generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(0,xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),xi(2,2) + xi(1,1),0 ,0,0),(xi(4,1),xi(4,2),0,xi(2,2) + xi(1,1),0,0),(xi(5,1),xi(5,2),xi(5,3),xi(4,2) ,xi(2,2) + 2*xi(1,1),0),(xi(6,1),xi(6,2),xi(6,3), - (xi(4,1) + xi(3,1)),0,2*xi(2 ,2) + xi(1,1)))$ [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 0 0 0 0] [ ] [0 0 1 1 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 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] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 -1 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,2):=0 delta:= [ 0 0 0 0 0 0] [ ] [ 0 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(5,3) 0 0 0] [ ] [xi(6,1) 0 xi(6,3) - xi(3,1) 0 0] We denote this delta by the shortform shortformdelta:={xi(3,1), xi(3,2), ss, xi(5,2), xi(5,3), ss, xi(6,1), xi(6,3)} paramindexeslist:={{3,1},{3,2},{5,2},{5,3},{6,1},{6,3}} In case 1 one has$ xi(3,1):=1$ xi(3,2):=1$ a neq {}$ b neq {}$ a:=a$ b:=b$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(1,1,0,0,0,0),(0,0,0,0,0,0),(0,0,a,0,0,0),(0,0,b ,-1,0,0))$ shortformdelta:={1,1,ss,0,a,ss,0,b}$ phase 1 de la resolution des equations$ 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(2,1) + d(1, 1) + d(0,0)$ Unknowns: {d(3,3),d(2,1),d(1,1),d(0,0)} Unknowns: {d(3,3),d(2,1),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,1) + 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(5,3) - d(4,0) + d(3, 1)*a$ Unknowns: {d(5,3),d(4,0),d(3,1),a} Unknowns: {d(5,3),d(4,0),d(3,1),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(4,0) + d(3,1)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,3) - d(4,1) + d(3, 1)*b$ Unknowns: {d(6,3),d(4,1),d(3,1),b} Unknowns: {d(6,3),d(4,1),d(3,1),b} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - d(4,1) + d(3,1)*b$ on resout l'equation {{0,2},3} qui est maintenant AA:=d(2,2) - d(2,1) + d(1,2) - d(1,1)$ Unknowns: {d(2,2),d(2,1),d(1,2),d(1,1)} Unknowns: {d(2,2),d(2,1),d(1,2),d(1,1)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=d(2,1) - d(1,2) + d(1,1)$ on resout l'equation {{0,2},4} qui est maintenant AA:=d(2,0) + d(1,0)$ Unknowns: {d(2,0),d(1,0)} Unknowns: {d(2,0),d(1,0)} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:= - d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=d(4,0) + d(3,2)*a - d(3,1 )*a$ Unknowns: {d(4,0),d(3,2),d(3,1),a} Unknowns: {d(4,0),d(3,2),d(3,1),a} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=a*( - d(3,2) + d(3,1))$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(4,2) + d(4,1) + d(3, 2)*a + d(3,2)*b - d(3,1)*a - d(3,1)*b - d(3,0)$ Unknowns: {d(4,2),d(4,1),d(3,2),d(3,1),d(3,0),a,b} Unknowns: {d(4,2),d(4,1),d(3,2),d(3,1),d(3,0),a,b} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:=d(4,1) + d(3,2)*a + d(3,2)*b - d(3,1)*a - d(3,1)*b - d(3 ,0)$ on resout l'equation {{0,3},0} qui est maintenant