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 {}$ a:=a$ b:=a**2$ 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**2)/4,0,0 ,0),(0,0,(a*(a + 4))/4,-1,0,0))$ shortformdelta:={1, 1, ss, 0, ( - a**2)/4, ss, 0, (a*(a + 4))/4}$ 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:=( - 4*d(5,3) - 4*d(4,0) - d(3,1)*a**2)/4$ 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:=( - 4*d(4,0) - d(3,1)*a**2)/4$ on resout l'equation {{0,1},6} qui est maintenant AA:=( - 4*d(6,3) - 4*d(4,1) + d(3,1)*a**2 + 4*d(3,1)*a)/4$ Unknowns: {d(6,3),d(4,1),d(3,1),a} Unknowns: {d(6,3),d(4,1),d(3,1),a} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=( - 4*d(4,1) + d(3,1)*a**2 + 4*d(3,1)*a)/4$ 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:=(4*d(4,0) - d(3,2)*a**2 + d(3,1)*a**2)/4$ 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**2*(d(3,2) - d(3,1)))/4$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(4,2) + d(4,1) + d(3, 2)*a - d(3,1)*a - d(3,0)$ Unknowns: {d(4,2),d(4,1),d(3,2),d(3,1),d(3,0),a} Unknowns: {d(4,2),d(4,1),d(3,2),d(3,1),d(3,0),a} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:=d(4,1) + d(3,2)*a - d(3,1)*a - d(3,0)$ on resout l'equation {{0,3},0} qui est maintenant AA:=(a*( - d(0,6)*a - 4*d(0,6 ) + d(0,5)*a))/4$ Unknowns: {d(0,6),d(0,5),a} Unknowns: {d(0,6),d(0,5),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(0,6) *a - 4*d(0,6) + d(0,5)*a))/4 on resout l'equation {{0,3},1} qui est maintenant AA:=(a*( - d(1,6)*a - 4*d(1,6 ) + d(1,5)*a))/4$ Unknowns: {d(1,6),d(1,5),a} Unknowns: {d(1,6),d(1,5),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(1,6) *a - 4*d(1,6) + d(1,5)*a))/4 on resout l'equation {{0,3},2} qui est maintenant AA:=(a*( - d(2,6)*a - 4*d(2,6 ) + d(2,5)*a))/4$ Unknowns: {d(2,6),d(2,5),a} Unknowns: {d(2,6),d(2,5),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(2,6) *a - 4*d(2,6) + d(2,5)*a))/4 on resout l'equation {{0,3},3} qui est maintenant AA:=(a*( - d(3,6)*a - 4*d(3,6 ) + d(3,5)*a))/4$ Unknowns: {d(3,6),d(3,5),a} Unknowns: {d(3,6),d(3,5),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(3,6) *a - 4*d(3,6) + d(3,5)*a))/4 on resout l'equation {{0,3},4} qui est maintenant AA:=(a*( - d(4,6)*a - 4*d(4,6 ) + d(4,5)*a))/4$ Unknowns: {d(4,6),d(4,5),a} Unknowns: {d(4,6),d(4,5),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(4,6) *a - 4*d(4,6) + d(4,5)*a))/4 on resout l'equation {{0,3},5} qui est maintenant AA:=(a*( - d(5,6)*a - 4*d(5,6 ) + d(5,5)*a - d(2,1)*a - d(1,1)*a - 2*d(0,0)*a))/4$ Unknowns: {d(5,6),d(5,5),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(5,6),d(5,5),d(2,1),d(1,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(5,6) *a - 4*d(5,6) + d(5,5)*a - d(2,1)*a - d(1,1)*a - 2*d(0,0)*a))/4 on resout l'equation {{0,3},6} qui est maintenant AA:=( - d(6,6)*a**2 - 4*d(6,6 )*a + d(6,5)*a**2 + d(2,1)*a**2 + 4*d(2,1)*a + d(1,1)*a**2 + 4*d(1,1)*a - 8*d(1, 0) + 2*d(0,0)*a**2 + 8*d(0,0)*a)/4$ Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(1,0),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(1,0),d(0,0),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=(a*( - d(6,6)*a - 4*d(6,6) + d(6,5)*a + d(2,1)*a + 4*d(2 ,1) + d(1,1)*a + 4*d(1,1) + 2*d(0,0)*a + 8*d(0,0)))/8$ 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)*a**2 - 4*d(6,6 )*a + d(6,5)*a**2 + 8*d(5,6) - 2*d(3,4)*a**2 + d(2,1)*a**2 + 4*d(2,1)*a + d(1,1) *a**2 + 