generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),3*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(2,1),xi(4,2 ),xi(2,2) + 2*xi(1,1),0),(xi(6,1),xi(6,2),xi(6,3),xi(5,2) - xi(3,1),xi(4,2),xi(2 ,2) + 3*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 1 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 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] [ ] [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 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] [ ] [-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(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,1):=0,xi(6,2):=0 delta:= [ 0 0 0 0 0 0] [ ] [xi(2,1) 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 xi(5,2) - xi(2,1) 0 0 0] [ ] [ 0 0 xi(6,3) xi(5,2) - xi(3,1) 0 0] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,2), ss, xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,2},{6,3}} a neq {}$ a:=a$ delta:= mat((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,a,-1,0,0,0),(0,0, 1,a,0,0))$ $ shortformdelta:={1,ss,0,0,ss,a,ss,1}$ on resout l'equation {{0,1},0} qui est maintenant AA:= - d(0,2)$ Unknown: d(0,2) Unknown: d(0,2) bonne inconnue W:=d(0,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},1} qui est maintenant AA:= - d(1,2)$ Unknown: d(1,2) Unknown: d(1,2) bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},2} qui est maintenant AA:= - d(2,2) + d(1,1) + d(0, 0)$ Unknowns: {d(2,2),d(1,1),d(0,0)} Unknowns: {d(2,2),d(1,1),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=d(1,1) + d(0,0)$ on resout l'equation {{0,1},3} qui est maintenant AA:= - d(3,2)$ Unknown: d(3,2) Unknown: d(3,2) bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,2) + d(2,0))$ Unknowns: {d(4,2),d(2,0)} Unknowns: {d(4,2),d(2,0)} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,2) - d(4,0) - d(3, 1) + d(2,1)*a$ Unknowns: {d(5,2),d(4,0),d(3,1),d(2,1),a} Unknowns: {d(5,2),d(4,0),d(3,1),d(2,1),a} bonne inconnue W:=d(5,2)$ sa valeur doit etre WW:= - d(4,0) - d(3,1) + d(2,1)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,2) - d(5,0) + d(4, 1)*a + d(3,1)$ Unknowns: {d(6,2),d(5,0),d(4,1),d(3,1),a} Unknowns: {d(6,2),d(5,0),d(4,1),d(3,1),a} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:= - d(5,0) + d(4,1)*a + d(3,1)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - d(0,5)*a$ Unknowns: {d(0,5),a} Unknowns: {d(0,5),a} pas de selection possible de variable a coefficient numerique dans - d(0,5)*a on resout l'equation {{0,2},1} qui est maintenant AA:= - d(1,5)*a$ Unknowns: {d(1,5),a} Unknowns: {d(1,5),a} pas de selection possible de variable a coefficient numerique dans - d(1,5)*a on resout l'equation {{0,2},2} qui est maintenant AA:= - d(2,5)*a$ Unknowns: {d(2,5),a} Unknowns: {d(2,5),a} pas de selection possible de variable a coefficient numerique dans - d(2,5)*a on resout l'equation {{0,2},3} 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 {{0,2},4} qui est maintenant AA:= - d(4,5)*a + d(1,0)$ Unknowns: {d(4,5),d(1,0),a} Unknowns: {d(4,5),d(1,0),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(4,5)*a$ on resout l'equation {{0,2},5} qui est maintenant AA:=a*( - d(5,5) + d(1,1) + 2 *d(0,0))$ Unknowns: {d(5,5),d(1,1),d(0,0),a} Unknowns: {d(5,5),d(1,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(5,5) + d(1,1) + 2*d(0,0)) on resout l'equation {{0,2},6} qui est maintenant AA:= - (d(6,5)*a + d(3,0) + d (2,0)*a)$ Unknowns: {d(6,5),d(3,0),d(2,0),a} Unknowns: {d(6,5),d(3,0),d(2,0),a} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:= - a*(d(6,5) + d(2,0))$ on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,6) + d(0,5)$ Unknowns: {d(0,6),d(0,5)} Unknowns: {d(0,6),d(0,5)} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:=d(0,5)$ on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,6) + d(1,5)$ Unknowns: {d(1,6),d(1,5)} Unknowns: {d(1,6),d(1,5)} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:=d(1,5)$ on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,6) + d(2,5) + d(1, 3)$ Unknowns: {d(2,6),d(2,5),d(1,3)} Unknowns: {d(2,6),d(2,5),d(1,3)} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:=d(2,5) + d(1,3)$ on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,6) + d(3,5)$ Unknowns: {d(3,6),d(3,5)} Unknowns: {d(3,6),d(3,5)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(3,5)$ on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,6) + d(4,5)$ Unknowns: {d(4,6),d(4,5)} Unknowns: {d(4,6),d(4,5)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=d(4,5)$ on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6) + d(5,5) - d(3, 3) + d(2,3)*a - d(0,0)$ Unknowns: {d(5,6),d(5,5),d(3,3),d(2,3),d(0,0),a} Unknowns: {d(5,6),d(5,5),d(3,3),d(2,3),d(0,0),a} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=d(5,5) - d(3,3) + d(2,3)*a - d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6) + d(6,5) + d(4, 3)*a + d(3,3) + d(2,0) + d(0,0)$ Unknowns: {d(6,6),d(6,5),d(4,3),d(3,3),d(2,0),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(4,3),d(3,3),d(2,0),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(6,5) + d(4,3)*a + d(3,3) + d(2,0) + d(0,0)$ on resout l'equation {{0,4},0} qui est maintenant AA:= - d(0,5)*a$ Unknowns: {d(0,5),a} Unknowns: {d(0,5),a} pas de selection possible de variable a coefficient numerique dans - d(0,5)*a on resout l'equation {{0,4},1} qui est maintenant AA:= - d(1,5)*a$ Unknowns: {d(1,5),a} Unknowns: {d(1,5),a} pas de selection possible de variable a coefficient numerique dans - d(1,5)*a on resout l'equation {{0,4},2} qui est maintenant AA:= - d(2,5)*a + d(1,4) - d( 1,3)*a$ Unknowns: {d(2,5),d(1,4),d(1,3),a} Unknowns: {d(2,5),d(1,4),d(1,3),a} bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:=a*(d(2,5) + d(1,3))$ on resout l'equation {{0,4},3} 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 {{0,4},4} qui est maintenant AA:= - d(4,5)*a$ Unknowns: {d(4,5),a} Unknowns: {d(4,5),a} pas de selection possible de variable a coefficient numerique dans - d(4,5)*a on resout l'equation {{0,4},5} qui est