!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 {}$ b neq {}$ a:=a$ b:=b$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,a,0,0,0,0),(0,0,0,0,0,0),(0,b,-1,0,0,0),(0,0, 1,b,1,0))$ shortformdelta:={1, ss, 0, a, ss, b, ss, 1}$ phase 1 de la resolution des equations$ on resout l'equation {{0,1},0} qui est maintenant AA:= - d(0,2)$ Unknown: d(0,2) Unknown: d(0,2) bonne inconnue W:=d(0,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},1} 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 {{0,1},2} 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 {{0,1},3} qui est maintenant AA:= - d(3,2) + d(2,1)*a$ Unknowns: {d(3,2),d(2,1),a} Unknowns: {d(3,2),d(2,1),a} bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:=d(2,1)*a$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,2) + d(2,0))$ Unknowns: {d(4,2),d(2,0)} Unknowns: {d(4,2),d(2,0)} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,2) - d(4,0) - d(3, 1) + d(2,1)*b$ Unknowns: {d(5,2),d(4,0),d(3,1),d(2,1),b} Unknowns: {d(5,2),d(4,0),d(3,1),d(2,1),b} bonne inconnue W:=d(5,2)$ sa valeur doit etre WW:= - d(4,0) - d(3,1) + d(2,1)*b$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,2) + d(5,1) - d(5, 0) + d(4,1)*b + d(3,1)$ Unknowns: {d(6,2),d(5,1),d(5,0),d(4,1),d(3,1),b} Unknowns: {d(6,2),d(5,1),d(5,0),d(4,1),d(3,1),b} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=d(5,1) - d(5,0) + d(4,1)*b + d(3,1)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,5)*b + d(0,3)*a)$ Unknowns: {d(0,5),d(0,3),a,b} Unknowns: {d(0,5),d(0,3),a,b} pas de selection possible de variable a coefficient numerique dans - (d(0,5)*b + d(0,3)*a) on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,5)*b + d(1,3)*a)$ Unknowns: {d(1,5),d(1,3),a,b} Unknowns: {d(1,5),d(1,3),a,b} pas de selection possible de variable a coefficient numerique dans - (d(1,5)*b + d(1,3)*a) on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,5)*b + d(2,3)*a)$ Unknowns: {d(2,5),d(2,3),a,b} Unknowns: {d(2,5),d(2,3),a,b} pas de selection possible de variable a coefficient numerique dans - (d(2,5)*b + d(2,3)*a) on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,5)*b - d(3,3)*a + d(1,1)*a + 2*d(0,0)*a$ Unknowns: {d(3,5),d(3,3),d(1,1),d(0,0),a,b} Unknowns: {d(3,5),d(3,3),d(1,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans - d(3,5)*b - d(3,3)*a + d(1,1)*a + 2*d(0,0)*a on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,5)*b - d(4,3)*a + d(1,0)$ Unknowns: {d(4,5),d(4,3),d(1,0),a,b} Unknowns: {d(4,5),d(4,3),d(1,0),a,b} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(4,5)*b + d(4,3)*a$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,5)*b - d(5,3)*a - d(2,1)*a + d(1,1)*b + 2*d(0,0)*b$ Unknowns: {d(5,5),d(5,3),d(2,1),d(1,1),d(0,0),a,b} Unknowns: {d(5,5),d(5,3),d(2,1),d(1,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans - d(5,5)*b - d(5,3)*a - d(2,1)*a + d(1,1)*b + 2*d(0,0)*b on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,5)*b - d(6,3)*a - 2*d(4,0) - d(3,1) - d(3,0) + d(2,1)*a + d(2,1)*b - d(2,0)*b$ Unknowns: {d(6,5),d(6,3),d(4,0),d(3,1),d(3,0),d(2,1),d(2,0),a,b} Unknowns: {d(6,5),d(6,3),d(4,0),d(3,1),d(3,0),d(2,1),d(2,0),a,b} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=( - d(6,5)*b - d(6,3)*a - d(3,1) - d(3,0) + d(2,1)*a + d (2,1)*b - d(2,0)*b)/2$ on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,6) + d(0,5)$ Unknowns: {d(0,6),d(0,5)} Unknowns: {d(0,6),d(0,5)} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:=d(0,5)$ on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,6) + d(1,5)$ Unknowns: {d(1,6),d(1,5)} Unknowns: {d(1,6),d(1,5)} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:=d(1,5)$ on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,6) + d(2,5) + d(1, 3)$ Unknowns: {d(2,6),d(2,5),d(1,3)} Unknowns: {d(2,6),d(2,5),d(1,3)} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:=d(2,5) + d(1,3)$ on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,6) + d(3,5) + d(2, 3)*a$ Unknowns: {d(3,6),d(3,5),d(2,3),a} Unknowns: {d(3,6),d(3,5),d(2,3),a} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(3,5) + d(2,3)*a$ on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,6) + d(4,5)$ Unknowns: {d(4,6),d(4,5)} Unknowns: {d(4,6),d(4,5)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=d(4,5)$ on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6) + d(5,5) - d(3, 3) + d(2,3)*b - d(0,0)$ Unknowns: {d(5,6),d(5,5),d(3,3),d(2,3),d(0,0),b} Unknowns: {d(5,6),d(5,5),d(3,3),d(2,3),d(0,0),b} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=d(5,5) - d(3,3) + d(2,3)*b - d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6) + d(6,5) + d(5, 3) + d(4,3)*b + d(3,3) + d(2,0) + d(0,0)$ Unknowns: {d(6,6),d(6,5),d(5,3),d(4,3),d(3,3),d(2,0),d(0,0),b} Unknowns: {d(6,6),d(6,5),d(5,3),d(4,3),d(3,3),d(2,0),d(0,0),b} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(6,5) + d(5,3) + d(4,3)*b + d(3,3) + d(2,0) + d(0,0)$ on resout l'equation {{0,4},0} qui est maintenant AA:= - d(0,5)*b$ Unknowns: {d(0,5),b} Unknowns: {d(0,5),b} pas de selection possible de variable a coefficient numerique dans - d(0,5)*b on resout l'equation {{0,4},1} qui est maintenant AA:= - d(1,5)*b$ Unknowns: {d(1,5),b} Unknowns: {d(1,5),b} pas de selection possible de variable a coefficient numerique dans - d(1,5)*b on resout l'equation {{0,4},2} qui est maintenant AA:= - d(2,5)*b + d(1,4) - d( 1,3)*b$ Unknowns: {d(2,5),d(1,4),d(1,3),b} Unknowns: {d(2,5),d(1,4),d(1,3),b} bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:=b*(d(2,5) + d(1,3))$ on resout l'equation {{0,4},3} qui est maintenant AA:= - d(3,5)*b + d(2,4)*a - d(2,3)*a*b$ Unknowns: {d(3,5),d(2,4),d(2,3),a,b} Unknowns: {d(3,5),d(2,4),d(2,3),a,b} pas de selection possible de variable a coefficient numerique dans - d(3,5)*b + d(2,4)*a - d(2,3)*a*b on resout l'equation {{0,4},4} qui est maintenant AA:= - d(4,5)*b$ Unknowns: {d(4,5),b} Unknowns: {d(4,5),b} pas de selection possible de variable a coefficient numerique dans - d(4,5)*b on resout l'equation {{0,4},5} qui est maintenant AA:= - d(5,5)*b + d(4,5)*b + d(4,3)*a - d(3,4) + d(3,3)*b + d(2,4)*b - d(2,3)*b**2 + d(0,0)*b$ Unknowns: {d(5,5),d(4,5),d(4,3),d(3,4),d(3,3),d(2,4),d(2,3),d(0,0),a,b} Unknowns: {d(5,5),d(4,5),d(4,3),d(3,4),d(3,3),d(2,4),d(2,3),d(0,0),a,b} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(5,5)*b + d(4,5)*b + d(4,3)*a + d(3,3)*b + d(2,4)*b - d(2,3)*b**2 + d(0,0)*b$ on resout l'equation {{0,4},6} qui est maintenant AA:= - d(6,5)*b - d(5,5)*b + d(5,4) - d(5,3)*b + d(4,5)*b + d(4,4)*b + d(4,3)*a - d(4,3)*b**2 + d(2,4)*b - d( 2,3)*b**2 - d(2,0)*b + d(2,0) + d(0,0)*b$ Unknowns: {d(6,5), d(5,5), d(5,4), d(5,3), d(4,5), d(4,4), d(4,3), d(2,4), d(2,3), d(2,0), d(0,0), a, b} Unknowns: {d(6,5), d(5,5), d(5,4), d(5,3), d(4,5), d(4,4), d(4,3), d(2,4), d(2,3), d(2,0), d(0,0), a, b} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(6,5)*b + d(5,5)*b + d(5,3)*b - d(4,5)*b - d(4,4)*b - d (4,3)*a + d(4,3)*b**2 - d(2,4)*b + d(2,3)*b**2 + d(2,0)*b - d(2,0) - d(0,0)*b$ on resout l'equation {{0,5},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 {{0,5},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 {{0,5},2} qui est maintenant AA:= - (d(2,5) + d(1,3))$ Unknowns: {d(2,5),d(1,3)} Unknowns: {d(2,5),d(1,3)} bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:= - d(1,3)$ on resout l'equation {{0,5},3} qui est maintenant AA:= - (d(3,5) + d(2,3)*a + d (1,3)*a)$ Unknowns: {d(3,5),d(2,3),d(1,3),a} Unknowns: {d(3,5),d(2,3),d(1,3),a} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - a*(d(2,3) + d(1,3))$ on resout l'equation {{0,5},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 {{0,5},5} qui est maintenant AA:= - d(5,5) + d(3,3) + d(2, 3)*a - d(2,3)*b + d(1,3)*a - d(1,3)*b + d(0,0)$ Unknowns: {d(5,5),d(3,3),d(2,3),d(1,3),d(0,0),a,b} Unknowns: {d(5,5),d(3,3),d(2,3),d(1,3),d(0,0),a,b} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(3,3) + d(2,3)*a - d(2,3)*b + d(1,3)*a - d(1,3)*b + d(0 ,0)$ on resout l'equation {{0,5},6} qui est maintenant AA:= - d(6,5) - d(5,3) + d(4, 3)*a - d(4,3)*b - d(2,3)*b - d(2,0) - d(1,3)*b + d(0,0)$ Unknowns: {d(6,5),d(5,3),d(4,3),d(2,3),d(2,0),d(1,3),d(0,0),a,b} Unknowns: {d(6,5),d(5,3),d(4,3),d(2,3),d(2,0),d(1,3),d(0,0),a,b} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(5,3) + d(4,3)*a - d(4,3)*b - d(2,3)*b - d(2,0) - d( 1,3)*b + d(0,0)$ on resout l'equation {{0,6},5} qui est maintenant AA:=d(1,3)*a$ Unknowns: {d(1,3),a} Unknowns: {d(1,3),a} pas de selection possible de variable a coefficient numerique dans d(1,3)*a on resout l'equation {{0,6},6} qui est maintenant AA:=d(2,3)*a - d(1,3)*b$ Unknowns: {d(2,3),d(1,3),a,b} Unknowns: {d(2,3),d(1,3),a,b} pas de selection possible de variable a coefficient numerique dans d(2,3)*a - d( 1,3)*b 