(0,2)*a - d(3,2)), 2 - (d(4,1) + d(3,1) - d(5,2)),d(4,2) - 2*d(0,2)*a , - 10*d(0,2))) {{x + 2, 1, [ - arbcomplex(219)*a] [ ] [ arbcomplex(219) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 4, 1, [ - arbcomplex(220)] [ ] [ 0 ] [ ] [ arbcomplex(220) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 6, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(221) ] [ ----------------- ] [ a ] [ ] [ - arbcomplex(221) ] [--------------------] [ 2 ] [ a ] [ ] [ arbcomplex(221) ] [ ] [ 0 ] }, {x + 3,1, [arbcomplex(222)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 7, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(223) ] [-----------------] [ a ] [ ] [ 0 ] [ ] [ arbcomplex(223) ] [ ] [ 0 ] }, {x + 8, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(224) ] [--------------------] [ 2 ] [ a ] [ ] [ arbcomplex(224) ] }, {x + 10,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(225)] }} Unknowns: {d(0,2),a} Unknowns: {d(0,2),a} commutant de t1 dans der(gtildedelta): mat(( - 3*d(0,2), - d(0,2)*a,d(0,2),0,0,0,0), (0, - 2*d(0,2),0,0,0,0,0), (0,0, - 4*d(0,2),0,0,0,0), (0,0,0, - 7*d(0,2), - d(0,2)*a,0,0), (0,0,0,0, - 6*d(0,2),0,0), 2 (0,0,0,d(0,2)*a, - d(0,2)*a , - 8*d(0,2),0), 3 2 (0,0,0,2*d(0,2)*a ,0, - 2*d(0,2)*a , - 10*d(0,2))) rank 1 with maximal torus t1 1 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 - a -1 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 - a 0 0] [ ] [0 0 0 0 1 0 0] [ ] [ 2 ] [0 0 0 a - a 1 0] [ ] [ 2 ] [0 0 0 0 0 - a 1] P**(-1)*t1*P:= [-3 0 0 0 0 0 0 ] [ ] [0 -2 0 0 0 0 0 ] [ ] [0 0 -4 0 0 0 0 ] [ ] [0 0 0 -7 0 0 0 ] [ ] [0 0 0 0 -6 0 0 ] [ ] [0 0 0 0 0 -8 0 ] [ ] [0 0 0 0 0 0 -10] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat(( - 3*d(0,2),0,0,0,0,0,0),(0, - 2*d(0,2),0,0,0,0,0),(0,0, - 4*d(0,2),0,0,0,0 ),(d(3,0), - (4*d(3,1) + d(3,0)*a), - (2*d(3,2) + d(3,0)), - 7*d(0,2),0,0,0),(0, d(4,1),d(4,2),0, - 6*d(0,2),0,0),( - (4*d(3,1) + d(3,0)*a),(3*d(3,1) + d(3,0)*a) *a + d(5,1),4*d(3,1) + d(3,0)*a - d(3,2)*a + d(5,2),0,d(4,2), - 8*d(0,2),0),( - ((4*d(3,1) + d(3,0)*a)*a**2 - d(6,0)),(((3*d(3,1) + d(3,0)*a)*a + d(5,1))*a - d( 6,0))*a + d(6,1),(4*d(3,1) + d(3,0)*a - d(3,2)*a + d(5,2))*a**2 - d(6,0) + d(6,2 ), - (2*d(3,2) + d(3,0)), - (d(4,1) + d(3,2)*a + d(3,1) - d(3,0)*a - d(5,2)),d(4 ,2), - 10*d(0,2)))$ PP:= [1 - a -1 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 - a 0 0] [ ] [0 0 0 0 1 0 0] [ ] [ 2 ] [0 0 0 a - a 1 0] [ ] [ 2 ] [0 0 0 0 0 - a 1] avec PP:=P*Q:= [1 - a -1 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 - a 0 0] [ ] [0 0 0 0 1 0 0] [ ] [ 2 ] [0 0 0 a - a 1 0] [ ] [ 2 ] [0 0 0 0 0 - a 1] MATDDIAGONALISE:= mat(( - 3*d(0,2),0,0,0,0,0,0), (0, - 2*d(0,2),0,0,0,0,0), (0,0, - 4*d(0,2),0,0,0,0), (d(3,0), - (4*d(3,1) + d(3,0)*a), - (2*d(3,2) + d(3,0)), - 7*d(0,2),0,0,0), (0,d(4,1),d(4,2),0, - 6*d(0,2),0,0), ( - (4*d(3,1) + d(3,0)*a),(3*d(3,1) + d(3,0)*a)*a + d(5,1), 4*d(3,1) + d(3,0)*a - d(3,2)*a + d(5,2),0,d(4,2), - 8*d(0,2),0), 2 ( - ((4*d(3,1) + d(3,0)*a)*a - d(6,0)), (((3*d(3,1) + d(3,0)*a)*a + d(5,1))*a - d(6,0))*a + d(6,1), 2 (4*d(3,1) + d(3,0)*a - d(3,2)*a + d(5,2))*a - d(6,0) + d(6,2), - (2*d(3,2) + d(3,0)), - (d(4,1) + d(3,2)*a + d(3,1) - d(3,0)*a - d(5,2)), d(4,2), - 10*d(0,2))) on voit apparaitre les poids sur la diagonale r(1) := - 3*d(0,2) r(2) := - 2*d(0,2) r(3) := - 4*d(0,2) r(4) := - 7*d(0,2) r(5) := - 6*d(0,2) r(6) := - 8*d(0,2) r(7) := - 10*d(0,2) 3*gamma1 r(1) := ---------- 2 r(2) := gamma1 r(3) := 2*gamma1 7*gamma1 r(4) := ---------- 2 r(5) := 3*gamma1 r(6) := 4*gamma1 r(7) := 5*gamma1 Le systeme de poids est le systeme 1.8 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},0}, {{0,2},x(5)*a + x(3)}, {{0,3},x(6)}, {{0,4},a*x(6)}, {{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)*a diaY(3):=x(2) - x(0) diaY(4):=x(5)*a + x(3) 2 diaY(5):= - x(5)*a + x(4) - x(3)*a 2 diaY(6):= - x(6)*a + x(5) diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},0}, {{1,3},diay(4)}, {{1,4},diay(7)}, {{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},0}, {{3,5},diay(7)*( - a + 1)}, {{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_{7,1.8}$ pour a neq{1}$ 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 :$ This isom computed by calculisom6_12III.red$ mat((0,0,1,0,0,0,0),(-1,0,0,0,0,0,0),(0,( - 1)/(a - 1),0,0,0,0,0),(0,0,0,0,1/(a - 1),0,0),(0,0,0,1/(a - 1),0,0,0),(0,0,0,0,0,( - 1)/(a - 1),0),(0,0,0,0,0,0,1/(a - 1)))$ det(isom):= 1/(a - 1)**5$ ZZ(1):= - diay(2)$ ZZ(2):=( - diay(3))/(a - 1)$ ZZ(3):=diay(1)$ ZZ(4):=diay(5)/(a - 1)$ ZZ(5):=diay(4)/(a - 1)$ ZZ(6):=( - diay(6))/(a - 1)$ ZZ(7):=diay(7)/(a - 1)$ 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},zz(7)}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4},0}$ {{3,5},zz(7)}$ {{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_{7,1.8}$ Et cela pour a:=a.$ Et cela pour a different de {1}.$ shortformdelta:={0,ss,0,1,ss,a,ss,1}$ 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,a,0,0,0,0),(0,0,1 ,a,0,0))$