generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),xi(2,3),0,0,0),( - xi(2,3),0,(xi(2,2) + xi(1,1))/2,0,0,0),(xi(4,1),xi(4,2),xi(4,3),xi(2,2) + xi(1,1),xi(2,3),0),(xi(5,1 ),0,xi(5,3),0,(xi(2,2) + 3*xi(1,1))/2,0),(xi(6,1),xi(6,2),xi(6,3),xi(4,2), - xi( 5,1) + xi(4,3),xi(2,2) + 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 1 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] adx2 := [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [0 0 0 0 0 0] [ ] [-1 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] adx4 := [ ] [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 0] adx5 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 -1 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(6,1):=0,xi(6,3):=0 delta:= [ 0 0 0 0 0 0] [ ] [ xi(2,1) 0 xi(2,3) 0 0 0] [ ] [ - xi(2,3) 0 0 0 0 0] [ ] [ 0 0 xi(4,3) 0 xi(2,3) 0] [ ] [ 0 0 xi(5,3) 0 0 0] [ ] [ 0 xi(6,2) 0 0 xi(4,3) 0] We denote this delta by the shortform shortformdelta:={xi(2,1), xi(2,3), ss, xi(4,3), ss, xi(5,3), ss, xi(6,2)} paramindexeslist:={{2,1},{2,3},{4,3},{5,3},{6,2}} a neq {}$ a:=a$ delta:= mat((0,0,0,0,0,0),(1,0,1,0,0,0),(-1,0,0,0,0,0),(0,0,0,0,1,0),(0,0,1,0,0,0),(0,a, 0,0,0,0))$ shortformdelta:={1,1,ss,0,ss,1,ss,a}$ on resout l'equation {{0,1},0} qui est maintenant AA:=d(0,3) - d(0,2)$ Unknowns: {d(0,3),d(0,2)} Unknowns: {d(0,3),d(0,2)} bonne inconnue W:=d(0,3)$ sa valeur doit etre WW:=d(0,2)$ on resout l'equation {{0,1},1} qui est maintenant AA:=d(1,3) - d(1,2)$ Unknowns: {d(1,3),d(1,2)} Unknowns: {d(1,3),d(1,2)} bonne inconnue W:=d(1,3)$ sa valeur doit etre WW:=d(1,2)$ on resout l'equation {{0,1},2} qui est maintenant AA:=d(3,1) + d(2,3) - d(2,2) + d(1,1) + d(0,0)$ Unknowns: {d(3,1),d(2,3),d(2,2),d(1,1),d(0,0)} Unknowns: {d(3,1),d(2,3),d(2,2),d(1,1),d(0,0)} bonne inconnue W:=d(3,1)$ sa valeur doit etre WW:= - d(2,3) + d(2,2) - d(1,1) - d(0,0)$ on resout l'equation {{0,1},3} qui est maintenant AA:=d(3,3) - d(3,2) - d(1,1) - d(0,0)$ Unknowns: {d(3,3),d(3,2),d(1,1),d(0,0)} Unknowns: {d(3,3),d(3,2),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(3,2) + d(1,1) + d(0,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:=d(5,1) + d(4,3) - d(4,2) - d(2,0)$ Unknowns: {d(5,1),d(4,3),d(4,2),d(2,0)} Unknowns: {d(5,1),d(4,3),d(4,2),d(2,0)} bonne inconnue W:=d(5,1)$ sa valeur doit etre WW:= - d(4,3) + d(4,2) + d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:=d(5,3) - d(5,2) - d(3,0) - d(2,3) + d(2,2) - d(1,1) - d(0,0)$ Unknowns: {d(5,3),d(5,2),d(3,0),d(2,3),d(2,2),d(1,1),d(0,0)} Unknowns: {d(5,3),d(5,2),d(3,0),d(2,3),d(2,2),d(1,1),d(0,0)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(5,2) + d(3,0) + d(2,3) - d(2,2) + d(1,1) + d(0,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:=d(6,3) - d(6,2) - d(4,0) + d(2,1)*a$ Unknowns: {d(6,3),d(6,2),d(4,0),d(2,1),a} Unknowns: {d(6,3),d(6,2),d(4,0),d(2,1),a} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(6,2) + d(4,0) - d(2,1)*a$ on resout l'equation {{0,2},0} qui