maintenant AA:= - d(5,5)*a + d(4,5)*a - d(3,4) + d(3,3)*a + d(2,4)*a - d(2,3)*a**2 + d(0,0)*a$ Unknowns: {d(5,5),d(4,5),d(3,4),d(3,3),d(2,4),d(2,3),d(0,0),a} Unknowns: {d(5,5),d(4,5),d(3,4),d(3,3),d(2,4),d(2,3),d(0,0),a} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=a*( - d(5,5) + d(4,5) + d(3,3) + d(2,4) - d(2,3)*a + d(0 ,0))$ on resout l'equation {{0,4},6} qui est maintenant AA:=a*( - d(6,5) - d(5,5) + d (4,5) + d(4,4) - d(4,3)*a + d(2,4) - d(2,3)*a - d(2,0) + d(0,0))$ Unknowns: {d(6,5),d(5,5),d(4,5),d(4,4),d(4,3),d(2,4),d(2,3),d(2,0),d(0,0),a} Unknowns: {d(6,5),d(5,5),d(4,5),d(4,4),d(4,3),d(2,4),d(2,3),d(2,0),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(6,5) - d(5,5) + d(4,5) + d(4,4) - d(4,3)*a + d(2,4) - d(2,3)*a - d(2,0) + d(0,0)) on resout l'equation {{0,5},2} qui est maintenant AA:=d(1,5)$ Unknown: d(1,5) Unknown: d(1,5) bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},5} qui est maintenant AA:= - d(3,5) + d(2,5)*a$ Unknowns: {d(3,5),d(2,5),a} Unknowns: {d(3,5),d(2,5),a} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:=d(2,5)*a$ on resout l'equation {{0,5},6} qui est maintenant AA:=a*(2*d(4,5) + d(2,5))$ Unknowns: {d(4,5),d(2,5),a} Unknowns: {d(4,5),d(2,5),a} pas de selection possible de variable a coefficient numerique dans a*(2*d(4,5) + d(2,5)) on resout l'equation {{0,6},5} qui est maintenant AA:=d(1,3)*a$ Unknowns: {d(1,3),a} Unknowns: {d(1,3),a} pas de selection possible de variable a coefficient numerique dans d(1,3)*a on resout l'equation {{0,6},6} qui est maintenant AA:=a*(d(4,5) + d(2,5))$ Unknowns: {d(4,5),d(2,5),a} Unknowns: {d(4,5),d(2,5),a} pas de selection possible de variable a coefficient numerique dans a*(d(4,5) + d (2,5)) 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:= - a*(d(2,5) + d(1,3))$ Unknowns: {d(2,5),d(1,3),a} Unknowns: {d(2,5),d(1,3),a} pas de selection possible de variable a coefficient numerique dans - a*(d(2,5) + d(1,3)) 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:=a*(d(5,5) - d(4,5) - d(3, 3) + d(2,3)*a - d(0,0))$ Unknowns: {d(5,5),d(4,5),d(3,3),d(2,3),d(0,0),a} Unknowns: {d(5,5),d(4,5),d(3,3),d(2,3),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*(d(5,5) - d (4,5) - d(3,3) + d(2,3)*a - d(0,0)) on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,4) + 2*d(1,1) + d( 0,0)$ Unknowns: {d(4,4),d(1,1),d(0,0)} Unknowns: {d(4,4),d(1,1),d(0,0)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=2*d(1,1) + d(0,0)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) - d(2,0) + d(0, 1)*a$ Unknowns: {d(5,4),d(2,0),d(0,1),a} Unknowns: {d(5,4),d(2,0),d(0,1),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(2,0) + d(0,1)*a$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,4) - d(4,0) - 2*d( 3,1) + d(2,1)*a$ Unknowns: {d(6,4),d(4,0),d(3,1),d(2,1),a} Unknowns: {d(6,4),d(4,0),d(3,1),d(2,1),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(4,0) - 2*d(3,1) + d(2,1)*a$ on resout l'equation {{1,3},2} qui est maintenant AA:= - d(0,3)$ Unknown: d(0,3) Unknown: d(0,3) bonne inconnue W:=d(0,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,3},4} qui est maintenant AA:=d(2,3)$ Unknown: d(2,3) Unknown: d(2,3) bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,3},5} qui est maintenant AA:=d(4,3) - d(0,1)$ Unknowns: {d(4,3),d(0,1)} Unknowns: {d(4,3),d(0,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,3},6} qui est maintenant AA:=d(5,3) + d(2,1) + d(0,1)$ Unknowns: {d(5,3),d(2,1),d(0,1)} Unknowns: {d(5,3),d(2,1),d(0,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - (d(2,1) + d(0,1))$ on resout l'equation {{1,4},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},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 {{1,4},4} qui est maintenant AA:= - d(4,5)$ Unknown: d(4,5) Unknown: d(4,5) bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,4},5} qui est maintenant AA:= - d(5,5) + 3*d(1,1) + d( 0,0)$ Unknowns: {d(5,5),d(1,1),d(0,0)} Unknowns: {d(5,5),d(1,1),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=3*d(1,1) + d(0,0)$ on resout l'equation {{1,4},6} qui est maintenant AA:= - d(6,5) - d(2,0) + 2*d( 0,1)*a$ Unknowns: {d(6,5),d(2,0),d(0,1),a} Unknowns: {d(6,5),d(2,0),d(0,1),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(2,0) + 2*d(0,1)*a$ on resout l'equation {{1,5},2} qui est maintenant AA:= - d(1,3)$ Unknown: d(1,3) Unknown: d(1,3) bonne inconnue W:=d(1,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,5},5} qui est maintenant AA:=d(3,3) - 3*d(1,1)$ Unknowns: {d(3,3),d(1,1)} Unknowns: {d(3,3),d(1,1)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=3*d(1,1)$ on resout l'equation {{1,5},6} qui est maintenant AA:=d(1,1) - 3*d(0,1)*a$ Unknowns: {d(1,1),d(0,1),a} Unknowns: {d(1,1),d(0,1),a} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:=3*d(0,1)*a$ 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},a*( - 6*d(0,1)*a + d(0,0))}, {{{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},a*( - 6*d(0,1)*a + d(0,0))}, {{{0,5},2},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},2},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},2},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},0},0}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},0},0}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},2},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},0},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,3},6},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,4},6},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},5},0}, {{{4,5},6},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ Il y a une phase 2$ 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},2},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},2},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},2},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},0},0}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},0},0}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},2},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},0},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,3},6},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,4},6},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},5},0}, {{{4,5},6},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ a neq {0}$ derivation generique de gtildedelta:$ MATD:= mat((6*d(0,1)*a,d(0,1),0,0,0,0,0),(0,3*d(0,1)*a,0,0,0,0,0),(d(2,0),d(2,1),9*d(0, 1)*a,0,0,0,0),( - 2*d(0,1)*a**2,d(3,1),0,9*d(0,1)*a,0,0,0),(d(4,0),d(4,1), - d(2 ,0),d(0,1),12*d(0,1)*a,0,0),(d(5,0),d(5,1), - (d(3,1) - d(2,1)*a + d(4,0)), - (d (2,1) + d(0,1)), - (d(2,0) - d(0,1)*a),15*d(0,1)*a,0),(d(6,0),d(6,1),d(4,1)*a + d(3,1) - d(5,0),d(6,3), - (2*d(3,1) - d(2,1)*a + d(4,0)), - (d(2,0) - 2*d(0,1)*a ),18*d(0,1)*a))$ $ pour delta:= [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 a -1 0 0 0] [ ] [0 0 1 a 0 0] pour shortformdelta:={1,ss,0,0,ss,a,ss,1} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(2,1), d(2,0), d(0,1), a} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(2,1), d(2,0), d(0,1), a} listeparametresMATD{d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(2,1), d(2,0), d(0,1)}$ dim Der(gtildedelta):=11$ t1:=D(0,1):= [ 6*a 1 0 0 0 0 0 ] [ ] [ 0 3*a 0 0 0 0 0 ] [ ] [ 0 0 9*a 0 0 0 0 ] [ ] [ 2 ] [ - 2*a 0 0 9*a 0 0 0 ] [ ] [ 0 0 0 1 12*a 0 0 ] [ ] [ 0 0 0 -1 a 15*a 0 ] [ ] [ 0 0 0 0 0 2*a 18*a] MATD:= mat((6*d(0,1)*a,d(0,1),0,0,0,0,0), (0,3*d(0,1)*a,0,0,0,0,0), (d(2,0),d(2,1),9*d(0,1)*a,0,0,0,0), 2 ( - 2*d(0,1)*a ,d(3,1),0,9*d(0,1)*a,0,0,0), (d(4,0),d(4,1), - d(2,0),d(0,1),12*d(0,1)*a,0,0), (d(5,0),d(5,1), - (d(3,1) - d(2,1)*a + d(4,0)), - (d(2,1) + d(0,1)), - (d(2,0) - d(0,1)*a),15*d(0,1)*a,0), (d(6,0),d(6,1),d(4,1)*a + d(3,1) - d(5,0),d(6,3), - (2*d(3,1) - d(2,1)*a + d(4,0)), - (d(2,0) - 2*d(0,1)*a),18*d(0,1)*a)) {{ - (3*a - x), 1, [ - 243*arbcomplex(283) ] [ ] [ 729*arbcomplex(283)*a ] [ ] [ 0 ] [ ] [ - 81*arbcomplex(283)*a] [ ] [ 9*arbcomplex(283) ] [ ] [ - 15*arbcomplex(283) ] [-----------------------] [ 2 ] [ ] [ arbcomplex(283) ] }, { - (18*a - x), 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(284)] }, { - (15*a - x), 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - 3*arbcomplex(285) ] [----------------------] [ 2 ] [ ] [ arbcomplex(285) ] }, { - (12*a - x), 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 9*arbcomplex(286) ] [ ] [ - 3*arbcomplex(286)] [ ] [ arbcomplex(286) ] }, { - (9*a - x), 2, [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(287) ] [ ] [ - 81*arbcomplex(288)*a ] [-------------------------] [ 4 ] [ ] [ 27*arbcomplex(288) ] [ -------------------- ] [ 4 ] [ ] [ - 9*arbcomplex(288) ] [ ---------------------- ] [ 2 ] [ ] [ arbcomplex(288) ] }, { - (6*a - x), 1, [ - 486*arbcomplex(289) ] [ ------------------------ ] [ 7 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - 324*arbcomplex(289)*a ] [--------------------------] [ 7 ] [ ] [ 54*arbcomplex(289) ] [ -------------------- ] [ 7 ] [ ] [ - 6*arbcomplex(289) ] [ ] [ arbcomplex(289) ] }} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(2,1), d(2,0), d(0,1), a} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(2,1), d(2,0), d(0,1), a} commutant de t1 dans der(gtildedelta): mat((6*d(0,1)*a,d(0,1),0,0,0,0,0), (0,3*d(0,1)*a,0,0,0,0,0), (d(2,0),d(2,1),9*d(0,1)*a,0,0,0,0), 2 ( - 2*d(0,1)*a ,d(3,1),0,9*d(0,1)*a,0,0,0), (d(4,0),d(4,1), - d(2,0),d(0,1),12*d(0,1)*a,0,0), - (6*d(3,1) - d(2,0) + 6*d(4,0)) (d(5,0),d(5,1),-----------------------------------, - (d(2,1) + d(0,1)), 6 - (d(2,0) - d(0,1)*a),15*d(0,1)*a,0), d(4,0) + 8*d(3,1) - 9*d(5,0) (d(6,0),d(6,1),------------------------------,d(6,3), 9 - (12*d(3,1) - d(2,0) + 6*d(4,0)) ------------------------------------, - (d(2,0) - 2*d(0,1)*a),18*d(0,1)*a)) 6 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 ] [ 1 ------ 0 0 0 0 0] [ 3*a ] [ ] [ 0 1 0 0 0 0 0] [ ] [ 0 0 1 0 0 0 0] [ ] [ 2*a - 1 ] [----- ------ 0 1 0 0 0] [ 3 9 ] [ ] [ - 1 1 - 1 ] [------ ------ 0 ------ 1 0 0] [ 9 81*a 3*a ] [ ] [ 7 - 5 2 - 1 ] [ ---- ------- 0 ----- ------ 1 0] [ 81 486*a 9*a 3 ] [ ] [ - 7 1 - 4 1 - 2 ] [------ ------- 0 ------ --- ------ 1] [ 486 729*a 81*a 9 3 ] P**(-1)*t1*P:= [6*a 0 0 0 0 0 0 ] [ ] [ 0 3*a 0 0 0 0 0 ] [ ] [ 0 0 9*a 0 0 0 0 ] [ ] [ 0 0 0 9*a 0 0 0 ] [ ] [ 0 0 0 0 12*a 0 0 ] [ ] [ 0 0 0 0 0 15*a 0 ] [ ] [ 0 0 0 0 0 0 18*a] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((6*d(0,1)*a,0,0,0,0,0,0),(0,3*d(0,1)*a,0,0,0,0,0),(d(2,0),( - d(2,0))/(6*a), 9*d(0,1)*a,0,0,0,0),(0,d(3,1),0,9*d(0,1)*a,0,0,0),(d(4,0),( - 2*(d(4,0) - d(3,1) ))/(9*a), - d(2,0),0,12*d(0,1)*a,0,0),((3*d(5,0) + d(4,0))/3,( - (2*(9*d(3,1) - d(2,0)) + 27*d(4,0) + 81*d(5,0)))/(324*a),( - (6*d(4,0) + 6*d(3,1) + d(2,0)))/6, d(2,0)/(6*a), - d(2,0),15*d(0,1)*a,0),((162*d(6,0) + 108*d(5,0) + 24*d(4,0) + 12 *d(3,1) + d(2,0))/162,( - (5*(12*d(3,1) + d(2,0)) + 72*d(4,0) + 432*d(5,0) + 648 *d(6,0)))/(2430*a),( - (9*d(5,0) + 5*d(4,0) - 2*d(3,1)))/9,(2*(d(4,0) + 2*d(3,1) ))/(9*a),( - (6*d(4,0) + 12*d(3,1) + d(2,0)))/6, - d(2,0),18*d(0,1)*a))$ $ PP:= [ - 1 ] [ 1 ------ 0 0 0 0 0] [ 3*a ] [ ] [ 0 1 0 0 0 0 0] [ ] [ 0 0 1 0 0 0 0] [ ] [ 2*a - 1 ] [----- ------ 0 1 0 0 0] [ 3 9 ] [ ] [ - 1 1 - 1 ] [------ ------ 0 ------ 1 0 0] [ 9 81*a 3*a ] [ ] [ 7 - 5 2 - 1 ] [ ---- ------- 0 ----- ------ 1 0] [ 81 486*a 9*a 3 ] [ ] [ - 7 1 - 4 1 - 2 ] [------ ------- 0 ------ --- ------ 1] [ 486 729*a 81*a 9 3 ] avec PP:=P*Q:= [ - 1 ] [ 1 ------ 0 0 0 0 0] [ 3*a ] [ ] [ 0 1 0 0 0 0 0] [ ] [ 0 0 1 0 0 0 0] [ ] [ 2*a - 1 ] [----- ------ 0 1 0 0 0] [ 3 9 ] [ ] [ - 1 1 - 1 ] [------ ------ 0 ------ 1 0 0] [ 9 81*a 3*a ] [ ] [ 7 - 5 2 - 1 ] [ ---- ------- 0 ----- ------ 1 0] [ 81 486*a 9*a 3 ] [ ] [ - 7 1 - 4 1 - 2 ] [------ ------- 0 ------ --- ------ 1] [ 486 729*a 81*a 9 3 ] MATDDIAGONALISE:= mat((6*d(0,1)*a,0,0,0,0,0,0), (0,3*d(0,1)*a,0,0,0,0,0), - d(2,0) (d(2,0),-----------,9*d(0,1)*a,0,0,0,0), 6*a (0,d(3,1),0,9*d(0,1)*a,0,0,0), - 2*(d(4,0) - d(3,1)) (d(4,0),------------------------, - d(2,0),0,12*d(0,1)*a,0,0), 9*a 3*d(5,0) + d(4,0) - (2*(9*d(3,1) - d(2,0)) + 27*d(4,0) + 