4*d(1,1)*a + 2*d(0,0)*a**2 + 8*d(0,0)*a)/8$ Unknowns: {d(6,6),d(6,5),d(5,6),d(3,4),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(5,6),d(3,4),d(2,1),d(1,1),d(0,0),a} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=(a*(d(6,6)*a + 4*d(6,6) - d(6,5)*a + 2*d(3,4)*a - d(2,1) *a - 4*d(2,1) - d(1,1)*a - 4*d(1,1) - 2*d(0,0)*a - 8*d(0,0)))/8$ on resout l'equation {{0,4},6} qui est maintenant AA:=(d(6,6)*a**2 + 4*d(6,6)*a + 8*d(6,6) - d(6,5)*a**2 - 8*d(4,4) + 2*d(3,4)*a**2 + 8*d(3,4)*a - d(2,1)*a**2 - 4*d(2,1)*a - d(1,1)*a**2 - 4*d(1,1)*a - 2*d(0,0)*a**2 - 8*d(0,0)*a - 8*d(0,0)) /8$ Unknowns: {d(6,6),d(6,5),d(4,4),d(3,4),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(4,4),d(3,4),d(2,1),d(1,1),d(0,0),a} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=(d(6,6)*a**2 + 4*d(6,6)*a + 8*d(6,6) - d(6,5)*a**2 + 2*d (3,4)*a**2 + 8*d(3,4)*a - d(2,1)*a**2 - 4*d(2,1)*a - d(1,1)*a**2 - 4*d(1,1)*a - 2*d(0,0)*a**2 - 8*d(0,0)*a - 8*d(0,0))/8$ 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**2)/4$ Unknowns: {d(3,5),a} Unknowns: {d(3,5),a} pas de selection possible de variable a coefficient numerique dans ( - d(3,5)*a **2)/4 on resout l'equation {{0,5},6} qui est maintenant AA:=( - 4*d(4,5) + d(3,5)*a** 2 + 4*d(3,5)*a)/4$ Unknowns: {d(4,5),d(3,5),a} Unknowns: {d(4,5),d(3,5),a} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=(d(3,5)*a*(a + 4))/4$ on resout l'equation {{0,6},5} qui est maintenant AA:=(a**2*(d(2,4) + d(1,4)))/ 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**2*(d(2,4) + d(1,4)))/4 on resout l'equation {{0,6},6} qui est maintenant AA:=(a*( - d(2,4)*a - 4*d(2,4 ) - d(1,4)*a - 4*d(1,4)))/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) *a - 4*d(2,4) - d(1,4)*a - 4*d(1,4)))/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)*a**2 - 4*d(6,6 )*a - 8*d(6,6) + d(6,5)*a**2 + d(2,1)*a**2 + 4*d(2,1)*a + 8*d(2,1) - 8*d(1,2) + d(1,1)*a**2 + 4*d(1,1)*a + 16*d(1,1) + 2*d(0,2)*a**2 + 8*d(0,2)*a - 2*d(0,1)*a** 2 - 8*d(0,1)*a + 2*d(0,0)*a**2 + 8*d(0,0)*a + 8*d(0,0))/8$ 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} 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} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=( - d(6,6)*a**2 - 4*d(6,6)*a - 8*d(6,6) + d(6,5)*a**2 + d(2,1)*a**2 + 4*d(2,1)*a + 8*d(2,1) + d(1,1)*a**2 + 4*d(1,1)*a + 16*d(1,1) + 2*d (0,2)*a**2 + 8*d(0,2)*a - 2*d(0,1)*a**2 - 8*d(0,1)*a + 2*d(0,0)*a**2 + 8*d(0,0)* a + 8*d(0,0))/8$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) + d(4,1) + d(3, 2)*a - d(3,1)*a - d(3,0)$ Unknowns: {d(5,4),d(4,1),d(3,2),d(3,1),d(3,0),a} Unknowns: {d(5,4),d(4,1),d(3,2),d(3,1),d(3,0),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,1) + d(3,2)*a - d(3,1)*a - 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:=(a*( - d(6,6)*a - 4*d(6,6 ) + d(6,5)*a + d(2,1)*a + 4*d(2,1) + d(1,1)*a + 4*d(1,1) - 2*d(0,1)*a + 2*d(0,0) *a + 8*d(0,0)))/8$ Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(0,1),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(2,1),d(1,1),d(0,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (a*( - d(6,6) *a - 4*d(6,6) + d(6,5)*a + d(2,1)*a + 4*d(2,1) + d(1,1)*a + 4*d(1,1) - 2*d(0,1)* a + 2*d(0,0)*a + 8*d(0,0)))/8 on resout l'equation {{1,3},6} qui est maintenant AA:=(4*d(2,1) + d(0,1)*a**2 + 4*d(0,1)*a)/4$ Unknowns: {d(2,1),d(0,1),a} Unknowns: {d(2,1),d(0,1),a} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=(d(0,1)*a*( - a - 4))/4$ 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:=(4*d(6,6)*a**2 + 16*d(6,6 )*a + 32*d(6,6) - 4*d(6,5)*a**2 - 32*d(5,5) - 4*d(1,1)*a**2 - 16*d(1,1)*a + 32*d (1,1) - 8*d(0,2)*a**2 - 32*d(0,2)*a + d(0,1)*a**4 + 8*d(0,1)*a**3 + 24*d(0,1)*a **2 + 32*d(0,1)*a - 8*d(0,0)*a**2 - 32*d(0,0)*a - 32*d(0,0))/32$ Unknowns: {d(6,6),d(6,5),d(5,5),d(1,1),d(0,2),d(0,1),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(5,5),d(1,1),d(0,2),d(0,1),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=(4*d(6,6)*a**2 + 16*d(6,6)*a + 32*d(6,6) - 4*d(6,5)*a**2 - 4*d(1,1)*a**2 - 16*d(1,1)*a + 32*d(1,1) - 8*d(0,2)*a**2 - 32*d(0,2)*a + d(0,1 )*a**4 + 8*d(0,1)*a**3 + 24*d(0,1)*a**2 + 32*d(0,1)*a - 8*d(0,0)*a**2 - 32*d(0,0 )*a - 32*d(0,0))/32$ on resout l'equation {{1,4},6} qui est maintenant AA:=( - 4*d(6,5) - d(0,1)*a** 2 - 4*d(0,1)*a - 4*d(0,1))/4$ Unknowns: {d(6,5),d(0,1),a} Unknowns: {d(6,5),d(0,1),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=(d(0,1)*( - a**2 - 4*a - 4))/4$ on resout l'equation {{2,3},5} qui est maintenant AA:=(a*( - 2*d(6,6)*a - 8*d(6 ,6) + 2*d(1,1)*a + 8*d(1,1) - d(0,1)*a**3 - 6*d(0,1)*a**2 - 14*d(0,1)*a + 4*d(0, 0)*a + 16*d(0,0)))/16$ Unknowns: {d(6,6),d(1,1),d(0,1),d(0,0),a} Unknowns: {d(6,6),d(1,1),d(0,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (a*( - 2*d(6, 6)*a - 8*d(6,6) + 2*d(1,1)*a + 8*d(1,1) - d(0,1)*a**3 - 6*d(0,1)*a**2 - 14*d(0,1 )*a + 4*d(0,0)*a + 16*d(0,0)))/16 on resout l'equation {{2,4},5} qui est maintenant AA:=( - 2*d(6,6)*a**2 - 8*d(6 ,6)*a - 8*d(6,6) + 2*d(1,1)*a**2 + 8*d(1,1)*a + 16*d(1,1) + 4*d(0,2)*a**2 + 8*d( 0,2)*a - d(0,1)*a**4 - 6*d(0,1)*a**3 - 16*d(0,1)*a**2 - 16*d(0,1)*a + 4*d(0,0)*a **2 + 16*d(0,0)*a + 8*d(0,0))/8$ Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a} Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans ( - 2*d(6,6)* a**2 - 8*d(6,6)*a - 8*d(6,6) + 2*d(1,1)*a**2 + 8*d(1,1)*a + 16*d(1,1) + 4*d(0,2) *a**2 + 8*d(0,2)*a - d(0,1)*a**4 - 6*d(0,1)*a**3 - 16*d(0,1)*a**2 - 16*d(0,1)*a + 4*d(0,0)*a**2 + 16*d(0,0)*a + 8*d(0,0))/8 on resout l'equation {{2,4},6} qui est maintenant AA:=(2*d(6,6)*a**2 + 8*d(6,6) *a + 8*d(6,6) - 2*d(1,1)*a**2 - 8*d(1,1)*a - 8*d(1,1) - 4*d(0,2)*a**2 - 16*d(0,2 )*a - 16*d(0,2) + d(0,1)*a**4 + 6*d(0,1)*a**3 + 14*d(0,1)*a**2 + 16*d(0,1)*a + 8 *d(0,1) - 4*d(0,0)*a**2 - 16*d(0,0)*a - 16*d(0,0))/8$ Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a} Unknowns: {d(6,6),d(1,1),d(0,2),d(0,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (2*d(6,6)*a** 2 + 8*d(6,6)*a + 8*d(6,6) - 2*d(1,1)*a**2 - 8*d(1,1)*a - 8*d(1,1) - 4*d(0,2)*a** 2 - 16*d(0,2)*a - 16*d(0,2) + d(0,1)*a**4 + 6*d(0,1)*a**3 + 14*d(0,1)*a**2 + 16* d(0,1)*a + 8*d(0,1) - 4*d(0,0)*a**2 - 16*d(0,0)*a - 16*d(0,0))/8 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}, (a**2*( - 2*d(6,6)*a - 4*d(6,6) + 2*d(1,1)*a + 8*d(1,1) - d(0,1)*a**3 - 5*d(0,1) *a**2 - 6*d(0,1)*a + 4*d(0,0)*a + 4*d(0,0)))/16}, {{{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}, (a*( - 2*d(6,6)*a - 8*d(6,6) + 2*d(1,1)*a + 8*d(1,1) - d(0,1)*a**3 - 6*d(0,1)*a **2 - 14*d(0,1)*a + 4*d(0,0)*a + 16*d(0,0)))/16}, {{{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}, (a*( - 2*d(6,6)*a - 8*d(6,6) + 2*d(1,1)*a + 8*d(1,1) - d(0,1)*a**3 - 6*d(0,1)*a **2 - 14*d(0,1)*a + 4*d(0,0)*a + 16*d(0,0)))/16}, {{{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}, ( - 