est maintenant AA:= - d(0,6)*a$ Unknowns: {d(0,6),a} Unknowns: {d(0,6),a} pas de selection possible de variable a coefficient numerique dans - d(0,6)*a on resout l'equation {{0,2},1} qui est maintenant AA:= - d(1,6)*a$ Unknowns: {d(1,6),a} Unknowns: {d(1,6),a} pas de selection possible de variable a coefficient numerique dans - d(1,6)*a on resout l'equation {{0,2},2} qui est maintenant AA:=d(3,2) - d(2,6)*a + d(1,2 )$ Unknowns: {d(3,2),d(2,6),d(1,2),a} Unknowns: {d(3,2),d(2,6),d(1,2),a} bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:=d(2,6)*a - d(1,2)$ on resout l'equation {{0,2},3} qui est maintenant AA:= - (d(3,6)*a + d(1,2))$ Unknowns: {d(3,6),d(1,2),a} Unknowns: {d(3,6),d(1,2),a} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:= - d(3,6)*a$ on resout l'equation {{0,2},4} qui est maintenant AA:=d(5,2) - d(4,6)*a + d(1,0 )$ Unknowns: {d(5,2),d(4,6),d(1,0),a} Unknowns: {d(5,2),d(4,6),d(1,0),a} bonne inconnue W:=d(5,2)$ sa valeur doit etre WW:=d(4,6)*a - d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=a*( - d(5,6) + d(3,6) + d (2,6))$ Unknowns: {d(5,6),d(3,6),d(2,6),a} Unknowns: {d(5,6),d(3,6),d(2,6),a} pas de selection possible de variable a coefficient numerique dans a*( - d(5,6) + d(3,6) + d(2,6)) on resout l'equation {{0,2},6} qui est maintenant AA:=a*( - d(6,6) + d(2,2) + d (0,0))$ Unknowns: {d(6,6),d(2,2),d(0,0),a} Unknowns: {d(6,6),d(2,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(6,6) + d(2,2) + d(0,0)) on resout l'equation {{0,3},0} qui est maintenant AA:= - (d(0,5) + d(0,2))$ Unknowns: {d(0,5),d(0,2)} Unknowns: {d(0,5),d(0,2)} bonne inconnue W:=d(0,5)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{0,3},1} qui est maintenant AA:=d(3,6)*a - d(1,5)$ Unknowns: {d(3,6),d(1,5),a} Unknowns: {d(3,6),d(1,5),a} bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:=d(3,6)*a$ on resout l'equation {{0,3},2} qui est maintenant AA:=d(2,6)*a - d(2,5) - d(2,2 ) + d(1,1) + 2*d(0,0)$ Unknowns: {d(2,6),d(2,5),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(2,6),d(2,5),d(2,2),d(1,1),d(0,0),a} bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:=d(2,6)*a - d(2,2) + d(1,1) + 2*d(0,0)$ on resout l'equation {{0,3},3} qui est maintenant AA:= - (d(3,5) + d(2,6)*a)$ Unknowns: {d(3,5),d(2,6),a} Unknowns: {d(3,5),d(2,6),a} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(2,6)*a$ on resout l'equation {{0,3},4} qui est maintenant AA:=d(4,6)*a - d(4,5) - d(4,2 ) + d(3,0) + d(2,3) - d(2,2) + d(1,1) - d(1,0) + d(0,0)$ Unknowns: {d(4,6),d(4,5),d(4,2),d(3,0),d(2,3),d(2,2),d(1,1),d(1,0),d(0,0),a} Unknowns: {d(4,6),d(4,5),d(4,2),d(3,0),d(2,3),d(2,2),d(1,1),d(1,0),d(0,0),a} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=d(4,6)*a - d(4,2) + d(3,0) + d(2,3) - d(2,2) + d(1,1) - d(1,0) + d(0,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,5) - d(4,6)*a + d( 3,6)*a + d(2,6)*a + d(1,1) + 2*d(1,0) + 2*d(0,0)$ Unknowns: {d(5,5),d(4,6),d(3,6),d(2,6),d(1,1),d(1,0),d(0,0),a} Unknowns: {d(5,5),d(4,6),d(3,6),d(2,6),d(1,1),d(1,0),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(4,6)*a + d(3,6)*a + d(2,6)*a + d(1,1) + 2*d(1,0) + 2*d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,5) - d(6,2) - d(5, 0) + d(2,3)*a$ Unknowns: {d(6,5),d(6,2),d(5,0),d(2,3),a} Unknowns: {d(6,5),d(6,2),d(5,0),d(2,3),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(6,2) - d(5,0) + d(2,3)*a$ on resout l'equation {{0,4},2} qui est maintenant AA:=d(3,4) + d(1,4)$ Unknowns: {d(3,4),d(1,4)} Unknowns: {d(3,4),d(1,4)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,4},3} 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 {{0,4},4} qui est maintenant AA:=d(5,4)$ Unknown: d(5,4) Unknown: d(5,4) bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(2,4)*a + d(1,0)$ Unknowns: {d(2,4),d(1,0),a} Unknowns: {d(2,4),d(1,0),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:= - d(2,4)*a$ on resout l'equation {{0,5},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 {{0,5},2} qui est maintenant AA:=d(3,6)*a - d(2,6)*a - d(2 ,4)$ Unknowns: {d(3,6),d(2,6),d(2,4),a} Unknowns: {d(3,6),d(2,6),d(2,4),a} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:=a*(d(3,6) - d(2,6))$ on resout l'equation {{0,5},3} qui est maintenant AA:= - d(3,6)*a$ Unknowns: {d(3,6),a} Unknowns: {d(3,6),a} pas de selection possible de variable a coefficient numerique dans - d(3,6)*a on resout l'equation {{0,5},4} qui est maintenant AA:= - d(4,6)*a - d(4,4) - 2* d(3,6)*a**2 + d(3,6)*a + 2*d(2,6)*a**2 + d(2,6)*a + d(1,1) + 3*d(0,0)$ Unknowns: {d(4,6),d(4,4),d(3,6),d(2,6),d(1,1),d(0,0),a} Unknowns: {d(4,6),d(4,4),d(3,6),d(2,6),d(1,1),d(0,0),a} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:= - d(4,6)*a - 2*d(3,6)*a**2 + d(3,6)*a + 2*d(2,6)*a**2 + d(2,6)*a + d(1,1) + 3*d(0,0)$ on resout l'equation {{0,5},5} qui est maintenant AA:= - d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans - d(2,6)*a on resout l'equation {{0,5},6} qui est maintenant AA:= - d(6,4) + d(3,0) + d(2, 6)*a**2 - d(2,2)*a + d(1,1)*a + 2*d(0,0)*a$ Unknowns: {d(6,4),d(3,0),d(2,6),d(2,2),d(1,1),d(0,0),a} Unknowns: {d(6,4),d(3,0),d(2,6),d(2,2),d(1,1),d(0,0),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(3,0) + d(2,6)*a**2 - d(2,2)*a + d(1,1)*a + 2*d(0,0)*a$ on resout l'equation {{0,6},2} qui est maintenant AA:=d(3,6) + d(1,6)$ Unknowns: {d(3,6),d(1,6)} Unknowns: {d(3,6),d(1,6)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(1,6)$ on resout l'equation {{0,6},3} 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 {{0,6},4} qui est maintenant AA:=d(5,6)$ Unknown: d(5,6) Unknown: d(5,6) bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,6},6} qui est maintenant AA:=d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans d(2,6)*a on resout l'equation {{1,2},2} qui est maintenant AA:=d(2,6)*a - d(0,2)$ Unknowns: {d(2,6),d(0,2),a} Unknowns: {d(2,6),d(0,2),a} bonne inconnue W:=d(0,2)$ sa valeur doit etre WW:=d(2,6)*a$ on resout l'equation {{1,2},3} qui est maintenant