AA:= - (d(0,6)*b + d(0,5)*a)$ Unknowns: {d(0,6),d(0,5),a,b} Unknowns: {d(0,6),d(0,5),a,b} pas de selection possible de variable a coefficient numerique dans - (d(0,6)*b + d(0,5)*a) on resout l'equation {{0,3},1} qui est maintenant AA:= - (d(1,6)*b + d(1,5)*a)$ Unknowns: {d(1,6),d(1,5),a,b} Unknowns: {d(1,6),d(1,5),a,b} pas de selection possible de variable a coefficient numerique dans - (d(1,6)*b + d(1,5)*a) on resout l'equation {{0,3},2} qui est maintenant AA:= - (d(2,6)*b + d(2,5)*a)$ Unknowns: {d(2,6),d(2,5),a,b} Unknowns: {d(2,6),d(2,5),a,b} pas de selection possible de variable a coefficient numerique dans - (d(2,6)*b + d(2,5)*a) on resout l'equation {{0,3},3} qui est maintenant AA:= - (d(3,6)*b + d(3,5)*a)$ Unknowns: {d(3,6),d(3,5),a,b} Unknowns: {d(3,6),d(3,5),a,b} pas de selection possible de variable a coefficient numerique dans - (d(3,6)*b + d(3,5)*a) on resout l'equation {{0,3},4} qui est maintenant AA:= - (d(4,6)*b + d(4,5)*a)$ Unknowns: {d(4,6),d(4,5),a,b} Unknowns: {d(4,6),d(4,5),a,b} pas de selection possible de variable a coefficient numerique dans - (d(4,6)*b + d(4,5)*a) on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6)*b - d(5,5)*a + d(2,1)*a + d(1,1)*a + 2*d(0,0)*a$ Unknowns: {d(5,6),d(5,5),d(2,1),d(1,1),d(0,0),a,b} Unknowns: {d(5,6),d(5,5),d(2,1),d(1,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans - d(5,6)*b - d(5,5)*a + d(2,1)*a + d(1,1)*a + 2*d(0,0)*a on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6)*b - d(6,5)*a + d(2,1)*b + d(1,1)*b - 2*d(1,0) + 2*d(0,0)*b$ Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(1,0),d(0,0),a,b} Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(1,0),d(0,0),a,b} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=( - d(6,6)*b - d(6,5)*a + d(2,1)*b + d(1,1)*b + 2*d(0,0) *b)/2$ on resout l'equation {{0,4},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 {{0,4},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 {{0,4},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 {{0,4},3} qui est maintenant AA:=d(3,6) + d(2,4) + d(1,4)$ Unknowns: {d(3,6),d(2,4),d(1,4)} Unknowns: {d(3,6),d(2,4),d(1,4)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - (d(2,4) + d(1,4))$ on resout l'equation {{0,4},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 {{0,4},5} qui est maintenant AA:=( - d(6,6)*b - d(6,5)*a + 2*d(5,6) + 2*d(3,4)*a + d(2,1)*b + d(1,1)*b + 2*d(0,0)*b)/2$ Unknowns: {d(6,6),d(6,5),d(5,6),d(3,4),d(2,1),d(1,1),d(0,0),a,b} Unknowns: {d(6,6),d(6,5),d(5,6),d(3,4),d(2,1),d(1,1),d(0,0),a,b} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=(d(6,6)*b + d(6,5)*a - 2*d(3,4)*a - d(2,1)*b - d(1,1)*b - 2*d(0,0)*b)/2$ on resout l'equation {{0,4},6} qui est maintenant AA:=(d(6,6)*b + 2*d(6,6) + d( 6,5)*a - 2*d(4,4) + 2*d(3,4)*b - d(2,1)*b - d(1,1)*b - 2*d(0,0)*b - 2*d(0,0))/2$ Unknowns: {d(6,6),d(6,5),d(4,4),d(3,4),d(2,1),d(1,1),d(0,0),a,b} Unknowns: {d(6,6),d(6,5),d(4,4),d(3,4),d(2,1),d(1,1),d(0,0),a,b} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=(d(6,6)*b + 2*d(6,6) + d(6,5)*a + 2*d(3,4)*b - d(2,1)*b - d(1,1)*b - 2*d(0,0)*b - 2*d(0,0))/2$ on resout l'equation {{0,5},3} qui est maintenant AA:=d(2,5) + d(1,5)$ Unknowns: {d(2,5),d(1,5)} Unknowns: {d(2,5),d(1,5)} bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:= - d(1,5)$ on resout l'equation {{0,5},5} qui est maintenant AA:=d(3,5)*a$ Unknowns: {d(3,5),a} Unknowns: {d(3,5),a} pas de selection possible de variable a coefficient numerique dans d(3,5)*a on resout l'equation {{0,5},6} qui est maintenant AA:= - d(4,5) + d(3,5)*b$ Unknowns: {d(4,5),d(3,5),b} Unknowns: {d(4,5),d(3,5),b} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=d(3,5)*b$ on resout l'equation {{0,6},5} qui est maintenant AA:= - a*(d(2,4) + d(1,4))$ Unknowns: {d(2,4),d(1,4),a} Unknowns: {d(2,4),d(1,4),a} pas de selection possible de variable a coefficient numerique dans - a*(d(2,4) + d(1,4)) on resout l'equation {{0,6},6} qui est maintenant AA:= - b*(d(2,4) + d(1,4))$ Unknowns: {d(2,4),d(1,4),b} Unknowns: {d(2,4),d(1,4),b} pas de selection possible de variable a coefficient numerique dans - b*(d(2,4) + d(1,4)) 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) + d(0, 1)$ Unknowns: {d(3,4),d(0,2),d(0,1)} Unknowns: {d(3,4),d(0,2),d(0,1)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(0,2) + d(0,1)$ on resout l'equation {{1,2},4} qui est maintenant AA:=( - d(6,6)*b - 2*d(6,6) - d(6,5)*a + d(2,1)*b + 2*d(2,1) - 2*d(1,2) + d(1,1)*b + 4*d(1,1) + 2*d(0,2)*b - 2*d(0,1)*b + 2*d(0,0)*b + 2*d(0,0))/2$ Unknowns: {d(6,6),d(6,5),d(2,1),d(1,2),d(1,1),d(0,2),d(0,1),d(0,0),a,b} Unknowns: {d(6,6),d(6,5),d(2,1),d(1,2),d(1,1),d(0,2),d(0,1),d(0,0),a,b} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=( - d(6,6)*b - 2*d(6,6) - d(6,5)*a + d(2,1)*b + 2*d(2,1) + d(1,1)*b + 4*d(1,1) + 2*d(0,2)*b - 2*d(0,1)*b + 2*d(0,0)*b + 2*d(0,0))/2$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) + d(4,1) + d(3, 2)*a + d(3,2)*b - d(3,1)*a - d(3,1)*b - d(3,0)$ Unknowns: {d(5,4),d(4,1),d(3,2),d(3,1),d(3,0),a,b} Unknowns: {d(5,4),d(4,1),d(3,2),d(3,1),d(3,0),a,b} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,1) + d(3,2)*a + d(3,2)*b - d(3,1)*a - d(3,1)*b - d(3 ,0)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - (d(6,4) + d(4,1) + d(3 ,1))$ Unknowns: {d(6,4),d(4,1),d(3,1)} Unknowns: {d(6,4),d(4,1),d(3,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - (d(4,1) + d(3,1))$ on resout l'equation {{1,3},5} qui est maintenant AA:=( - d(6,6)*b - d(6,5)*a + d(2,1)*b + d(1,1)*b + 2*d(0,1)*a + 2*d(0,0)*b)/2$ Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(0,1),d(0,0),a,b} Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(0,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans ( - d(6,6)*b - d(6,5)*a + d(2,1)*b + d(1,1)*b + 2*d(0,1)*a + 2*d(0,0)*b)/2 on resout