2*d(6,6)*a**2 - 8*d(6,6)*a - 8*d(6,6) + 2*d(1,1)*a**2 + 8*d(1,1)*a + 16*d(1, 1) + 4*d(0,2)*a**2 + 8*d(0,2)*a - d(0,1)*a**4 - 6*d(0,1)*a**3 - 16*d(0,1)*a**2 - 16*d(0,1)*a + 4*d(0,0)*a**2 + 16*d(0,0)*a + 8*d(0,0))/8}, {{{2,4},6}, (2*d(6,6)*a**2 + 8*d(6,6)*a + 8*d(6,6) - 2*d(1,1)*a**2 - 8*d(1,1)*a - 8*d(1,1) - 4*d(0,2)*a**2 - 16*d(0,2)*a - 16*d(0,2) + d(0,1)*a**4 + 6*d(0,1)*a**3 + 14*d(0, 1)*a**2 + 16*d(0,1)*a + 8*d(0,1) - 4*d(0,0)*a**2 - 16*d(0,0)*a - 16*d(0,0))/8}, {{{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$ 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 {-2,0,-4}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(0,1),( - (a - 2)*d(0,1))/(a + 4),0,0,0,0),((d(0,1)*a**2)/4,(4*(a + 4)*d(0,0) + (a + 2)*(a - 2)*d(0,1)*a)/(4*(a + 4)),((a - 2)*d(0,1)*a**2)/(4*(a + 4)),0,0,0,0),(( - d(0,1)*a**2)/4,( - (a + 4)*d(0,1)*a)/4,( - (d(0,1)*a**3 + 6*d (0,1)*a**2 + 20*d(0,1)*a - 4*d(0,0)*a - 16*d(0,0)))/(4*(a + 4)),0,0,0,0),(d(3,0) ,d(3,1),d(3,2),( - (2*d(0,1)*a**2 + 5*d(0,1)*a - 2*d(0,0)*a - 8*d(0,0)))/(a + 4) ,(2*(a + 1)*d(0,1))/(a + 4),0,0),(((d(3,2) - d(3,1))*a**2)/4,d(4,1), - (d(3,1)*a + d(3,0) - d(3,2)*a - d(4,1)),(d(0,1)*a**2)/4,( - (3*d(0,1)*a - 4*d(0,0)))/2,0, 0),(d(5,0),d(5,1),d(5,2),( - d(3,2)*a**2)/4, - (d(3,1)*a + d(3,0) - d(3,2)*a - d (4,1)),((a**2 - 6*a - 28)*d(0,1)*a + 12*(a + 4)*d(0,0))/(4*(a + 4)),((a - 2)*d(0 ,1)*a**2)/(4*(a + 4))),(d(6,0),d(6,1),d(6,2),((a + 4)*d(3,1)*a - 4*d(4,1))/4, - (d(4,1) + d(3,1)),( - (a + 2)**2*d(0,1))/4,( - (d(0,1)*a**2 + 8*d(0,1)*a - 12*d( 0,0)))/4))$ 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] [ ] [ 2 ] [ - a ] [0 0 ------- 0 0 0] [ 4 ] [ ] [ (a + 4)*a ] [0 0 ----------- -1 0 0] [ 4 ] pour shortformdelta:={1, 1, ss, 0, 2 - a -------, 4 ss, 0, (a + 4)*a -----------} 4 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(0,1), d(0,0), a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(0,1), d(0,0), a} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(0,1), d(0,0)}$ dim Der(gtildedelta):=12$ 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:= - (a - 2)*d(0,1) mat((d(0,0),d(0,1),-------------------,0,0,0,0), a + 4 2 d(0,1)*a 4*(a + 4)*d(0,0) + (a + 2)*(a - 2)*d(0,1)*a (-----------,---------------------------------------------, 4 4*(a + 4) 2 (a - 2)*d(0,1)*a -------------------,0,0,0,0), 4*(a + 4) 2 - d(0,1)*a - (a + 4)*d(0,1)*a (--------------,---------------------, 4 4 2 - ((a + 6*a + 20)*d(0,1)*a - 4*(a + 4)*d(0,0)) --------------------------------------------------,0,0,0,0), 4*(a + 4) - ((2*a + 5)*d(0,1)*a - 2*(a + 4)*d(0,0)) (d(3,0),d(3,1),d(3,2),--------------------------------------------, a + 4 2*(a + 1)*d(0,1) ------------------,0,0), a + 4 2 (d(3,2) - d(3,1))*a (----------------------,d(4,1), - (d(3,1)*a + d(3,0) - d(3,2)*a - d(4,1)), 4 2 d(0,1)*a - (3*d(0,1)*a - 4*d(0,0)) -----------,----------------------------,0,0), 4 2 2 - d(3,2)*a (d(5,0),d(5,1),d(5,2),--------------, 4 - (d(3,1)*a + d(3,0) - d(3,2)*a - d(4,1)), 2 2 (a - 6*a - 28)*d(0,1)*a + 12*(a + 4)*d(0,0) (a - 2)*d(0,1)*a ----------------------------------------------,-------------------), 4*(a + 4) 4*(a + 4) (a + 4)*d(3,1)*a - 4*d(4,1) (d(6,0),d(6,1),d(6,2),-----------------------------, - (d(4,1) + d(3,1)), 4 2 2 - (a + 2) *d(0,1) - (d(0,1)*a + 8*d(0,1)*a - 12*d(0,0)) --------------------,-----------------------------------------)) 4 4 Unknowns: {d(0,1),d(0,0),a} Unknowns: {d(0,1),d(0,0),a} commutant de t1 dans der(gtildedelta): - (a - 2)*d(0,1) mat((d(0,0),d(0,1),-------------------,0,0,0,0), a + 4 2 d(0,1)*a 4*(a + 4)*d(0,0) + (a + 2)*(a - 2)*d(0,1)*a (-----------,---------------------------------------------, 4 4*(a + 4) 2 (a - 2)*d(0,1)*a -------------------,0,0,0,0), 4*(a + 4) 2 - d(0,1)*a - (a + 4)*d(0,1)*a (--------------,---------------------, 4 4 2 - ((a + 6*a + 20)*d(0,1)*a - 4*(a + 4)*d(0,0)) --------------------------------------------------,0,0,0,0), 4*(a + 4) - ((2*a + 5)*d(0,1)*a - 2*(a + 4)*d(0,0)) 2*(a + 1)*d(0,1) (0,0,0,--------------------------------------------,------------------,0,0), a + 4 a + 4 2 d(0,1)*a - (3*d(0,1)*a - 4*d(0,0)) (0,0,0,-----------,----------------------------,0,0), 4 2 2 (a - 6*a - 28)*d(0,1)*a + 12*(a + 4)*d(0,0) (0,0,0,0,0,----------------------------------------------, 4*(a + 4) 2 (a - 2)*d(0,1)*a -------------------), 4*(a + 4) 2 2 - (a + 2) *d(0,1) - (d(0,1)*a + 8*d(0,1)*a - 12*d(0,0)) (0,0,0,0,0,--------------------,-----------------------------------------)) 4 4 Unknowns: {d(0,1),d(0,0),a} Unknowns: {d(0,1),d(0,0),a} t2:=D(0,1):= - (a - 2) mat((0,1,------------,0,0,0,0), a + 4 2 2 a (a + 2)*(a - 2)*a (a - 2)*a (----,-------------------,------------,0,0,0,0), 4 4*(a + 4) 4*(a + 4) 2 2 - a - (a + 4)*a - (a + 6*a + 20)*a (-------,--------------,----------------------,0,0,0,0), 4 4 4*(a + 4) - (2*a + 5)*a 2*(a + 1) (0,0,0,----------------,-----------,0,0), a + 4 a + 4 2 a - 3*a (0,0,0,----,--------,0,0), 4 2 2 2 (a - 6*a - 28)*a (a - 2)*a (0,0,0,0,0,-------------------,------------), 4*(a + 4) 4*(a + 4) 2 - (a + 2) - (a + 8)*a (0,0,0,0,0,-------------,--------------)) 4 4 Unknowns: {d(0,1),d(0,0),a} Unknowns: {d(0,1),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),d(0,1),( - (a - 2)*d(0,1))/(a + 4),0,0,0,0),((d(0,1)*a**2)/4,(4*(a + 4)*d(0,0) + (a + 2)*(a - 2)*d(0,1)*a)/(4*(a + 4)),((a - 2)*d(0,1)*a**2)/(4*(a + 4)),0,0,0,0),(( - d(0,1)*a**2)/4,( - (a + 4)*d(0,1)*a)/4,( - ((a**2 + 6*a + 20) *d(0,1)*a - 4*(a + 4)*d(0,0)))/(4*(a + 4)),0,0,0,0),(0,0,0,( - ((2*a + 5)*d(0,1) *a - 2*(a + 4)*d(0,0)))/(a + 4),(2*(a + 1)*d(0,1))/(a + 4),0,0),(0,0,0,(d(0,1)*a **2)/4,( - (3*d(0,1)*a - 4*d(0,0)))/2,0,0),(0,0,0,0,0,((a**2 - 6*a - 28)*d(0,1)* a + 12*(a + 4)*d(0,0))/(4*(a + 4)),((a - 2)*d(0,1)*a**2)/(4*(a + 4))),(0,0,0,0,0 ,( - (a + 2)**2*d(0,1))/4,( - (d(0,1)*a**2 + 8*d(0,1)*a - 12*d(0,0)))/4))$ - (a - 2) t2 := mat((0,1,------------,0,0,0,0), a + 4 2 2 a (a + 2)*(a - 2)*a (a - 2)*a (----,-------------------,------------,0,0,0,0), 4 4*(a + 4) 4*(a + 4) 2 2 - a - (a + 4)*a - (a + 6*a + 20)*a (-------,--------------,----------------------,0,0,0,0), 4 4 4*(a + 4) - (2*a + 5)*a 2*(a + 1) (0,0,0,----------------,-----------,0,0), a + 4 a + 4 2 a - 3*a (0,0,0,----,--------,0,0), 4 2 2 2 (a - 6*a - 28)*a (a - 2)*a (0,0,0,0,0,-------------------,------------), 4*(a + 4) 4*(a + 4) 2 - (a + 2) - (a + 8)*a (0,0,0,0,0,-------------,--------------)) 4 4 rank 2 with maximal torus t1,t2 2 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:= - 4*(a + 1) a - 2 mat((1,--------------,-----------,0,0,0,0), (a + 4)*a (a + 4)*a a - (a + 1)*(a - 2) (---,1,--------------------,0,0,0,0), 2 2 (a + 4) - a (------,-1,1,0,0,0,0), 2 - 4*(a + 1) (0,0,0,1,--------------,0,0), (a + 4)*a a (0,0,0,---,1,0,0), 2 - (a - 2)*a (0,0,0,0,0,1,-----------------), (a + 4)*(a + 2) - (a + 2) (0,0,0,0,0,------------,1)) a 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] P**(-1)*t2*P:= (a + 1)*a mat((-----------,0,0,0,0,0,0), a + 4 - a (0,------,0,0,0,0,0), 2 - (2*a + 5)*a (0,0,----------------,0,0,0,0), a + 4 (0,0,0, - a,0,0,0), - (5*a + 14)*a (0,0,0,0,-----------------,0,0), 2*(a + 4) - 3*a (0,0,0,0,0,--------,0), 2 - 3*(a + 3)*a (0,0,0,0,0,0,----------------)) a + 4 matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((((a + 4)*d(0,0) + (a + 1)*d(0,1)*a)/(a + 4),0,0,0,0,0,0),(0,( - (d(0,1)*a - 2*d(0,0)))/2,0,0,0,0,0),(0,0,( - ((2*a + 5)*d(0,1)*a - (a + 4)*d(0,0)))/(a + 4) ,0,0,0,0),((d(3,1)*a + 2*d(3,0) - d(3,2)*a)/2,( - 3*(d(3,2) - d(3,1))*(a + 2))/( a + 4),( - (((a + 4)*d(3,0) + 2*(a + 1)*d(3,1)*a - (2*a + 5)*d(3,2)*a)*(a + 4) - 12*(a + 1)*d(4,1)))/((a + 4)**2*a), - (d(0,1)*a - 2*d(0,0)),0,0,0),(0,(d(3,1)*a + 2*d(3,0) - d(3,2)*a)/2,( - ((a + 10)*d(3,1)*a + 2*(a + 4)*d(3,0) - (a + 4)*d( 3,2)*a - 12*d(4,1)))/(4*(a + 4)),0,( - ((5*a + 14)*d(0,1)*a - 4*(a + 4)*d(0,0))) /(2*(a + 4)),0,0),(((d(5,1)*a + 2*d(5,0) - d(5,2)*a)*(a + 4)*(a + 2) + 2*(a - 2) *d(6,0)*a + (a - 2)*d(6,1)*a**2 - (a - 2)*d(6,2)*a**2)/(12*(a + 2)),(((a + 4)*d( 5,1)*a - 4*(a + 1)*d(5,0) - (a + 4)*d(5,2)*a)*(a + 4)*(a + 2) - 4*(a + 1)*(a - 2 )*d(6,0)*a + (a + 4)*(a - 2)*d(6,1)*a**2 - (a + 4)*(a - 2)*d(6,2)*a**2)/(6*(a + 4)*(a + 2)*a),(((((a + 4)*d(5,0) - (a + 1)*d(5,1)*a)*(a - 2) + (a + 4)**2*d(5,2) *a)*(a + 2) + (a - 2)**2*d(6,0)*a)*(a + 4) - (a + 1)*(a - 2)**2*d(6,1)*a**2 + (a + 4)**2*(a - 2)*d(6,2)*a**2)/(6*(a + 4)**2*(a + 2)*a),( - ((a + 10)*d(3,1)*a + 2*(a + 4)*d(3,0) - (a + 4)*d(3,2)*a - 12*d(4,1))*a)/24,( - (((a + 4)*d(3,0) + 2* (a + 1)*d(3,1)*a - (2*a + 5)*d(3,2)*a)*(a + 4) - 12*(a + 1)*d(4,1)))/(6*(a + 4)) ,( - 3*(d(0,1)*a - 2*d(0,0)))/2,0),((((d(5,1)*a + 2*d(5,0) - d(5,2)*a)*(a + 2) + 2*d(6,0)*a + d(6,1)*a**2 - d(6,2)*a**2)*(a + 4))/(12*a),(((a + 4)*d(5,1)*a - 4* (a + 1)*d(5,0) - (a + 4)*d(5,2)*a)*(a + 2) - 4*(a + 1)*d(6,0)*a + (a + 4)*d(6,1) *a**2 - (a + 4)*d(6,2)*a**2)/(6*a**2),((((a + 4)*d(5,0) - (a + 1)*d(5,1)*a)*(a - 2) + (a + 4)**2*d(5,2)*a)*(a + 2) + (a + 4)*(a - 2)*d(6,0)*a - (a + 1)*(a - 2)* d(6,1)*a**2 + (a + 4)**2*d(6,2)*a**2)/(6*(a + 4)*a**2),( - (d(3,1)*a + 2*d(3,0) - d(3,2)*a)*(a + 4)*(a + 2))/24,( - ((2*d(3,1)*a + d(3,0))*(a + 4) - (2*a + 5)*d (3,2)*a - 6*d(4,1))*(a + 2))/(6*a),0,(3*((a + 4)*d(0,0) - (a + 3)*d(0,1)*a))/(a + 4)))$ PP:= - 4*(a + 1) a - 2 mat((1,--------------,-----------,0,0,0,0), (a + 4)*a (a + 4)*a a - (a + 1)*(a - 2) (---,1,--------------------,0,0,0,0), 2 2 (a + 4) - a (------,-1,1,0,0,0,0), 2 - 4*(a + 1) (0,0,0,1,--------------,0,0), (a + 4)*a a (0,0,0,---,1,0,0), 2 - (a - 2)*a (0,0,0,0,0,1,-----------------), (a + 4)*(a + 2) - (a + 2) (0,0,0,0,0,------------,1)) a avec PP:=P*Q:= - 4*(a + 1) a - 2 mat((1,--------------,-----------,0,0,0,0), (a + 4)*a (a + 4)*a a - (a + 1)*(a - 2) (---,1,--------------------,0,0,0,0), 2 2 (a + 4) - a (------,-1,1,0,0,0,0), 2 - 4*(a + 1) (0,0,0,1,--------------,0,0), (a + 4)*a a (0,0,0,---,1,0,0), 2 - (a - 2)*a (0,0,0,0,0,1,-----------------), (a + 4)*(a + 2) - (a + 2) (0,0,0,0,0,------------,1)) a MATDDIAGONALISE:= (a + 4)*d(0,0) + (a + 1)*d(0,1)*a mat((-----------------------------------,0,0,0,0,0,0), a + 4 - (d(0,1)*a - 2*d(0,0)) (0,--------------------------,0,0,0,0,0), 2 - ((2*a + 5)*d(0,1)*a - (a + 