AA:=d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans d(2,6)*a on resout l'equation {{1,2},4} qui est maintenant AA:=d(4,6)*a - 2*d(2,6)*a**2 - d(2,6)*a + d(2,2) - 3*d(0,0)$ Unknowns: {d(4,6),d(2,6),d(2,2),d(0,0),a} Unknowns: {d(4,6),d(2,6),d(2,2),d(0,0),a} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:= - d(4,6)*a + 2*d(2,6)*a**2 + d(2,6)*a + 3*d(0,0)$ on resout l'equation {{1,2},5} qui est maintenant AA:=d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans d(2,6)*a on resout l'equation {{1,2},6} qui est maintenant AA:= - d(4,6)*a**2 + d(4,2) - d(3,0) + 2*d(2,6)*a**3 - d(1,1)*a + d(0,1)*a + d(0,0)*a$ Unknowns: {d(4,6),d(4,2),d(3,0),d(2,6),d(1,1),d(0,1),d(0,0),a} Unknowns: {d(4,6),d(4,2),d(3,0),d(2,6),d(1,1),d(0,1),d(0,0),a} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:=d(4,6)*a**2 + d(3,0) - 2*d(2,6)*a**3 + d(1,1)*a - d(0,1) *a - d(0,0)*a$ on resout l'equation {{1,3},0} qui est maintenant AA:=d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans d(2,6)*a on resout l'equation {{1,3},2} qui est maintenant AA:= - d(4,6)*a + 2*d(2,6)*a **2 - d(2,6)*a - d(1,1) + d(0,1) + d(0,0)$ Unknowns: {d(4,6),d(2,6),d(1,1),d(0,1),d(0,0),a} Unknowns: {d(4,6),d(2,6),d(1,1),d(0,1),d(0,0),a} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:= - d(4,6)*a + 2*d(2,6)*a**2 - d(2,6)*a + d(0,1) + d(0,0) $ on resout l'equation {{1,3},3} qui est maintenant AA:=2*d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans 2*d(2,6)*a on resout l'equation {{1,3},4} qui est maintenant AA:= - d(4,6)*a + 2*d(2,6)*a - d(0,1) + d(0,0)$ Unknowns: {d(4,6),d(2,6),d(0,1),d(0,0),a} Unknowns: {d(4,6),d(2,6),d(0,1),d(0,0),a} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:= - d(4,6)*a + 2*d(2,6)*a + d(0,0)$ on resout l'equation {{1,3},5} qui est maintenant AA:= - 2*d(4,6)*a + 3*d(2,6)* a + 2*d(0,0)$ Unknowns: {d(4,6),d(2,6),d(0,0),a} Unknowns: {d(4,6),d(2,6),d(0,0),a} bonne inconnue W:=d(0,0)$ sa valeur doit etre WW:=(a*(2*d(4,6) - 3*d(2,6)))/2$ on resout l'equation {{1,3},6} qui est maintenant AA:=d(6,2) + d(5,0) + 2*d(4,3 ) - d(3,0) + d(2,6)*a**2 - d(2,3)*a - d(2,0)$ Unknowns: {d(6,2),d(5,0),d(4,3),d(3,0),d(2,6),d(2,3),d(2,0),a} Unknowns: {d(6,2),d(5,0),d(4,3),d(3,0),d(2,6),d(2,3),d(2,0),a} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:= - d(5,0) - 2*d(4,3) + d(3,0) - d(2,6)*a**2 + d(2,3)*a + d(2,0)$ on resout l'equation {{1,4},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 {{1,4},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 {{1,4},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 {{1,4},6} qui est maintenant AA:= - d(6,6)$ Unknown: d(6,6) Unknown: d(6,6) bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=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},0}, {{{0,3},6},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},4},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},0},0}, {{{1,3},1},0}, {{{1,3},2},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},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},2},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},2},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,3},6},0}, {{{2,4},4},0}, {{{2,4},6},0}, {{{2,5},4},0}, {{{2,5},6},0}, {{{2,6},4},0}, {{{2,6},6},0}, {{{3,4},2},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},0},0}, {{{3,5},1},0}, {{{3,5},2},0}, {{{3,5},3},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},2},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},4},0}, {{{4,5},6},0}, {{{4,6},6},0}, {{{5,6},4},0}, {{{5,6},6},0}}$ Il n'y a pas de phase 2$ 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,d(2,3),0,0,0),(d(3,0), - d( 2,3),0,0,0,0,0),(d(4,0),d(4,1),d(3,0),d(4,3),0,d(2,3),0),(d(5,0), - d(4,3) + d(3 ,0) + d(2,0),0,d(3,0) + d(2,3),0,0,0),(d(6,0),d(6,1), - d(5,0) - 2*d(4,3) + d(3, 0) + d(2,3)*a + d(2,0), - d(5,0) - 2*d(4,3) + d(4,0) + d(3,0) + d(2,3)*a - d(2,1 )*a + d(2,0),d(3,0),2*d(4,3) - d(3,0) - d(2,0),0))$ pour delta:= [0 0 0 0 0 0] [ ] [1 0 1 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 1 0] [ ] [0 0 1 0 0 0] [ ] [0 a 0 0 0 0] pour shortformdelta:={1,1,ss,0,ss,1,ss,a} Unknowns: {d(6,1), d(6,0), d(5,0), d(4,3), d(4,1), d(4,0), d(3,0), d(2,3), d(2,1), d(2,0), a} Unknowns: {d(6,1), d(6,0), d(5,0), d(4,3), d(4,1), d(4,0), d(3,0), d(2,3), d(2,1), d(2,0), a} listeparametresMATD{d(6,1), d(6,0), d(5,0), d(4,3), d(4,1), d(4,0), d(3,0), d(2,3), 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,d(2,3),0,0,0), (d(3,0), - d(2,3),0,0,0,0,0), (d(4,0),d(4,1),d(3,0),d(4,3),0,d(2,3),0), (d(5,0), - d(4,3) + d(3,0) + d(2,0),0,d(3,0) + d(2,3),0,0,0), (d(6,0),d(6,1), - d(5,0) - 2*d(4,3) + d(3,0) + d(2,3)*a + d(2,0), - d(5,0) - 2*d(4,3) + d(4,0) + d(3,0) + d(2,3)*a - d(2,1)*a + d(2,0), d(3,0),2*d(4,3) - d(3,0) - 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,d(2,3),0,0,0), (d(3,0), - d(2,3),0,0,0,0,0), (d(4,0),d(4,1),d(3,0),d(4,3),0,d(2,3),0), (d(5,0), - d(4,3) + d(3,0) + d(2,0),0,d(3,0) + d(2,3),0,0,0), (d(6,0),d(6,1), - d(5,0) - 2*d(4,3) + d(3,0) + d(2,3)*a + d(2,0), - d(5,0) - 2*d(4,3) + d(4,0) + d(3,0) + d(2,3)*a - d(2,1)*a + d(2,0), d(3,0),2*d(4,3) - d(3,0) - d(2,0),0)) MATD**2:= mat((0,0,0,0,0,0,0), (0,0,0,0,0,0,0), 2 (d(3,0)*d(2,3), - d(2,3) ,0,0,0,0,0), (0,0,0,0,0,0,0), (d(5,0)*d(2,3) + d(4,3)*d(3,0) + d(3,0)*d(2,0), - 2*d(4,3)*d(2,3) + d(3,0)*d(2,3) + d(3,0)*d(2,1) + d(2,3)*d(2,0),0, d(2,3)*(2*d(3,0) + d(2,3)),0,0,0), (d(3,0)*(d(3,0) + d(2,3)), - d(2,3)*(d(3,0) + d(2,3)),0,0,0,0,0), (2*d(5,0)*d(4,3) - 2*d(5,0)*d(3,0) - 2*d(5,0)*d(2,0) - 2*d(4,3)*d(3,0) 2 - 2*d(4,3)*d(2,0) + 2*d(4,0)*d(3,0) + d(3,0) + d(3,0)*d(2,3)*a 2 - d(3,0)*d(2,1)*a + 2*d(3,0)*d(2,0) + d(2,3)*d(2,0)*a + d(2,0) , 2 d(5,0)*d(2,3) - d(5,0)*d(2,1) - 2*d(4,3) + 3*d(4,3)*d(3,0) + 2*d(4,3)*d(2,3) - 2*d(4,3)*d(2,1) + 3*d(4,3)*d(2,0) + d(4,1)*d(3,0) 2 - d(4,0)*d(2,3) - d(3,0) - d(3,0)*d(2,3) + d(3,0)*d(2,1) 2 - 2*d(3,0)*d(2,0) - d(2,3) *a + 2*d(2,3)*d(2,1)*a - d(2,3)*d(2,0) 2 2 + d(2,1)*d(2,0) - d(2,0) ,d(3,0) , 