81*d(5,0)) (-------------------,----------------------------------------------------, 3 324*a - (6*d(4,0) + 6*d(3,1) + d(2,0)) d(2,0) -----------------------------------,--------, - d(2,0),15*d(0,1)*a,0), 6 6*a 162*d(6,0) + 108*d(5,0) + 24*d(4,0) + 12*d(3,1) + d(2,0) (----------------------------------------------------------, 162 - (5*(12*d(3,1) + d(2,0)) + 72*d(4,0) + 432*d(5,0) + 648*d(6,0)) -------------------------------------------------------------------, 2430*a - (9*d(5,0) + 5*d(4,0) - 2*d(3,1)) 2*(d(4,0) + 2*d(3,1)) -------------------------------------,-----------------------, 9 9*a - (6*d(4,0) + 12*d(3,1) + d(2,0)) ------------------------------------, - d(2,0),18*d(0,1)*a)) 6 on voit apparaitre les poids sur la diagonale r(1) := 6*d(0,1)*a r(2) := 3*d(0,1)*a r(3) := 9*d(0,1)*a r(4) := 9*d(0,1)*a r(5) := 12*d(0,1)*a r(6) := 15*d(0,1)*a r(7) := 18*d(0,1)*a r(1) := 2*gamma1 r(2) := gamma1 r(3) := 3*gamma1 r(4) := 3*gamma1 r(5) := 4*gamma1 r(6) := 5*gamma1 r(7) := 6*gamma1 Le systeme de poids est le systeme 1.11 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(2)}, {{0,2},a*x(5)}, {{0,3},x(6) - x(5)}, {{0,4},a*x(6)}, {{0,5},0}, {{0,6},0}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},x(6)}, {{1,6},0}, {{2,3},x(6)}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{3,4},0}, {{3,5},0}, {{3,6},0}, {{4,5},0}, {{4,6},0}, {{5,6},0}} - 7*x(6) + 42*x(5) - 54*x(4) + 324*x(3)*a + 486*x(0) diaY(1):=------------------------------------------------------- 486 2*x(6) - 15*x(5) + 18*x(4) - 162*x(3)*a + 1458*x(1)*a - 486*x(0) diaY(2):=------------------------------------------------------------------ 1458*a diaY(3):=x(2) - 4*x(6) + 18*x(5) - 27*x(4) + 81*x(3)*a diaY(4):=------------------------------------------- 81*a x(6) - 3*x(5) + 9*x(4) diaY(5):=------------------------ 9 - 2*x(6) + 3*x(5) diaY(6):=-------------------- 3 diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},diay(3)}, {{1,3},diay(6)*a}, {{1,4}, - diay(6)}, {{1,5},diay(7)*a}, {{1,6},0}, {{1,7},0}, {{2,3},diay(5)}, {{2,4},0}, {{2,5},diay(6)}, {{2,6},diay(7)}, {{2,7},0}, {{3,4},diay(7)}, {{3,5},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_{7,1.11}$ 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 :$ This isomorphism computed by calculisom6_11III.red$ mat((0,1/a,0,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,( - 1)/a,0,0,0),(0,0,1,0,0,0,0),(0, 0,0,0,( - 1)/a,0,0),(0,0,0,0,0,( - 1)/a,0),(0,0,0,0,0,0,( - 1)/a))$ $ det(isom):= 1/a**5$ ZZ(1):=diay(2)$ ZZ(2):=diay(1)/a$ ZZ(3):=diay(4)$ ZZ(4):=( - diay(3))/a$ ZZ(5):=( - diay(5))/a$ ZZ(6):=( - diay(6))/a$ ZZ(7):=( - diay(7))/a$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},0}$ {{1,4},zz(5)}$ {{1,5},zz(6)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(6)}$ {{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_{7,1.11}$ Et cela pour a:=a$ and that for a neq {0}$ shortformdelta:={1,ss,0,0,ss,a,ss,1}$ delta:= mat((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,a,-1,0,0,0),(0,0, 1,a,0,0))$ $