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:=a$ b:=0$ delta:= mat((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,a,0,0,0,0),(0,0,0 ,a - 1,0,0))$ $ shortformdelta:={0,ss,1,0,ss,a,ss,0}$ 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(1,1) + d(0, 0)$ Unknowns: {d(3,3),d(1,1),d(0,0)} Unknowns: {d(3,3),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(1,1) + d(0,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,3) + d(2,0))$ Unknowns: {d(4,3),d(2,0)} Unknowns: {d(4,3),d(2,0)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,3) - d(4,0) + d(2, 1)*a$ Unknowns: {d(5,3),d(4,0),d(2,1),a} Unknowns: {d(5,3),d(4,0),d(2,1),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(4,0) + d(2,1)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,3) - d(5,0) + d(4, 1)*a - d(4,1)$ Unknowns: {d(6,3),d(5,0),d(4,1),a} Unknowns: {d(6,3),d(5,0),d(4,1),a} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - d(5,0) + d(4,1)*a - d(4,1)$ 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 + d(1,2)$ Unknowns: {d(3,5),d(1,2),a} Unknowns: {d(3,5),d(1,2),a} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=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(2,2) + d (0,0))$ Unknowns: {d(5,5),d(2,2),d(0,0),a} Unknowns: {d(5,5),d(2,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(5,5) + d(2,2) + d(0,0)) on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,5)*a + d(4,2)*a - d(4,2) - d(3,0)$ Unknowns: {d(6,5),d(4,2),d(3,0),a} Unknowns: {d(6,5),d(4,2),d(3,0),a} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:= - d(6,5)*a + d(4,2)*a - d(4,2)$ on resout l'equation {{0,3},6} qui est maintenant AA:=d(2,0)*( - a + 2)$ Unknowns: {d(2,0),a} Unknowns: {d(2,0),a} pas de selection possible de variable a coefficient numerique dans d(2,0)*( - a + 2) on resout l'equation {{0,4},0} qui est maintenant AA:=d(0,6)*( - a + 1)$ Unknowns: {d(0,6),a} Unknowns: {d(0,6),a} pas de selection possible de variable a coefficient numerique dans d(0,6)*( - a + 1) on resout l'equation {{0,4},1} qui est maintenant AA:=d(1,6)*( - a + 1)$ Unknowns: {d(1,6),a} Unknowns: {d(1,6),a} pas de selection possible de variable a coefficient numerique dans d(1,6)*( - a + 1) on resout l'equation {{0,4},2} qui est maintenant AA:=d(2,6)*( - a + 1)$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans d(2,6)*( - a + 1) on resout l'equation {{0,4},3} qui est maintenant AA:= - d(3,6)*a + d(3,6) + d( 1,4)$ Unknowns: {d(3,6),d(1,4),a} Unknowns: {d(3,6),d(1,4),a} bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:=d(3,6)*(a - 1)$ on resout l'equation {{0,4},4} qui est maintenant AA:=d(4,6)*( - a + 1)$ Unknowns: {d(4,6),a} Unknowns: {d(4,6),a} pas de selection possible de variable a coefficient numerique dans d(4,6)*( - a + 1) on resout l'equation {{0,4},5} qui est maintenant AA:= - d(5,6)*a + d(5,6) + d( 4,5)*a + d(2,4)*a$ Unknowns: {d(5,6),d(4,5),d(2,4),a} Unknowns: {d(5,6),d(4,5),d(2,4),a} pas de selection possible de variable a coefficient numerique dans - d(5,6)*a + d(5,6) + d(4,5)*a + d(2,4)*a on resout l'equation {{0,4},6} qui est maintenant AA:= - d(6,6)*a + d(6,6) + d( 4,4)*a - d(4,4) + d(0,0)*a - d(0,0)$ Unknowns: {d(6,6),d(4,4),d(0,0),a} Unknowns: {d(6,6),d(4,4),d(0,0),a} pas de selection possible de variable a coefficient numerique dans - d(6,6)*a + d(6,6) + d(4,4)*a - d(4,4) + d(0,0)*a - d(0,0) on resout l'equation {{0,5},3} 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(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 {{0,5},6} qui est maintenant AA:=d(4,5)*(2*a - 1)$ Unknowns: {d(4,5),a} Unknowns: {d(4,5),a} pas de selection possible de variable a coefficient numerique dans d(4,5)*(2*a - 1) 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},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 {{0,6},6} qui est maintenant AA:=d(4,6)*(a - 1)$ Unknowns: {d(4,6),a} Unknowns: {d(4,6),a} pas de selection possible de variable a coefficient numerique dans d(4,6)*(a - 1 ) 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(3,6)*( - a + 1)$ Unknowns: {d(3,6),a} Unknowns: {d(3,6),a} pas de selection possible de variable a coefficient numerique dans d(3,6)*( - a + 1) 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))$ Unknowns: {d(3,4),d(0,2)} Unknowns: {d(3,4),d(0,2)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,4) + d(2,2) + d(1, 1)$ Unknowns: {d(4,4),d(2,2),d(1,1)} Unknowns: {d(4,4),d(2,2),d(1,1)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) + d(4,2) + d(0, 1)*a$ Unknowns: {d(5,4),d(4,2),d(0,1),a} Unknowns: {d(5,4),d(4,2),d(0,1),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) + d(0,1)*a$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,4) + d(5,2) - d(3, 1)$ Unknowns: {d(6,4),d(5,2),d(3,1)} Unknowns: {d(6,4),d(5,2),d(3,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(5,2) - d(3,1)$ on resout l'equation {{1,3},5} qui est maintenant AA:= - d(2,0)$ Unknown: d(2,0) Unknown: d(2,0) bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,3},6} qui est maintenant AA:= - d(4,0) + d(2,1)*a + d( 2,1)$ Unknowns: {d(4,0),d(2,1),a} Unknowns: {d(4,0),d(2,1),a} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(2,1)*(a + 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},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},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) + d(2,2) + 2*d( 1,1)$ Unknowns: {d(5,5),d(2,2),d(1,1)} Unknowns: {d(5,5),d(2,2),d(1,1)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,2) + 2*d(1,1)$ on resout l'equation {{1,4},6} qui est maintenant AA:= - d(6,5) + d(4,2) + 2*d( 0,1)*a - d(0,1)$ Unknowns: {d(6,5),d(4,2),d(0,1),a} Unknowns: {d(6,5),d(4,2),d(0,1),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(4,2) + 2*d(0,1)*a - d(0,1)$ 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},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 {{1,5},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(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 {{1,5},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(2,2) + 3*d( 1,1)$ Unknowns: {d(6,6),d(2,2),d(1,1)} Unknowns: {d(6,6),d(2,2),d(1,1)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(2,2) + 3*d(1,1)$ on resout l'equation {{2,3},6} qui est maintenant AA:= - 2*d(1,1) + d(0,0)$ Unknowns: {d(1,1),d(0,0)} Unknowns: {d(1,1),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:=d(0,0)/2$ on resout l'equation {{2,4},6} qui est maintenant AA:=d(0,2)*(a - 2)$ Unknowns: {d(0,2),a} Unknowns: {d(0,2),a} pas de selection possible de variable a coefficient numerique dans d(0,2)*(a - 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},3},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},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},4},0}, {{{2,4},5},0}, {{{2,4},6},d(0,2)*(a - 2)}, {{{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},6},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},3},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},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},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},6},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 {2}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(0,1),0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,d(2,1),d(2,2),0,0,0,0),( - ((2*a - 1)*d(0,1)*a + d(4,2)),d(3,1),d(3,2),(3*d(0,0))/2,0,0,0),((a + 1)*d(2, 1),d(4,1),d(4,2),0,(2*d(2,2) + d(0,0))/2,0,0),(d(5,0),d(5,1),d(5,2), - d(2,1),d( 4,2) + d(0,1)*a,d(2,2) + d(0,0),0),(d(6,0),d(6,1),d(6,2),(a - 1)*d(4,1) - d(5,0) ,d(5,2) - d(3,1),(2*a - 1)*d(0,1) + d(4,2),(2*d(2,2) + 3*d(0,0))/2))$ $ pour delta:= [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 a 0 0 0 0] [ ] [0 0 0 a - 1 0 0] pour shortformdelta:={0,ss,1,0,ss,a,ss,0} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,2), d(3,1), d(2,2), d(2,1), 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(4,2), d(4,1), d(3,2), d(3,1), d(2,2), d(2,1), 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(4,2), d(4,1), d(3,2), d(3,1), d(2,2), d(2,1), d(0,1), d(0,0)}$ dim Der(gtildedelta):=14$ t1:=D(0,0):= [1 0 0 0 0 0 0 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [0 0 0 0 0 0 0 ] [ ] [ 3 ] [0 0 0 --- 0 0 0 ] [ 2 ] [ ] [ 1 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 1 0 ] [ ] [ 3 ] [0 0 0 0 0 0 ---] [ 2 ] MATD:= mat((d(0,0),d(0,1),0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 (0,d(2,1),d(2,2),0,0,0,0), 3*d(0,0) ( - ((2*a - 1)*d(0,1)*a + d(4,2)),d(3,1),d(3,2),----------,0,0,0), 2 2*d(2,2) + d(0,0) ((a + 1)*d(2,1),d(4,1),d(4,2),0,-------------------,0,0), 2 (d(5,0),d(5,1),d(5,2), - d(2,1),d(4,2) + d(0,1)*a,d(2,2) + d(0,0),0), (d(6,0),d(6,1),d(6,2),(a - 1)*d(4,1) - d(5,0),d(5,2) - d(3,1), 2*d(2,2) + 3*d(0,0) (2*a - 1)*d(0,1) + d(4,2),---------------------)) 2 Unknowns: {d(5,0),d(4,1),d(2,2),d(0,0),a} Unknowns: {d(5,0),d(4,1),d(2,2),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 (0,0,d(2,2),0,0,0,0), 3*d(0,0) (0,0,0,----------,0,0,0), 2 2*d(2,2) + d(0,0) (0,d(4,1),0,0,-------------------,0,0), 2 (d(5,0),0,0,0,0,d(2,2) + d(0,0),0), 2*d(2,2) + 3*d(0,0) (0,0,0,(a - 1)*d(4,1) - d(5,0),0,0,---------------------)) 2 Unknowns: {d(5,0),d(4,1),d(2,2),d(0,0),a} Unknowns: {d(5,0),d(4,1),d(2,2),d(0,0),a} t2:=D(2,2):= [0 0 0 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 1 0 0] [ ] [0 0 0 0 0 1 0] [ ] [0 0 0 0 0 0 1] Unknowns: {d(2,2),d(0,0)} Unknowns: {d(2,2),d(0,0)} commutant simultane de t1,t2 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 (0,0,d(2,2),0,0,0,0), 3*d(0,0) (0,0,0,----------,0,0,0), 2 2*d(2,2) + d(0,0) (0,0,0,0,-------------------,0,0), 