4)*d(0,0)) (0,0,------------------------------------------,0,0,0,0), a + 4 d(3,1)*a + 2*d(3,0) - d(3,2)*a - 3*(d(3,2) - d(3,1))*(a + 2) (--------------------------------,--------------------------------,( - ( 2 a + 4 ((a + 4)*d(3,0) + 2*(a + 1)*d(3,1)*a - (2*a + 5)*d(3,2)*a)*(a + 4) 2 - 12*(a + 1)*d(4,1)))/((a + 4) *a), - (d(0,1)*a - 2*d(0,0)),0,0,0), d(3,1)*a + 2*d(3,0) - d(3,2)*a (0,--------------------------------, 2 - ((a + 10)*d(3,1)*a + 2*(a + 4)*d(3,0) - (a + 4)*d(3,2)*a - 12*d(4,1)) --------------------------------------------------------------------------, 4*(a + 4) - ((5*a + 14)*d(0,1)*a - 4*(a + 4)*d(0,0)) 0,---------------------------------------------,0,0), 2*(a + 4) (((d(5,1)*a + 2*d(5,0) - d(5,2)*a)*(a + 4)*(a + 2) + 2*(a - 2)*d(6,0)*a 2 2 + (a - 2)*d(6,1)*a - (a - 2)*d(6,2)*a )/(12*(a + 2)),( ((a + 4)*d(5,1)*a - 4*(a + 1)*d(5,0) - (a + 4)*d(5,2)*a)*(a + 4)*(a + 2) 2 - 4*(a + 1)*(a - 2)*d(6,0)*a + (a + 4)*(a - 2)*d(6,1)*a 2 - (a + 4)*(a - 2)*d(6,2)*a )/(6*(a + 4)*(a + 2)*a),(( 2 (((a + 4)*d(5,0) - (a + 1)*d(5,1)*a)*(a - 2) + (a + 4) *d(5,2)*a) 2 2 2 *(a + 2) + (a - 2) *d(6,0)*a)*(a + 4) - (a + 1)*(a - 2) *d(6,1)*a 2 2 2 + (a + 4) *(a - 2)*d(6,2)*a )/(6*(a + 4) *(a + 2)*a),( - ((a + 10)*d(3,1)*a + 2*(a + 4)*d(3,0) - (a + 4)*d(3,2)*a - 12*d(4,1))*a) /24,( - (((a + 4)*d(3,0) + 2*(a + 1)*d(3,1)*a - (2*a + 5)*d(3,2)*a)*(a + 4) - 3*(d(0,1)*a - 2*d(0,0)) - 12*(a + 1)*d(4,1)))/(6*(a + 4)),----------------------------,0) 2 , 2 ((((d(5,1)*a + 2*d(5,0) - d(5,2)*a)*(a + 2) + 2*d(6,0)*a + d(6,1)*a 2 - d(6,2)*a )*(a + 4))/(12*a),( ((a + 4)*d(5,1)*a - 4*(a + 1)*d(5,0) - (a + 4)*d(5,2)*a)*(a + 2) 2 2 2 - 4*(a + 1)*d(6,0)*a + (a + 4)*d(6,1)*a - (a + 4)*d(6,2)*a )/(6*a ),( 2 (((a + 4)*d(5,0) - (a + 1)*d(5,1)*a)*(a - 2) + (a + 4) *d(5,2)*a) 2 *(a + 2) + (a + 4)*(a - 2)*d(6,0)*a - (a + 1)*(a - 2)*d(6,1)*a 2 2 2 + (a + 4) *d(6,2)*a )/(6*(a + 4)*a ), - (d(3,1)*a + 2*d(3,0) - d(3,2)*a)*(a + 4)*(a + 2) -----------------------------------------------------,( - 24 ((2*d(3,1)*a + d(3,0))*(a + 4) - (2*a + 5)*d(3,2)*a - 6*d(4,1))*(a + 2)) 3*((a + 4)*d(0,0) - (a + 3)*d(0,1)*a) /(6*a),0,---------------------------------------)) a + 4 on voit apparaitre les poids sur la diagonale (a + 4)*d(0,0) + (a + 1)*d(0,1)*a r(1) := ----------------------------------- a + 4 - (d(0,1)*a - 2*d(0,0)) r(2) := -------------------------- 2 - ((2*a + 5)*d(0,1)*a - (a + 4)*d(0,0)) r(3) := ------------------------------------------ a + 4 r(4) := - (d(0,1)*a - 2*d(0,0)) - ((5*a + 14)*d(0,1)*a - 4*(a + 4)*d(0,0)) r(5) := --------------------------------------------- 2*(a + 4) - 3*(d(0,1)*a - 2*d(0,0)) r(6) := ---------------------------- 2 3*((a + 4)*d(0,0) - (a + 3)*d(0,1)*a) r(7) := --------------------------------------- a + 4 (a + 4)*d(0,0) + (a + 1)*d(0,1)*a r(1) := ----------------------------------- a + 4 - (d(0,1)*a - 2*d(0,0)) r(2) := -------------------------- 2 - ((2*a + 5)*d(0,1)*a - (a + 4)*d(0,0)) r(3) := ------------------------------------------ a + 4 r(4) := - (d(0,1)*a - 2*d(0,0)) - ((5*a + 14)*d(0,1)*a - 4*(a + 4)*d(0,0)) r(5) := --------------------------------------------- 2*(a + 4) - 3*(d(0,1)*a - 2*d(0,0)) r(6) := ---------------------------- 2 3*((a + 4)*d(0,0) - (a + 3)*d(0,1)*a) r(7) := --------------------------------------- a + 4 r(4)-(r(1)+r(3)):= 0 r(2)-(r(1)+r(3))/2:= 0 r(5)-(3*r(3)+r(1))/2:= 0 r(6)-3*(r(3)+r(1))/2:= 0 