l'equation {{1,3},6} qui est maintenant AA:=d(2,1) + d(0,1)*b$ Unknowns: {d(2,1),d(0,1),b} Unknowns: {d(2,1),d(0,1),b} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:= - d(0,1)*b$ 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},5} qui est maintenant AA:=(d(6,6)*b + 2*d(6,6) + d( 6,5)*a - 2*d(5,5) - d(1,1)*b + 2*d(1,1) - 2*d(0,2)*b + d(0,1)*b**2 + 2*d(0,1)*b - 2*d(0,0)*b - 2*d(0,0))/2$ Unknowns: {d(6,6),d(6,5),d(5,5),d(1,1),d(0,2),d(0,1),d(0,0),a,b} Unknowns: {d(6,6),d(6,5),d(5,5),d(1,1),d(0,2),d(0,1),d(0,0),a,b} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=(d(6,6)*b + 2*d(6,6) + d(6,5)*a - d(1,1)*b + 2*d(1,1) - 2*d(0,2)*b + d(0,1)*b**2 + 2*d(0,1)*b - 2*d(0,0)*b - 2*d(0,0))/2$ on resout l'equation {{1,4},6} qui est maintenant AA:= - (d(6,5) + d(0,1)*b + d (0,1))$ Unknowns: {d(6,5),d(0,1),b} Unknowns: {d(6,5),d(0,1),b} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(0,1)*(b + 1)$ on resout l'equation {{2,3},5} qui est maintenant AA:=( - d(6,6)*b + d(1,1)*b + d(0,1)*a*b + 3*d(0,1)*a - d(0,1)*b**2 + 2*d(0,0)*b)/2$ Unknowns: {d(6,6),d(1,1),d(0,1),d(0,0),a,b} Unknowns: {d(6,6),d(1,1),d(0,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans ( - d(6,6)*b + d(1,1)*b + d(0,1)*a*b + 3*d(0,1)*a - d(0,1)*b**2 + 2*d(0,0)*b)/2 on resout l'equation {{2,4},5} qui est maintenant AA:= - d(6,6)*b - d(6,6) + d( 1,1)*b + 2*d(1,1) - d(0,2)*a + d(0,2)*b + d(0,1)*a*b + 2*d(0,1)*a - d(0,1)*b**2 - 2*d(0,1)*b + 2*d(0,0)*b + d(0,0)$ Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a,b} Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans - d(6,6)*b - d(6,6) + d(1,1)*b + 2*d(1,1) - d(0,2)*a + d(0,2)*b + d(0,1)*a*b + 2*d(0,1)*a - d(0,1)*b**2 - 2*d(0,1)*b + 2*d(0,0)*b + d(0,0) on resout l'equation {{2,4},6} qui est maintenant AA:=d(6,6)*b + d(6,6) - d(1,1 )*b - d(1,1) - 2*d(0,2)*b - 2*d(0,2) - d(0,1)*a*b - d(0,1)*a + d(0,1)*b**2 + 2*d (0,1)*b + d(0,1) - 2*d(0,0)*b - 2*d(0,0)$ Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a,b} Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a,b} pas de selection possible de variable a coefficient numerique dans d(6,6)*b + d( 6,6) - d(1,1)*b - d(1,1) - 2*d(0,2)*b - 2*d(0,2) - d(0,1)*a*b - d(0,1)*a + d(0,1 )*b**2 + 2*d(0,1)*b + d(0,1) - 2*d(0,0)*b - 2*d(0,0) 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},0},0}, {{{0,3},1},0}, {{{0,3},2},0}, {{{0,3},3},0}, {{{0,3},4},0}, {{{0,3},5}, ( - (d(6,6)*a*b + 2*d(6,6)*a + d(6,6)*b**2 - d(1,1)*a*b - d(1,1)*b**2 - d(0,1)*a **2*b - d(0,1)*a**2 + d(0,1)*a*b + d(0,1)*b**3 - 2*d(0,0)*a*b - 6*d(0,0)*a - 2*d (0,0)*b**2))/2}, {{{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},3},0}, {{{1,3},4},0}, {{{1,3},5}, ( - (d(6,6)*b - d(1,1)*b - d(0,1)*a*b - 3*d(0,1)*a + d(0,1)*b**2 - 2*d(0,0)*b))/ 2}, {{{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},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}, ( - (d(6,6)*b - d(1,1)*b - d(0,1)*a*b - 