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),xi(3,2),xi(2,2) + 2*xi(1,1),0,0),(xi(5,1),xi(5,2), - xi(3 ,1),0,2*xi(2,2) + xi(1,1),0),(xi(6,1),xi(6,2),xi(5,2) - xi(4,1), - xi(3,1),xi(3, 2),2*(xi(2,2) + xi(1,1))))$ $ [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] adx1 := [ ] [0 0 1 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] [ ] [-1 0 0 0 0 0] adx2 := [ ] [0 0 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] adx3 := [ ] [-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 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [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] adx5 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 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(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(6,1):=0,xi(6,2):=0 delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 xi(4,2) 0 0 0 0] [ ] [xi(5,1) xi(5,2) 0 0 0 0] [ ] [ 0 0 xi(5,2) 0 0 0] We denote this delta by the shortform shortformdelta:={xi(4,2), ss, xi(5,1), xi(5,2)} paramindexeslist:={{4,2},{5,1},{5,2}} a neq {}$ a:=a$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(1,a,0,0,0,0),(0,0,a ,0,0,0))$ $ shortformdelta:={1,ss,1,a}$ on resout l'equation {{0,1},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 {{0,1},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 {{0,1},2} qui est maintenant AA:= - d(2,5)$ Unknown: d(2,5) Unknown: d(2,5) bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},3} qui est maintenant AA:= - (d(3,5) + d(2,0))$ Unknowns: {d(3,5),d(2,0)} Unknowns: {d(3,5),d(2,0)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - d(4,5) - d(3,0) + d(2, 1)$ Unknowns: {d(4,5),d(3,0),d(2,1)} Unknowns: {d(4,5),d(3,0),d(2,1)} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:= - d(3,0) + d(2,1)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,5) + d(2,1)*a + d( 1,1) + d(0,0)$ Unknowns: {d(5,5),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(5,5),d(2,1),d(1,1),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,1)*a + d(1,1) + d(0,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,5) - d(5,0) + d(3, 1)*a$ Unknowns: {d(6,5),d(5,0),d(3,1),a} Unknowns: {d(6,5),d(5,0),d(3,1),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(5,0) + d(3,1)*a$ on resout l'equation {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},3} qui est maintenant AA:= - d(3,4) + d(2,0)*a + d( 1,0)$ Unknowns: {d(3,4),d(2,0),d(1,0),a} Unknowns: {d(3,4),d(2,0),d(1,0),a} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=d(2,0)*a + d(1,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,4) + d(3,0)*a + d( 2,2) - d(2,1)*a + d(0,0)$ Unknowns: {d(4,4),d(3,0),d(2,2),d(2,1),d(0,0),a} Unknowns: {d(4,4),d(3,0),d(2,2),d(2,1),d(0,0),a} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(3,0)*a + d(2,2) - d(2,1)*a + d(0,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,4) - d(3,0) + d(2, 2)*a - d(2,1)*a**2 + d(1,2) - d(1,1)*a$ Unknowns: {d(5,4),d(3,0),d(2,2),d(2,1),d(1,2),d(1,1),a} Unknowns: {d(5,4),d(3,0),d(2,2),d(2,1),d(1,2),d(1,1),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(3,0) + d(2,2)*a - d(2,1)*a**2 + d(1,2) - d(1,1)*a$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,4) + d(5,0)*a - d( 4,0) + d(3,2)*a - d(3,1)*a**2$ Unknowns: {d(6,4),d(5,0),d(4,0),d(3,2),d(3,1),a} Unknowns: {d(6,4),d(5,0),d(4,0),d(3,2),d(3,1),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(5,0)*a - d(4,0) + d(3,2)*a - d(3,1)*a**2$ on resout l'equation {{0,3},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 {{0,3},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 {{0,3},2} 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 {{0,3},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 {{0,3},4} qui est maintenant AA:= - d(4,6)*a + d(2,3) + d( 1,0)$ Unknowns: {d(4,6),d(2,3),d(1,0),a} Unknowns: {d(4,6),d(2,3),d(1,0),a} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=d(4,6)*a - d(1,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6)*a + d(4,6)*a**2 + d(2,0) + d(1,3) - d(1,0)*a$ Unknowns: {d(5,6),d(4,6),d(2,0),d(1,3),d(1,0),a} Unknowns: {d(5,6),d(4,6),d(2,0),d(1,3),d(1,0),a} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=d(5,6)*a - d(4,6)*a**2 - d(1,3) + d(1,0)*a$ on resout l'equation {{0,3},6} qui est maintenant AA:=a*( - d(6,6) + d(3,3) + d (0,0))$ Unknowns: {d(6,6),d(3,3),d(0,0),a} Unknowns: {d(6,6),d(3,3),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(6,6) + d(3,3) + d(0,0)) on resout l'equation {{0,4},6} qui est maintenant AA:=d(5,6)*a**3 + d(5,6)*a - d(4,6)*a**4 - d(4,6)*a**2 - d(1,3)*a**2 - d(1,3) + d(1,0)*a**3 + 2*d(1,0)*a$ Unknowns: {d(5,6),d(4,6),d(1,3),d(1,0),a} Unknowns: {d(5,6),d(4,6),d(1,3),d(1,0),a} pas de selection possible de variable a coefficient numerique dans d(5,6)*a**3 + d(5,6)*a - d(4,6)*a**4 - d(4,6)*a**2 - d(1,3)*a**2 - d(1,3) + d(1,0)*a**3 + 2*d (1,0)*a on resout l'equation {{0,5},6} qui est maintenant AA:= - d(5,6)*a**2 + d(4,6)*a **3 + d(1,3)*a - d(1,0)*a**2 + d(1,0)$ Unknowns: {d(5,6),d(4,6),d(1,3),d(1,0),a} Unknowns: {d(5,6),d(4,6),d(1,3),d(1,0),a} pas de selection possible de variable a coefficient numerique dans - d(5,6)*a** 2 + d(4,6)*a**3 + d(1,3)*a - d(1,0)*a**2 + d(1,0) on resout l'equation {{0,6},4} 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 {{0,6},5} 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},6} 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 {{1,2},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 {{1,2},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 {{1,2},2} qui est maintenant AA:= - d(4,6)*a + d(1,0)$ Unknowns: {d(4,6),d(1,0),a} Unknowns: {d(4,6),d(1,0),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(4,6)*a$ on resout l'equation {{1,2},3} qui est maintenant AA:= - d(3,3) + d(2,2) + d(1, 1)$ Unknowns: {d(3,3),d(2,2),d(1,1)} Unknowns: {d(3,3),d(2,2),d(1,1)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,3) + d(3,2) + d(0, 1)$ Unknowns: {d(4,3),d(3,2),d(0,1)} Unknowns: {d(4,3),d(3,2),d(0,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(3,2) + d(0,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,3) - d(3,1) - d(0, 2) + d(0,1)*a$ Unknowns: {d(5,3),d(3,1),d(0,2),d(0,1),a} Unknowns: {d(5,3),d(3,1),d(0,2),d(0,1),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(3,1) - d(0,2) + d(0,1)*a$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,3) + d(5,2) - d(4, 1)$ Unknowns: {d(6,3),d(5,2),d(4,1)} Unknowns: {d(6,3),d(5,2),d(4,1)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,2) - d(4,1)$ on resout l'equation {{1,3},3} qui est maintenant AA:= - a*(d(5,6)*a + d(4,6))$ Unknowns: {d(5,6),d(4,6),a} Unknowns: {d(5,6),d(4,6),a} pas de selection possible de variable a coefficient numerique dans - a*(d(5,6)* a + d(4,6)) on resout l'equation {{1,3},4} qui est maintenant AA:= - d(3,0)*a + d(2,1)*a + 2*d(1,1) - d(0,0)$ Unknowns: {d(3,0),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(3,0),d(2,1),d(1,1),d(0,0),a} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:=(d(3,0)*a - d(2,1)*a + d(0,0))/2$ on resout l'equation {{1,3},5} qui est maintenant AA:=(d(3,0)*a**2 + 2*d(3,0) - 2*d(2,2)*a + d(2,1)*a**2 + 2*d(2,1) - 2*d(1,2) + d(0,0)*a)/2$ Unknowns: {d(3,0),d(2,2),d(2,1),d(1,2),d(0,0),a} Unknowns: {d(3,0),d(2,2),d(2,1),d(1,2),d(0,0),a} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=(d(3,0)*a**2 + 2*d(3,0) - 2*d(2,2)*a + d(2,1)*a**2 + 2*d (2,1) + d(0,0)*a)/2$ on resout l'equation {{1,3},6} qui est maintenant AA:= - d(5,0)*a + d(4,0) - d( 3,2)*a + d(3,1)*a**2 - d(3,1) - d(0,2) + 2*d(0,1)*a$ Unknowns: {d(5,0),d(4,0),d(3,2),d(3,1),d(0,2),d(0,1),a} Unknowns: {d(5,0),d(4,0),d(3,2),d(3,1),d(0,2),d(0,1),a} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(5,0)*a + d(3,2)*a - d(3,1)*a**2 + d(3,1) + d(0,2) - 2* d(0,1)*a$ on resout l'equation {{1,4},4} qui est maintenant AA:=a*(d(5,6)*a + d(4,6))$ Unknowns: {d(5,6),d(4,6),a} Unknowns: {d(5,6),d(4,6),a} pas de selection possible de variable a coefficient numerique dans a*(d(5,6)*a + d(4,6)) on resout l'equation {{1,4},6} qui est maintenant AA:=2*d(2,1)$ Unknown: d(2,1) Unknown: d(2,1) bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,5},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 {{1,5},3} qui est maintenant AA:= - d(3,6)$ Unknown: d(3,6) Unknown: d(3,6) bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,5},4} qui est maintenant AA:= - (d(5,6)*a + d(4,6))$ Unknowns: {d(5,6),d(4,6),a} Unknowns: {d(5,6),d(4,6),a} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(5,6)*a$ on resout l'equation {{1,5},5} 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 {{1,5},6} qui est maintenant AA:= - d(6,6) + d(3,0)*a + 2* d(0,0)$ Unknowns: {d(6,6),d(3,0),d(0,0),a} Unknowns: {d(6,6),d(3,0),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(3,0)*a + 2*d(0,0)$ on resout l'equation {{2,3},4} qui est maintenant AA:=(d(3,0)*a**2 + 4*d(3,0) - 2*d(2,2)*a + d(0,0)*a)/2$ Unknowns: {d(3,0),d(2,2),d(0,0),a} Unknowns: {d(3,0),d(2,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (d(3,0)*a**2 + 4*d(3,0) - 2*d(2,2)*a + d(0,0)*a)/2 on resout l'equation {{2,3},5} qui est maintenant AA:=2*d(2,2) - d(0,0)$ Unknowns: {d(2,2),d(0,0)} Unknowns: {d(2,2),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=d(0,0)/2$ on resout l'equation {{2,3},6} qui est maintenant AA:=d(5,0) + d(3,2) - d(3,1)* a + d(0,2)*a + d(0,1)$ Unknowns: {d(5,0),d(3,2),d(3,1),d(0,2),d(0,1),a} Unknowns: {d(5,0),d(3,2),d(3,1),d(0,2),d(0,1),a} bonne inconnue W:=d(5,0)$ sa valeur doit etre WW:= - d(3,2) + d(3,1)*a - d(0,2)*a - d(0,1)$ on resout l'equation {{2,5},6} qui est maintenant AA:=(d(3,0)*a**2)/2$ Unknowns: {d(3,0),a} Unknowns: {d(3,0),a} pas de selection possible de variable a coefficient numerique dans (d(3,0)*a**2) /2 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},( - d(3,0)*a**2)/2}, {{{0,4},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},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},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},3},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},(d(3,0)*(a**2 + 4))/2}, {{{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},(d(3,0)*a**2)/2}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},4},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},4},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},6},0}, {{{4,6},6},0}, {{{5,6},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},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},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},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},3},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},4},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},4},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},6},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ a neq {0}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(0,1),d(0,2),0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,0,d(0,0)/2,0,0,0,0) ,(0,d(3,1),d(3,2),d(0,0),0,0,0),( - (d(0,2)*a**2 - d(0,2) + 3*d(0,1)*a - d(3,1)) ,d(4,1),d(4,2),d(3,2) + d(0,1),(3*d(0,0))/2,0,0),( - (d(0,2)*a + d(0,1) - d(3,1) *a + d(3,2)),d(5,1),d(5,2), - (d(0,2) - d(0,1)*a + d(3,1)),0,(3*d(0,0))/2,0),(d( 6,0),d(6,1),d(6,2),d(5,2) - d(4,1), - (d(0,2) - 2*d(0,1)*a + d(3,1)),d(0,2)*a + d(0,1) + d(3,2),2*d(0,0)))$ $ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] [ ] [1 a 0 0 0 0] [ ] [0 0 a 0 0 0] pour shortformdelta:={1,ss,1,a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,1), d(0,0), a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,1), d(0,0), a} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,1), d(0,0)}$ dim Der(gtildedelta):=12$ t1:=D(0,0):= [1 0 0 0 0 0 0] [ ] [ 1 ] [0 --- 0 0 0 0 0] [ 2 ] [ ] [ 1 ] [0 0 --- 0 0 0 0] [ 2 ] [ ] [0 0 0 1 0 0 0] [ ] [ 3 ] [0 0 0 0 --- 0 0] [ 2 ] [ ] [ 3 ] [0 0 0 0 0 --- 0] [ 2 ] [ ] [0 0 0 0 0 0 2] Unknown: d(0,0) Unknown: d(0,0) commutant de t1 dans der(gtildedelta): [d(0,0) 0 0 0 0 0 0 ] [ ] [ d(0,0) ] [ 0 -------- 0 0 0 0 0 ] [ 2 ] [ ] [ d(0,0) ] [ 0 0 -------- 0 0 0 0 ] [ 2 ] [ ] [ 0 0 0 d(0,0) 0 0 0 ] [ ] [ 3*d(0,0) ] [ 0 0 0 0 ---------- 0 0 ] [ 2 ] [ ] [ 3*d(0,0) ] [ 0 0 0 0 0 ---------- 0 ] [ 2 ] [ ] [ 0 0 0 0 0 0 2*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] [ ] [ 1 ] [0 --- 0 0 0 0 0] [ 2 ] [ ] [ 1 ] [0 0 --- 0 0 0 0] [ 2 ] [ ] [0 0 0 1 0 0 0] [ ] [ 3 ] [0 0 0 0 --- 0 0] [ 2 ] [ ] [ 3 ] [0 0 0 0 0 --- 0] [ 2 ] [ ] [0 0 0 0 0 0 2] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((d(0,0),d(0,1),d(0,2),0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,0,d(0,0)/2,0,0,0,0) ,(0,d(3,1),d(3,2),d(0,0),0,0,0),( - (d(0,2)*a**2 - d(0,2) + 3*d(0,1)*a - d(3,1)) ,d(4,1),d(4,2),d(3,2) + d(0,1),(3*d(0,0))/2,0,0),( - (d(0,2)*a + d(0,1) - d(3,1) *a + d(3,2)),d(5,1),d(5,2), - (d(0,2) - d(0,1)*a + d(3,1)),0,(3*d(0,0))/2,0),(d( 6,0),d(6,1),d(6,2),d(5,2) - d(4,1), - (d(0,2) - 2*d(0,1)*a + d(3,1)),d(0,2)*a + d(0,1) + d(3,2),2*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),d(0,1),d(0,2),0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 d(0,0) (0,0,--------,0,0,0,0), 2 (0,d(3,1),d(3,2),d(0,0),0,0,0), 2 ( - (d(0,2)*a - d(0,2) + 3*d(0,1)*a - d(3,1)),d(4,1),d(4,2),d(3,2) + d(0,1) 3*d(0,0) ,----------,0,0), 2 ( - (d(0,2)*a + d(0,1) - d(3,1)*a + d(3,2)),d(5,1),d(5,2), 3*d(0,0) - (d(0,2) - d(0,1)*a + d(3,1)),0,----------,0), 2 (d(6,0),d(6,1),d(6,2),d(5,2) - d(4,1), - (d(0,2) - 2*d(0,1)*a + d(3,1)), d(0,2)*a + d(0,1) + d(3,2),2*d(0,0))) on voit apparaitre les poids sur la diagonale r(1) := d(0,0) d(0,0) r(2) := -------- 2 d(0,0) r(3) := -------- 2 r(4) := d(0,0) 3*d(0,0) r(5) := ---------- 2 3*d(0,0) r(6) := ---------- 2 r(7) := 2*d(0,0) r(1) := 2*gamma1 r(2) := gamma1 r(3) := gamma1 r(4) := 2*gamma1 r(5) := 3*gamma1 r(6) := 3*gamma1 r(7) := 4*gamma1 Le systeme de poids est le systeme 1.2 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(5)}, {{0,2},x(5)*a + x(4)}, {{0,3},a*x(6)}, {{0,4},0}, {{0,5},0}, {{0,6},0}, {{1,2},x(3)}, {{1,3},x(4)}, {{1,4},0}, {{1,5},x(6)}, {{1,6},0}, {{2,3},x(5)}, {{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(6)}, {{1,3},diay(6)*a + diay(5)}, {{1,4},diay(7)*a}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},diay(4)}, {{2,4},diay(5)}, {{2,5},0}, {{2,6},diay(7)}, {{2,7},0}, {{3,4},diay(6)}, {{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_{7,1.2}$ (iL)$ and that for a neq{2*i,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((0,0,(4*i)/(a - 2*i),0,0,0,0),(1,i,0,0,0,0,0),(i,1,0,0,0,0,0),(0,0,(2*i*a)/( a - 2*i),2,0,0,0),(0,0,0,0,2,2*i,0),(0,0,0,0,2*i,2,0),(0,0,0,0,0,0,4*i))$ $ det(isom):= ( - 512)/(a - 2*i)$ ZZ(1):=i*diay(3) + diay(2)$ ZZ(2):=diay(3) + i*diay(2)$ ZZ(3):=(2*i*(diay(4)*a + 2*diay(1)))/(a - 2*i)$ ZZ(4):=2*diay(4)$ ZZ(5):=2*(i*diay(6) + diay(5))$ ZZ(6):=2*(diay(6) + i*diay(5))$ ZZ(7):=4*i*diay(7)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(6)}$ {{1,4},zz(5)}$ {{1,5},zz(7)}$ {{1,6},0}$ {{1,7},0}$ {{2,3}, ( - zz(5)*a**2 - 4*i*zz(5)*a + 4*zz(5))/(a**2 + 4)}$ {{2,4},zz(6)}$ {{2,5},0}$ {{2,6},zz(7)}$ {{2,7},0}$ {{3,4},(2*zz(7)*a**2 + 4*i*zz(7)*a)/(a**2 + 4)}$ {{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_{7,1.2}$ (iL)$ with L:=( - (a + 2*i))/(a - 2*i)$ since we note that :$ ( - a**2 - 4*i*a + 4)/(a**2 + 4) -L:=0$ (2*a**2 + 4*i*a)/(a**2 + 4) -(1-L):=0$ Et cela pour a:=a$ and that for a neq {2*i,0}$ shortformdelta:={1,ss,1,a}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(1,a,0,0,0,0),(0,0,a ,0,0,0))$ $