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},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(4,3)*a + d(2,3)*a*b + d(1,3)*a*b - d(1,3)*b**2 + d(0,1)*a$ Unknowns: {d(4,3),d(2,3),d(1,3),d(0,1),a,b} Unknowns: {d(4,3),d(2,3),d(1,3),d(0,1),a,b} pas de selection possible de variable a coefficient numerique dans - d(4,3)*a + d(2,3)*a*b + d(1,3)*a*b - d(1,3)*b**2 + d(0,1)*a 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(4,3)*a*b + d(4,3)*a - d(3,3)*b - d(2,3)*a*b + d(2,3)*b**2 - d(1,3)*a*b + 2*d(1,3)*b**2 + 2*d(1,1)*b + d(0,1)*b$ Unknowns: {d(4,3),d(3,3),d(2,3),d(1,3),d(1,1),d(0,1),a,b} Unknowns: {d(4,3),d(3,3),d(2,3),d(1,3),d(1,1),d(0,1),a,b} pas de selection possible de variable a coefficient numerique dans - d(4,3)*a*b + d(4,3)*a - d(3,3)*b - d(2,3)*a*b + d(2,3)*b**2 - d(1,3)*a*b + 2*d(1,3)*b**2 + 2*d(1,1)*b + d(0,1)*b on resout l'equation {{1,2},6} qui est maintenant AA:=( - 2*d(6,4) + d(6,3)*a - d(5,3)*b + d(4,3)*a*b - d(4,3)*b**2 - 2*d(4,1) - 3*d(3,1) + d(3,0) - d(2,3)*b** 2 - d(2,1)*a + d(2,1)*b - d(1,3)*b**2 + d(0,0)*b)/2$ Unknowns: {d(6,4), d(6,3), d(5,3), d(4,3), d(4,1), d(3,1), d(3,0), d(2,3), d(2,1), d(1,3), d(0,0), a, b} Unknowns: {d(6,4), d(6,3), d(5,3), d(4,3), d(4,1), d(3,1), d(3,0), d(2,3), d(2,1), d(1,3), d(0,0), a, b} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=(d(6,3)*a - d(5,3)*b + d(4,3)*a*b - d(4,3)*b**2 - 2*d(4, 1) - 3*d(3,1) + d(3,0) - d(2,3)*b**2 - d(2,1)*a + d(2,1)*b - d(1,3)*b**2 + d(0,0 )*b)/2$ on resout l'equation {{1,3},2} 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 {{1,3},4} 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 {{1,3},5} qui est maintenant AA:=d(4,3) - d(0,1)$ Unknowns: {d(4,3),d(0,1)} Unknowns: {d(4,3),d(0,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,3},6} qui est maintenant AA:=d(5,3) + d(2,1) + d(0,1)$ Unknowns: {d(5,3),d(2,1),d(0,1)} Unknowns: {d(5,3),d(2,1),d(0,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - (d(2,1) + d(0,1))$ on resout l'equation {{1,4},2} 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 {{1,4},5} qui est maintenant AA:= - d(3,3) + 3*d(1,1)$ Unknowns: {d(3,3),d(1,1)} Unknowns: {d(3,3),d(1,1)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=3*d(1,1)$ on resout l'equation {{1,4},6} qui est maintenant AA:=d(1,1)*b + d(0,1)*a*b - 2 *d(0,1)*a + 2*d(0,1)*b - d(0,1) - d(0,0)$ Unknowns: {d(1,1),d(0,1),d(0,0),a,b} Unknowns: {d(1,1),d(0,1),d(0,0),a,b} bonne inconnue W:=d(0,0)$ sa valeur doit etre WW:=d(1,1)*b + d(0,1)*a*b - 2*d(0,1)*a + 2*d(0,1)*b - d(0,1) $ on resout l'equation {{1,5},6} qui est maintenant AA:= - d(1,1)*b + d(1,1) - d( 0,1)*a*b + d(0,1)*a - 2*d(0,1)*b + 2*d(0,1)$ Unknowns: {d(1,1),d(0,1),a,b} Unknowns: {d(1,1),d(0,1),a,b} pas de selection possible de variable a coefficient numerique dans - d(1,1)*b + d(1,1) - d(0,1)*a*b + d(0,1)*a - 2*d(0,1)*b + 2*d(0,1) on resout l'equation {{2,3},6} qui est maintenant AA:= - d(1,1)*b + d(1,1) - d( 0,1)*a*b + d(0,1)*a - 2*d(0,1)*b + 2*d(0,1)$ Unknowns: {d(1,1),d(0,1),a,b} Unknowns: {d(1,1),d(0,1),a,b} pas de selection possible de variable a coefficient numerique dans - d(1,1)*b + d(1,1) - d(0,1)*a*b + d(0,1)*a - 2*d(0,1)*b + 2*d(0,1) 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}, 2*((2*b - 1 + (b - 2)*a)*d(0,1) + (b - 1)*d(1,1))*a}, {{{0,2},4},0}, {{{0,2},5}, (2*b - 1 + (b - 2)*a)*d(0,1)*b + d(1,1)*b**2 - 2*d(1,1)*b + d(0,1)*a}, {{{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},0},0}, {{{0,5},1},0}, {{{0,5},2},0}, {{{0,5},3},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},2},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}, - (d(1,1)*b + d(0,1)*a*b - d(0,1)*a - d(0,1)*b)}, {{{1,2},6},0}, {{{1,3},2},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}, - (d(1,1) + d(0,1)*a + 2*d(0,1))*(b - 1)}, {{{1,6},2},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}, - (d(1,1) + d(0,1)*a + 2*d(0,1))*(b - 1)}, {{{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},5},0}, {{{3,5},6},0}, {{{3,6},5},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$ b neq {1}$ a neq { - 3*b}$ collect_eq:={{{{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},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},0},0}, {{{0,5},1},0}, {{{0,5},2},0}, {{{0,5},3},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},2},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},2},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},2},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},5},0}, {{{3,5},6},0}, {{{3,6},5},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((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(d(2,0),d(2,1),0,0,0,0,0),(d(3,0),d(3,1),d(2 ,1)*a,0,0,0,0),(( - (d(3,0) - d(2,1)*a + d(3,1) + d(6,3)*a))/2,d(4,1), - d(2,0), 0,0,0,0),(d(5,0),d(5,1),( - ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a))/2, - d(2,1), - d(2,0),0,0),(d(6,0),d(6,1),d(4,1)*b + d(3,1) - d(5,0) + d(5,1),d(6,3 ),( - ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a))/2,d(2,1) - d (2,0),0))$ pour delta:= [0 0 0 0 0 0] [ ] [1 0 0 0 0 0] [ ] [0 a 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 b -1 0 0 0] [ ] [0 0 1 b 1 0] pour shortformdelta:={1, ss, 0, a, ss, b, ss, 1} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(3,1), d(3,0), d(2,1), d(2,0), a, b} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(3,1), d(3,0), d(2,1), d(2,0), a, b} listeparametresMATD{d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(3,1), d(3,0), d(2,1), d(2,0)}$ dim Der(gtildedelta):=10$ MATD:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (d(2,0),d(2,1),0,0,0,0,0), (d(3,0),d(3,1),d(2,1)*a,0,0,0,0), - (d(3,0) - d(2,1)*a + d(3,1) + d(6,3)*a) (--------------------------------------------,d(4,1), - d(2,0),0,0,0,0), 2 - ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a) (d(5,0),d(5,1),----------------------------------------------------, 2 - d(2,1), - d(2,0),0,0), (d(6,0),d(6,1),d(4,1)*b + d(3,1) - d(5,0) + d(5,1),d(6,3), - ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a) -----------------------------------------------------------------, 2 d(2,1) - d(2,0),0)) *********** gtildedelta est caracteristiquement nilpotente MATD**1:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (d(2,0),d(2,1),0,0,0,0,0), (d(3,0),d(3,1),d(2,1)*a,0,0,0,0), - (d(3,0) - d(2,1)*a + d(3,1) + d(6,3)*a) (--------------------------------------------,d(4,1), - d(2,0),0,0,0,0), 2 - ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a) (d(5,0),d(5,1),----------------------------------------------------, 2 - d(2,1), - d(2,0),0,0), (d(6,0),d(6,1),d(4,1)*b + d(3,1) - d(5,0) + d(5,1),d(6,3), - ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a) -----------------------------------------------------------------, 2 d(2,1) - d(2,0),0)) MATD**2:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), 2 (d(2,1)*d(2,0)*a,d(2,1) *a,0,0,0,0,0), 2 ( - d(2,0) , - d(2,1)*d(2,0),0,0,0,0,0), (d(6,3)*d(2,0)*a - d(3,0)*d(2,1) + d(3,0)*d(2,0) - d(2,1)*d(2,0)*a + d(2,1)*d(2,0)*b,( - ( ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a)*d(2,1) 2 2 + 2*(d(4,1)*d(2,0) + d(3,1)*d(2,1))))/2, - (d(2,1) *a - d(2,0) ),0,0 ,0,0), ((((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a) *(d(3,0) - d(2,1)*a + d(3,1) + d(6,3)*a) + 4*( (d(4,1)*b + d(3,1) - d(5,0) + d(5,1))*d(2,0) + (d(2,1) - d(2,0))*d(5,0) + d(6,3)*d(3,0)))/4,( - ( ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a)*d(4,1) - 2*((d(4,1)*b + d(3,1) - d(5,0) + d(5,1))*d(2,1) + (d(2,1) - d(2,0))*d(5,1) + d(6,3)*d(3,1))))/2,( ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a)*d(2,0) - ( ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a)*(d(2,1) - d(2,0)) - 2*d(6,3)*d(2,1)*a))/2, - (d(2,1) - d(2,0))*d(2,1), - (d(2,1) - d(2,0))*d(2,0),0,0)) MATD**3:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), 2 2 2 2 ( - (d(2,1) *a - d(2,0) )*d(2,0), - (d(2,1) *a - d(2,0) )*d(2,1),0,0,0,0,0), 2 ((((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a)*d(2,0) + 2*( 2 2 2*d(6,3)*d(2,1)*d(2,0)*a - d(6,3)*d(2,0) *a - d(3,0)*d(2,1) 2 2 + 2*d(3,0)*d(2,1)*d(2,0) - d(3,0)*d(2,0) - d(2,1) *d(2,0)*a 2 2 2 + d(2,1) *d(2,0)*b + d(2,1)*d(2,0) *a - d(2,1)*d(2,0) *b))/2,( ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a)*d(2,1) *d(2,0) - (((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a) *(d(2,1) - d(2,0))*d(2,1) + 2*( (d(4,1)*d(2,0) + d(3,1)*d(2,1))*(d(2,1) - d(2,0)) 2 - d(6,3)*d(2,1) *a)))/2, 2 2 - (d(2,1) *a - d(2,0) )*(d(2,1) - d(2,0)),0,0,0,0)) MATD**4:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (0,0,0,0,0,0,0), 2 2 ( - (d(2,1) *a - d(2,0) )*(d(2,1) - d(2,0))*d(2,0), 2 2 - (d(2,1) *a - d(2,0) )*(d(2,1) - d(2,0))*d(2,1),0,0,0,0,0)) MATD**5:= [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 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] MATD**6:= [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 