2 (0,0,0,0,0,d(2,2) + d(0,0),0), 2*d(2,2) + 3*d(0,0) (0,0,0,0,0,0,---------------------)) 2 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:= [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 ] [ ] [0 0 0 0 0 0 0 ] [ ] [ 3 ] [0 0 0 --- 0 0 0 ] [ 2 ] [ ] [ 1 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 1 0 ] [ ] [ 3 ] [0 0 0 0 0 0 ---] [ 2 ] P**(-1)*t2*P:= [0 0 0 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 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((d(0,0),d(0,1),0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,d(2,1),d(2,2),0,0,0,0),( - ((2*a - 1)*d(0,1)*a + d(4,2)),d(3,1),d(3,2),(3*d(0,0))/2,0,0,0),((a + 1)*d(2, 1),d(4,1),d(4,2),0,(2*d(2,2) + d(0,0))/2,0,0),(d(5,0),d(5,1),d(5,2), - d(2,1),d( 4,2) + d(0,1)*a,d(2,2) + d(0,0),0),(d(6,0),d(6,1),d(6,2),(a - 1)*d(4,1) - d(5,0) ,d(5,2) - d(3,1),(2*a - 1)*d(0,1) + d(4,2),(2*d(2,2) + 3*d(0,0))/2))$ $ 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),0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 (0,d(2,1),d(2,2),0,0,0,0), 3*d(0,0) ( - ((2*a - 1)*d(0,1)*a + d(4,2)),d(3,1),d(3,2),----------,0,0,0), 2 2*d(2,2) + d(0,0) ((a + 1)*d(2,1),d(4,1),d(4,2),0,-------------------,0,0), 2 (d(5,0),d(5,1),d(5,2), - d(2,1),d(4,2) + d(0,1)*a,d(2,2) + d(0,0),0), (d(6,0),d(6,1),d(6,2),(a - 1)*d(4,1) - d(5,0),d(5,2) - d(3,1), 2*d(2,2) + 3*d(0,0) (2*a - 1)*d(0,1) + d(4,2),---------------------)) 2 on voit apparaitre les poids sur la diagonale r(1) := d(0,0) d(0,0) r(2) := -------- 2 r(3) := d(2,2) 3*d(0,0) r(4) := ---------- 2 2*d(2,2) + d(0,0) r(5) := ------------------- 2 r(6) := d(2,2) + d(0,0) 2*d(2,2) + 3*d(0,0) r(7) := --------------------- 2 r(1) := 2*gamma1 r(2) := gamma1 r(3) := gamma2 r(4) := 3*gamma1 r(5) := gamma1 + gamma2 r(6) := 2*gamma1 + gamma2 r(7) := 3*gamma1 + gamma2 Le systeme de poids est le systeme 2.1 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(3)}, {{0,2},a*x(5)}, {{0,3},0}, {{0,4},x(6)*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}} diaY(1):=x(0) diaY(2):=x(1) diaY(3):=x(2) diaY(4):=x(3) diaY(5):=x(4) diaY(6):=x(5) diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},diay(4)}, {{1,3},diay(6)*a}, {{1,4},0}, {{1,5},diay(7)*(a - 1)}, {{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_{7,2.1}$ (iL)$ and that for a neq{0,2}$ 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,( - u**2)/a,0,0,0,0),(u,0,0,0,0,0,0),(0,v,0,0,0,0,0),(0,0,0,0,u**3/a,0, 0),(0,0,0,u*v,0,0,0),(0,0,0,0,0,u**2*v,0),(0,0,0,0,0,0,u**3*v))$ $ det(isom):= (u**12*v**4)/a**2$ ZZ(1):=diay(2)*u$ ZZ(2):=diay(3)*v$ ZZ(3):=( - diay(1)*u**2)/a$ ZZ(4):=diay(5)*u*v$ ZZ(5):=(diay(4)*u**3)/a$ ZZ(6):=diay(6)*u**2*v$ ZZ(7):=diay(7)*u**3*v$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(5)}$ {{1,4},zz(6)}$ {{1,5},0}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(6)}$ {{2,4},0}$ {{2,5},zz(7)/a}$ {{2,6},0}$ {{2,7},0}$ {{3,4},( - zz(7)*a + zz(7))/a}$ {{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_{7,2.1}$ (iL)$ with L:=1/a$ Et cela pour a:=a, b:=0.$ and that for a neq {0,2}$ shortformdelta:={0,ss,1,0,ss,a,ss,0}$ delta:= mat((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,a,0,0,0,0),(0,0,0 ,a - 1,0,0))$ $