3*d(0,1)*a + d(0,1)*b**2 - 2*d(0,0)*b))/ 2}, {{{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}, - (d(6,6)*b + d(6,6) - d(1,1)*b - 2*d(1,1) + d(0,2)*a - d(0,2)*b - d(0,1)*a*b - 2*d(0,1)*a + d(0,1)*b**2 + 2*d(0,1)*b - 2*d(0,0)*b - d(0,0))}, {{{2,4},6}, (d(6,6) - d(1,1) - 2*d(0,2) - d(0,1)*a + d(0,1)*b + d(0,1) - 2*d(0,0))*(b + 1)}, {{{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},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}}$ Il y a une phase 2$ 24|6 gives if b neq -1:$ d(6,6):=d(0,1)*a - d(0,1)*b - d(0,1) + 2*d(0,0) + 2*d(0,2) + d(1,1)$ Then 13|5 reads:$ {{{1,3},5},( - (2*d(0,2)*b - 3*d(0,1)*a - d(0,1)*b))/2},$ and gives for b neq 0:$ d(0,2):=((3*a + b)*d(0,1))/(2*b)$ conditionssurb:={0,-1}$ Then 03|5 reads:$ {{{0,3},5},(! -! (d(1,1)*b! +! 2*d(0,1)*a*b! +! 3*d(0,1)*a! +! d(0,1)*b**2! -! d (0,0)*b)\ *a)/b},$ be careful. Here one has to suppose a neq 0, otherwise equation is zero.$ and gives for a,b neq 0:$ d(1,1):=( - (2*d(0,1)*a*b + 3*d(0,1)*a + d(0,1)*b**2 - d(0,0)*b))/b$ conditionssura:={0}$ Then we are left with 24|5 which reads:$ {{{2,4},5},( - 3*(a**2 + 2*a*b + 4*a + b**2)*d(0,1))/(2*b)},$ Hence if (a+b)**2 +4*a neq 0 , then d(0,1):=0$ d(0,1):=0$ conditionssurb:={2*i*sqrt(a) - a,0,-1}$ with sqrt(a) (defined modulo + or -) is any square root of a$ 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},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},3},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},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},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},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}}$ a neq {0}$ b neq {2*i*sqrt(a) - a,0,-1}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),0,0,0,0,0,0),(0,d(0,0),0,0,0,0,0),(0,0,d(0,0),0,0,0,0),(d(3,0),d(3,1 ),d(3,2),2*d(0,0),0,0,0),( - (d(3,2) - d(3,1))*a,d(4,1), - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),0,2*d(0,0),0,0),(d(5,0),d(5,1),d(5,2),d(3,2) *a, - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),3*d(0,0),0),(d(6, 0),d(6,1),d(6,2), - (d(4,1) - d(3,1)*b), - (d(4,1) + d(3,1)),0,3*d(0,0)))$ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [1 1 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 a 0 0 0] [ ] [0 0 b -1 0 0] pour shortformdelta:={1,1,ss,0,a,ss,0,b} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(0,0), a, b} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(0,0), a, b} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(0,0)}$ dim Der(gtildedelta):=11$ t1:=D(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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 3] MATD:= mat((d(0,0),0,0,0,0,0,0), (0,d(0,0),0,0,0,0,0), (0,0,d(0,0),0,0,0,0), (d(3,0),d(3,1),d(3,2),2*d(0,0),0,0,0), ( - (d(3,2) - d(3,1))*a,d(4,1), - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),0,2*d(0,0),0,0) , (d(5,0),d(5,1),d(5,2),d(3,2)*a, - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),3*d(0,0),0), (d(6,0),d(6,1),d(6,2), - (d(4,1) - d(3,1)*b), - (d(4,1) + d(3,1)),0,3*d(0,0) )) Unknown: d(0,0) Unknown: d(0,0) commutant de t1 dans der(gtildedelta): [d(0,0) 0 0 0 0 0 0 ] [ ] [ 0 d(0,0) 0 0 0 0 0 ] [ ] [ 0 0 d(0,0) 0 0 0 0 ] [ ] [ 0 0 0 2*d(0,0) 0 0 0 ] [ ] [ 0 0 0 0 2*d(0,0) 0 0 ] [ ] [ 0 0 0 0 0 3*d(0,0) 0 ] [ ] [ 0 0 0 0 0 0 3*d(0,0)] 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 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] P**(-1)*t1*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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 3] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((d(0,0),0,0,0,0,0,0),(0,d(0,0),0,0,0,0,0),(0,0,d(0,0),0,0,0,0),(d(3,0),d(3,1 ),d(3,2),2*d(0,0),0,0,0),( - (d(3,2) - d(3,1))*a,d(4,1), - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),0,2*d(0,0),0,0),(d(5,0),d(5,1),d(5,2),d(3,2) *a, - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),3*d(0,0),0),(d(6, 0),d(6,1),d(6,2), - (d(4,1) - d(3,1)*b), - (d(4,1) + d(3,1)),0,3*d(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((d(0,0),0,0,0,0,0,0), (0,d(0,0),0,0,0,0,0), (0,0,d(0,0),0,0,0,0), (d(3,0),d(3,1),d(3,2),2*d(0,0),0,0,0), ( - (d(3,2) - d(3,1))*a,d(4,1), - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),0,2*d(0,0),0,0) , (d(5,0),d(5,1),d(5,2),d(3,2)*a, - (d(3,1)*a + d(3,1)*b + d(3,0) - (a + b)*d(3,2) - d(4,1)),3*d(0,0),0), (d(6,0),d(6,1),d(6,2), - (d(4,1) - d(3,1)*b), - (d(4,1) + d(3,1)),0,3*d(0,0) )) on voit apparaitre les poids sur la diagonale r(1) := d(0,0) r(2) := d(0,0) r(3) := d(0,0) r(4) := 2*d(0,0) r(5) := 2*d(0,0) r(6) := 3*d(0,0) r(7) := 3*d(0,0) r(1) := gamma1 r(2) := gamma1 r(3) := gamma1 r(4) := 2*gamma1 r(5) := 2*gamma1 r(6) := 3*gamma1 r(7) := 3*gamma1 Le systeme de poids est le systeme 1.19 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(3)}, {{0,2},x(3)}, {{0,3},x(6)*b + x(5)*a}, {{0,4}, - x(6)}, {{0,5},0}, {{0,6},0}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},0}, {{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(4)}, {{1,3},diay(4)}, {{1,4},diay(7)*b + diay(6)*a}, {{1,5}, - diay(7)}, {{1,6},0}, {{1,7},0}, {{2,3},diay(5)}, {{2,4},0}, {{2,5},diay(6)}, {{2,6},0}, {{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_{7,1.19}$ a neq {0}$ pour a neq{0}$ pour b neq{2*i*sqrt(a) - a,0,-1}$ !The! computation! of! an! isomorphism! is! rather! involved! and! the! final! e xpression of! isom! is! rather! cumbersome,! but! there! exists! an! isomorphism.! $ !We! refer! to! the! program! calculisom6_5case1!I.red! and! its! results! rcalc ulisom6_5\ case1.r$ Et cela pour a:=a.$ Et cela pour b:=b.$ Et cela pour a different de {0}.$ Et cela pour b different de {2*i*sqrt(a) - a,0,-1}.$ shortformdelta:={1,1,ss,0,a,ss,0,b}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(1,1,0,0,0,0),(0,0,0,0,0,0),(0,0,a,0,0,0),(0,0,b ,-1,0,0))$