r(7)-(2*r(3)+r(1)):= 0 r(1)-(2*r(2)-r(3)):= 0 r(4)-(2*r(2)):= 0 r(5)-(r(3)+r(2)):= 0 r(6)-3*r(2):= 0 r(7)-(r(3)+2*r(2)):= 0 Le systeme de poids est le systeme 2.35 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(3)}, {{0,2},x(3)}, a*(x(6)*a + 4*x(6) - x(5)*a) {{0,3},------------------------------}, 4 {{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}} - x(2)*a + x(1)*a + 2*x(0) diaY(1):=----------------------------- 2 2 2 - x(2)*a - 4*x(2)*a + x(1)*a + 4*x(1)*a - 4*x(0)*a - 4*x(0) diaY(2):=---------------------------------------------------------------- a*(a + 4) 3 2 3 2 diaY(3):=(x(2)*a + 8*x(2)*a + 16*x(2)*a - x(1)*a + x(1)*a + 2*x(1)*a 2 2 + x(0)*a + 2*x(0)*a - 8*x(0))/(a*(a + 8*a + 16)) x(4)*a + 2*x(3) diaY(4):=----------------- 2 2 x(4)*a + 4*x(4)*a - 4*x(3)*a - 4*x(3) diaY(5):=---------------------------------------- a*(a + 4) - x(6)*a - 2*x(6) + x(5)*a diaY(6):=----------------------------- a 2 2 x(6)*a + 6*x(6)*a + 8*x(6) - x(5)*a + 2*x(5)*a diaY(7):=-------------------------------------------------- 2 a + 6*a + 8 liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},0}, {{1,3},(9*diay(4)*(a + 2))/(a**2 + 8*a + 16)}, {{1,4},0}, {{1,5},(3*diay(6)*a*(a + 2))/(2*(a + 4))}, {{1,6},0}, {{1,7},0}, {{2,3},(9*diay(5)*(a**2 + 6*a + 8))/(a**3 + 12*a**2 + 48*a + 64)}, {{2,4},(3*diay(6)*a*(a + 2))/(2*(a + 4))}, {{2,5},(3*diay(7)*(a**3 + 8*a**2 + 20*a + 16))/(a*(a**2 + 8*a + 16))}, {{2,6},0}, {{2,7},0}, {{3,4},(3*diay(7)*(a**3 + 8*a**2 + 20*a + 16))/(4*(a**2 + 8*a + 16))}, {{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_{7,2.35}$ and that for a neq{-2,0,-4}$ i.e. we go from the basis diaY(i) to the new basis ZZ(i) defined by the matrix:$ on pose :$ avec comme matrice de changement de base :$ mat((0,0,4/a,0,0,0,0),(0,1,0,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,0,(36*( - a - 2))/( a*(a**2 + 8*a + 16)),0,0),(0,0,0,(9*( - a - 2))/(a**2 + 8*a + 16),0,0,0),(0,0,0, 0,0,0,(54*( - a**2 - 4*a - 4))/(a**3 + 12*a**2 + 48*a + 64)),(0,0,0,0,0,(27*( - a**3 - 6*a**2 - 12*a - 8))/(a*(a**3 + 12*a**2 + 48*a + 64)),0))$ det(isom):= ( - 1889568*(a + 2)**7)/((a + 4)**10*a**3)$ ZZ(1):=diay(3)$ ZZ(2):=diay(2)$ ZZ(3):=(4*diay(1))/a$ ZZ(4):=( - 9*(a + 2)*diay(5))/(a + 4)**2$ ZZ(5):=( - 36*(a + 2)*diay(4))/((a + 4)**2*a)$ ZZ(6):=( - 27*(a + 2)**3*diay(7))/((a + 4)**3*a)$ ZZ(7):=( - 54*(a + 2)**2*diay(6))/(a + 4)**3$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(5)}$ {{1,4},0}$ {{1,5},zz(6)}$ {{1,6},0}$ {{1,7},0}$ {{2,3},0}$ {{2,4},zz(6)}$ {{2,5},zz(7)}$ {{2,6},0}$ {{2,7},0}$ {{3,4},zz(7)}$ {{3,5},0}$ {{3,6},0}$ {{3,7},0}$ {{4,5},0}$ {{4,6},0}$ {{4,7},0}$ {{5,6},0}$ {{5,7},0}$ {{6,7},0}$ We get the commutation relations of$ g_{7,2.35}$ Et cela pour a:=a, b:=a**2.$ and that for a neq{-2,0,-4}$ shortformdelta:={1, 1, ss, 0, ( - a**2)/4, ss, 0, ((a + 4)*a)/4}$ 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**2)/4,0,0 ,0),(0,0,((a + 4)*a)/4,-1,0,0))$ ADDED: ********* singular cases: For a:=0 we get :$ shortformdelta:={1,1,ss,0,0,ss,0,0}$ which has already occured as case1I3.$ ADDED: ********* singular cases: For a:=-2 we get :$ shortformdelta:={1,1,ss,0,-1,ss,0,-1}$ which! is! projectively! equivalent! under! some! second! kind! automorphism! to ! the! pr\ eceding.$ ADDED: ********* singular cases: For a:=-4 we get :$ shortformdelta:={1,1,ss,0,-4,ss,0,0}$ which has already occured as case1I4.$