2 2 - d(5,0)*d(2,3) + 3*d(4,3)*d(3,0) - d(3,0) - d(3,0)*d(2,0) + d(2,3) *a,0, d(3,0)*d(2,3),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), 2 (d(3,0)*d(2,3)*(2*d(3,0) + d(2,3)),d(2,3) *( - 2*d(3,0) - d(2,3)),0,0,0,0,0) , (0,0,0,0,0,0,0), 2 2 2 (d(3,0)*(3*d(4,3)*d(3,0) - d(3,0) + d(2,3) *a),d(5,0)*d(2,3) 2 2 - 4*d(4,3)*d(3,0)*d(2,3) + 2*d(3,0) *d(2,3) + d(3,0) *d(2,1) 3 + 2*d(3,0)*d(2,3)*d(2,0) - d(2,3) *a,0,d(3,0)*d(2,3)*(2*d(3,0) + d(2,3)),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(3,0) *d(2,3)*(2*d(3,0) + d(2,3)),d(3,0)*d(2,3) *( - 2*d(3,0) - d(2,3)),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,d(2,3),0,0,0),(d(3,0), - d( 2,3),0,0,0,0,0),(d(4,0),d(4,1),d(3,0),d(4,3),0,d(2,3),0),(d(5,0), - d(4,3) + d(3 ,0) + d(2,0),0,d(3,0) + d(2,3),0,0,0),(d(6,0),d(6,1), - d(5,0) - 2*d(4,3) + d(3, 0) + d(2,3)*a + d(2,0), - d(5,0) - 2*d(4,3) + d(4,0) + d(3,0) + d(2,3)*a - d(2,1 )*a + d(2,0),d(3,0),2*d(4,3) - d(3,0) - 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,d(2,3),0,0,0), (d(3,0), - d(2,3),0,0,0,0,0), (d(4,0),d(4,1),d(3,0),d(4,3),0,d(2,3),0), (d(5,0), - d(4,3) + d(3,0) + d(2,0),0,d(3,0) + d(2,3),0,0,0), (d(6,0),d(6,1), - d(5,0) - 2*d(4,3) + d(3,0) + d(2,3)*a + d(2,0), - d(5,0) - 2*d(4,3) + d(4,0) + d(3,0) + d(2,3)*a - d(2,1)*a + d(2,0), d(3,0),2*d(4,3) - d(3,0) - 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 0.4 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1}, - x(3) + x(2)}, {{0,2},a*x(6)}, {{0,3},x(5) + x(2)}, {{0,4},0}, {{0,5},x(4)}, {{0,6},0}, {{1,2},x(4)}, {{1,3},x(5)}, {{1,4},x(6)}, {{1,5},0}, {{1,6},0}, {{2,3},0}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{3,4},0}, {{3,5},x(6)}, {{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) + diay(3)}, {{1,3},diay(7)*a}, {{1,4},diay(6) + diay(3)}, {{1,5},0}, {{1,6},diay(5)}, {{1,7},0}, {{2,3},diay(5)}, {{2,4},diay(6)}, {{2,5},diay(7)}, {{2,6},0}, {{2,7},0}, {{3,4},0}, {{3,5},0}, {{3,6},0}, {{3,7},0}, {{4,5},0}, {{4,6},diay(7)}, {{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}$ (L)$ and that for a neq{0}$ 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((i*a, - a,0,0,0,0,0),( - i*a,0,0,0,0,0,0),(0,0, - i*a**2, - a**3, - i*a**3,0 ,0),(0,0,i*a**2,0,0,0,0),(0,0,0, - a**3,0,i*a**4,0),(0,0,0,0, - i*a**3,0,0),(0,0 ,0,a**4,i*a**4,i*a**5,a**5))$ det(isom):= - a**19$ ZZ(1):= - i*(diay(2) - diay(1))*a$ ZZ(2):= - diay(1)*a$ ZZ(3):=i*(diay(4) - diay(3))*a**2$ ZZ(4):= - (diay(5) + diay(3) - diay(7)*a)*a**3$ ZZ(5):=i*(diay(7)*a - diay(6) - diay(3))*a**3$ ZZ(6):=i*(diay(7)*a + diay(5))*a**4$ ZZ(7):=diay(7)*a**5$ listcommutateursdesZZ:=$ {{1,2},zz(3)}$ {{1,3},zz(4)}$ {{1,4},( - 2*i*zz(7)*a + i*zz(7) + zz(6)*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}$ We get the commutation relations of$ g_{7,0.4}$ (L)$ with L:=( - 2*i*a + i)/a$ Et cela pour a:=a$ and that for a neq {0}$ shortformdelta:={1,1,ss,0,ss,1,ss,a}$ delta:= mat((0,0,0,0,0,0),(1,0,1,0,0,0),(-1,0,0,0,0,0),(0,0,0,0,1,0),(0,0,1,0,0,0),(0,a, 0,0,0,0))$