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] MATD**7:= [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 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] matd**j est nul pour j geq 5 rank 0 :gtildedelta is characteristically nilpotent rkgtildedelta 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 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 0] [ ] [0 0 0 0 0 0 1] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(d(2,0),d(2,1),0,0,0,0,0),(d(3,0),d(3,1),d(2 ,1)*a,0,0,0,0),(( - (d(3,0) - d(2,1)*a + d(3,1) + d(6,3)*a))/2,d(4,1), - d(2,0), 0,0,0,0),(d(5,0),d(5,1),( - ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a))/2, - d(2,1), - d(2,0),0,0),(d(6,0),d(6,1),d(4,1)*b + d(3,1) - d(5,0) + d(5,1),d(6,3 ),( - ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a))/2,d(2,1) - d (2,0),0))$ PP:= [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 0] [ ] [0 0 0 0 0 0 1] avec PP:=P*Q:= [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 0] [ ] [0 0 0 0 0 0 1] MATDDIAGONALISE:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), (d(2,0),d(2,1),0,0,0,0,0), (d(3,0),d(3,1),d(2,1)*a,0,0,0,0), - (d(3,0) - d(2,1)*a + d(3,1) + d(6,3)*a) (--------------------------------------------,d(4,1), - d(2,0),0,0,0,0), 2 - ((a - 2*b)*d(2,1) - d(3,0) + d(3,1) - d(6,3)*a) (d(5,0),d(5,1),----------------------------------------------------, 2 - d(2,1), - d(2,0),0,0), (d(6,0),d(6,1),d(4,1)*b + d(3,1) - d(5,0) + d(5,1),d(6,3), - ((a - 2*b)*d(2,1) - d(3,0) + 3*d(3,1) + 2*d(4,1) - d(6,3)*a) -----------------------------------------------------------------, 2 d(2,1) - d(2,0),0)) on voit apparaitre les poids sur la diagonale r(1) := 0 r(2) := 0 r(3) := 0 r(4) := 0 r(5) := 0 r(6) := 0 r(7) := 0 r(1) := 0 r(2) := 0 r(3) := 0 r(4) := 0 r(5) := 0 r(6) := 0 r(7) := 0 Le systeme de poids est le systeme 0.0 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(2)}, {{0,2},x(5)*b + x(3)*a}, {{0,3},x(6) - x(5)}, {{0,4},b*x(6)}, {{0,5},x(6)}, {{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) diaY(3):=x(2) 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},diay(3)}, {{1,3},diay(6)*b + diay(4)*a}, {{1,4},diay(7) - diay(6)}, {{1,5},diay(7)*b}, {{1,6},diay(7)}, {{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},diay(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}}$ Now we make explicit the isomorphism with an algebra of the book:$ Namely g_{7,0.4}$ (iL) with L given below$ a neq {0,1, - 3*b}$ pour a neq{0,1, - 3*b}$ pour b 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_12I.red$ mat(((a*( - i*b + i))/(sqrt(a)*(a**2 - 2*a + 1)),(a*( - b + 1))/(a**2 - 2*a + 1) ,0,0,0,0,0),((a**2*(i*b - i))/(sqrt(a)*(a**2 - 2*a + 1)),(a*(b - 1))/(a**2 - 2*a + 1),0,0,0,0,0),(0,0,(a**2*(i*b**2 - 2*i*b + i))/(sqrt(a)*(a**3 - 3*a**2 + 3*a - 1)),0,0,0,0),(0,0,0,(a**3*(b**3 - 3*b**2 + 3*b - 1))/(a**5 - 5*a**4 + 10*a**3 - 10*a**2 + 5*a - 1),(a**4*( - i*b**3 + 3*i*b**2 - 3*i*b + i))/(sqrt(a)*(a**5 - 5*a**4 + 10*a**3 - 10*a**2 + 5*a - 1)),0,0),(0,0,0,(a**3*( - b**3 + 3*b**2 - 3*b + 1))/(a**5 - 5*a**4 + 10*a**3 - 10*a**2 + 5*a - 1),(a**3*(i*b**3 - 3*i*b**2 + 3*i*b - i))/(sqrt(a)*(a**5 - 5*a**4 + 10*a**3 - 10*a**2 + 5*a - 1)),0,0),(0,0,0, (a**2*b*(b**3 - 3*b**2 + 3*b - 1))/(a**5 - 5*a**4 + 10*a**3 - 10*a**2 + 5*a - 1) ,(a**3*b*( - i*b**3 + 3*i*b**2 - 3*i*b + i))/(sqrt(a)*(a**5 - 5*a**4 + 10*a**3 - 10*a**2 + 5*a - 1)),(a**4*( - i*b**4 + 4*i*b**3 - 6*i*b**2 + 4*i*b - i))/(sqrt( a)*(a**6 - 6*a**5 + 15*a**4 - 20*a**3 + 15*a**2 - 6*a + 1)),0),(0,0,0,0,0,(a**4* (i*a*b**4 - 4*i*a*b**3 + 6*i*a*b**2 - 4*i*a*b + i*a - i*b**5 + 4*i*b**4 - 6*i*b **3 + 4*i*b**2 - i*b))/(sqrt(a)*(a**7 - 7*a**6 + 21*a**5 - 35*a**4 + 35*a**3 - 21*a**2 + 7*a - 1)),(a**4*(b**5 - 5*b**4 + 10*b**3 - 10*b**2 + 5*b - 1))/(a**7 - 7*a**6 + 21*a**5 - 35*a**4 + 35*a**3 - 21*a**2 + 7*a - 1)))$ det(isom):= ((b - 1)**19*a**16)/(a - 1)**28$ ZZ(1):=(i*(diay(2)*a - diay(1))*(b - 1)*a)/(sqrt(a)*(a - 1)**2)$ ZZ(2):=((diay(2) - diay(1))*(b - 1)*a)/(a - 1)**2$ ZZ(3):=(i*(b - 1)**2*diay(3)*a**2)/(sqrt(a)*(a - 1)**3)$ ZZ(4):=( - ((diay(5) - diay(4))*a - diay(6)*b)*(b - 1)**3*a**2)/(a - 1)**5$ ZZ(5):=( - i*(diay(6)*b - diay(5) + diay(4)*a)*(b - 1)**3*a**3)/(sqrt(a)*(a - 1) **5)$ ZZ(6):=(i*(diay(7)*a - diay(7)*b - diay(6)*a + diay(6))*(b - 1)**4*a**4)/(sqrt(a )*(a - 1)**7)$ ZZ(7):=((b - 1)**5*diay(7)*a**4)/(a - 1)**7$ listcommutateursdesZZ:=$ {{1,2},zz(3)}$ {{1,3},zz(4)}$ {{1,4}, ( - i*zz(7)*a**2 + 3*i*zz(7)*a*b - i*zz(7)*a - i*zz(7)*b + sqrt(a)*zz(6)*(b - 1) *a)/(sqrt(a)*a*b - sqrt(a)*a)}$ {{1,5},zz(7)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(5)}$ {{2,4},zz(7)}$ {{2,5},zz(6)}$ {{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}$ !The! expression! for! the! commutator! {zz(1),zz(4)}! appears! not! to! simplif y! automa\ tically$ it is:$ tobesimplified := {{1,4}, ( - i*(zz(7)*a**2 - 3*zz(7)*a*b + zz(7)*a + zz(7)*b + i*sqrt(a)*zz(6)*a*b - i* sqrt(a)*zz(6)*a))/(sqrt(a)*(b - 1)*a)}$ thews:=( - i*(zz(7)*a**2 - 3*zz(7)*a*b + zz(7)*a + zz(7)*b + i*sqrt(a)*zz(6)*a*b - i*sqrt(a)*zz(6)*a))/(sqrt(a)*(b - 1)*a)$ Df(thews,zz(7)):=( - i*(a**2 - 3*a*b + a + b))/(sqrt(a)*(b - 1)*a)$ Df(thews,zz(6)):=1$ thews-( DF(thews,zz(7))*zz(7) +DF(thews,zz(6))*zz(6) ):=0$ On obtient donc les relations de commutations de g_{7,0.4 }$ (iL) with L:=( - i*(a**2 - 3*a*b + a + b))/(sqrt(a)*(b - 1)*a)$ Et cela pour a:=a.$ Et cela pour b:=b.$ Et cela pour a different de {0,1, - 3*b}.$ Et cela pour b different de {1}.$ shortformdelta:={1, ss, 0, a, ss, b, ss, 1}$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,a,0,0,0,0),(0,0,0,0,0,0),(0,b,-1,0,0,0),(0,0, 1,b,1,0))$ $$$$$$$$$$$$$ WITH HINDSIGHT ONES SEES THAT THE COMPUTED ISOMORPHISM IS VALID$ UNDER THE CONDITIONS a NEQ 0,1 AND b NEQ 1. $$$$$$$$$$$$$ The last condition a neq -3*b is a technical condition related to$ the calculation of DER(gtildedelta) and appears in fact as irrelevant.$