generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(0,2*xi(1,1),0,0,0,0),(xi(3,1),xi(3,2),3*xi(1,1),0,0,0), (xi(4,1),xi(4,2),xi(3,2),4*xi(1,1),0,0),(xi(5,1),xi(5,2),xi(4,2) - xi(3,1),xi(3, 2),5*xi(1,1),0),(xi(6,1),xi(6,2),xi(5,2) - xi(4,1),xi(4,2) - xi(3,1),xi(3,2),6* 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 1 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] [ ] [-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] 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 by subtracting adjoints one then may suppose: xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0,xi(6,1):=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] [ ] [0 xi(5,2) xi(4,2) 0 0 0] [ ] [0 xi(6,2) xi(5,2) xi(4,2) 0 0] We denote this delta by the shortform shortformdelta:={xi(4,2), ss, xi(5,2), ss, xi(6,2)} paramindexeslist:={{4,2},{5,2},{6,2}} 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),(0,a,1,0,0,0),(0,1,a ,1,0,0))$ $ shortformdelta:={1,ss,a,ss,1}$ Comm. relations of g_{6,19} :$ x(1)*x(2):=x(3)$ x(1)*x(3):=x(4)$ x(1)*x(4):=x(5)$ x(1)*x(5):=x(6)$ x(2)*x(3):=x(5)$ x(2)*x(4):=x(6)$ on resout l'equation {{0,1},3} 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 {{0,1},4} qui est maintenant AA:= - d(3,0) + d(2,1)$ Unknowns: {d(3,0),d(2,1)} Unknowns: {d(3,0),d(2,1)} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(4,0) + d(3,1) + d(2, 1)*a$ Unknowns: {d(4,0),d(3,1),d(2,1),a} Unknowns: {d(4,0),d(3,1),d(2,1),a} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(3,1) + d(2,1)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(5,0) + d(4,1) + d(3, 1)*a + d(2,1)$ Unknowns: {d(5,0),d(4,1),d(3,1),d(2,1),a} Unknowns: {d(5,0),d(4,1),d(3,1),d(2,1),a} bonne inconnue W:=d(5,0)$ sa valeur doit etre WW:=d(4,1) + d(3,1)*a + d(2,1)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,6) + d(0,5)*a + d (0,4))$ Unknowns: {d(0,6),d(0,5),d(0,4),a} Unknowns: {d(0,6),d(0,5),d(0,4),a} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:= - (d(0,5)*a + d(0,4))$ on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,6) + d(1,5)*a + d (1,4))$ Unknowns: {d(1,6),d(1,5),d(1,4),a} Unknowns: {d(1,6),d(1,5),d(1,4),a} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:= - (d(1,5)*a + d(1,4))$ on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,6) + d(2,5)*a + d (2,4))$ Unknowns: {d(2,6),d(2,5),d(2,4),a} Unknowns: {d(2,6),d(2,5),d(2,4),a} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:= - (d(2,5)*a + d(2,4))$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,6) - d(3,5)*a - d( 3,4) + d(1,0)$ Unknowns: {d(3,6),d(3,5),d(3,4),d(1,0),a} Unknowns: {d(3,6),d(3,5),d(3,4),d(1,0),a} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(3,5)*a - d(3,4) + d(1,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,6) - d(4,5)*a - d( 4,4) + d(2,2) + d(0,0)$ Unknowns: {d(4,6),d(4,5),d(4,4),d(2,2),d(0,0),a} Unknowns: {d(4,6),d(4,5),d(4,4),d(2,2),d(0,0),a} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(4,5)*a - d(4,4) + d(2,2) + d(0,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,6) - d(5,5)*a - d( 5,4) + d(3,2) + d(2,2)*a - d(2,1) + d(0,0)*a$ Unknowns: {d(5,6),d(5,5),d(5,4),d(3,2),d(2,2),d(2,1),d(0,0),a} Unknowns: {d(5,6),d(5,5),d(5,4),d(3,2),d(2,2),d(2,1),d(0,0),a} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - d(5,5)*a - d(5,4) + d(3,2) + d(2,2)*a - d(2,1) + d(0, 0)*a$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,6) - d(6,5)*a - d( 6,4) + d(4,2) + d(3,2)*a - d(3,1) + d(2,2) - d(2,1)*a + d(0,0)$ Unknowns: {d(6,6),d(6,5),d(6,4),d(4,2),d(3,2),d(3,1),d(2,2),d(2,1),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(6,4),d(4,2),d(3,2),d(3,1),d(2,2),d(2,1),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(6,5)*a - d(6,4) + d(4,2) + d(3,2)*a - d(3,1) + d(2, 2) - d(2,1)*a + d(0,0)$ on resout l'equation {{0,3},0} qui est maintenant AA:=d(0,5)*a**2 - d(0,5) + d( 0,4)*a$ Unknowns: {d(0,5),d(0,4),a} Unknowns: {d(0,5),d(0,4),a} pas de selection possible de variable a coefficient numerique dans d(0,5)*a**2 - d(0,5) + d(0,4)*a on resout l'equation {{0,3},1} qui est maintenant AA:=d(1,5)*a**2 - d(1,5) + d( 1,4)*a$ Unknowns: {d(1,5),d(1,4),a} Unknowns: {d(1,5),d(1,4),a} pas de selection possible de variable a coefficient numerique dans d(1,5)*a**2 - d(1,5) + d(1,4)*a on resout l'equation {{0,3},2} qui est maintenant AA:=d(2,5)*a**2 - d(2,5) + d( 2,4)*a$ Unknowns: {d(2,5),d(2,4),a} Unknowns: {d(2,5),d(2,4),a} pas de selection possible de variable a coefficient numerique dans d(2,5)*a**2 - d(2,5) + d(2,4)*a on resout l'equation {{0,3},3} qui est maintenant AA:=d(3,5)*a**2 - d(3,5) + d( 3,4)*a - d(1,0)*a$ Unknowns: {d(3,5),d(3,4),d(1,0),a} Unknowns: {d(3,5),d(3,4),d(1,0),a} pas de selection possible de variable a coefficient numerique dans d(3,5)*a**2 - d(3,5) + d(3,4)*a - d(1,0)*a on resout l'equation {{0,3},4} qui est maintenant AA:=d(4,5)*a**2 - d(4,5) + d( 4,4)*a + d(2,3) - d(2,2)*a + d(1,0) - d(0,0)*a$ Unknowns: {d(4,5),d(4,4),d(2,3),d(2,2),d(1,0),d(0,0),a} Unknowns: {d(4,5),d(4,4),d(2,3),d(2,2),d(1,0),d(0,0),a} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:= - d(4,5)*a**2 + d(4,5) - d(4,4)*a + d(2,2)*a - d(1,0) + d(0,0)*a$ on resout l'equation {{0,3},5} qui est maintenant AA:=d(5,5)*a**2 - d(5,5) + d( 5,4)*a - d(4,5)*a**3 + d(4,5)*a - d(4,4)*a**2 + d(3,3) - d(3,2)*a + d(2,1)*a - d (1,0)*a + d(0,0)$ Unknowns: {d(5,5),d(5,4),d(4,5),d(4,4),d(3,3),d(3,2),d(2,1),d(1,0),d(0,0),a} Unknowns: {d(5,5),d(5,4),d(4,5),d(4,4),d(3,3),d(3,2),d(2,1),d(1,0),d(0,0),a} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:= - d(5,5)*a**2 + d(5,5) - d(5,4)*a + d(4,5)*a**3 - d(4,5 )*a + d(4,4)*a**2 + d(3,2)*a - d(2,1)*a + d(1,0)*a - d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:=d(6,5)*a**2 - d(6,5) + d( 6,4)*a - d(5,5)*a**3 + d(5,5)*a - d(5,4)*a**2 + d(4,5)*a**4 - 2*d(4,5)*a**2 + d( 4,5) + d(4,4)*a**3 - d(4,4)*a + d(4,3) - d(4,2)*a + d(3,1)*a + d(1,0)*a**2 - d(1 ,0)$ Unknowns: {d(6,5), d(6,4), d(5,5), d(5,4), d(4,5), d(4,4), d(4,3), d(4,2), d(3,1), d(1,0), a} Unknowns: {d(6,5), d(6,4), d(5,5), d(5,4), d(4,5), d(4,4), d(4,3), d(4,2), d(3,1), d(1,0), a} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(6,5)*a**2 + d(6,5) - d(6,4)*a + d(5,5)*a**3 - d(5,5 )*a + d(5,4)*a**2 - d(4,5)*a**4 + 2*d(4,5)*a**2 - d(4,5) - d(4,4)*a**3 + d(4,4)* a + d(4,2)*a - d(3,1)*a - d(1,0)*a**2 + d(1,0)$ on resout l'equation {{0,4},0} qui est maintenant AA:=d(0,5)*a + d(0,4)$ Unknowns: {d(0,5),d(0,4),a} Unknowns: {d(0,5),d(0,4),a} bonne inconnue W:=d(0,4)$ sa valeur doit etre WW:= - d(0,5)*a$ on resout l'equation {{0,4},1} qui est maintenant AA:=d(1,5)*a + d(1,4)$ Unknowns: {d(1,5),d(1,4),a} Unknowns: {d(1,5),d(1,4),a} bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:= - d(1,5)*a$ on resout l'equation {{0,4},2} qui est maintenant AA:=d(2,5)*a + d(2,4)$ Unknowns: {d(2,5),d(2,4),a} Unknowns: {d(2,5),d(2,4),a} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:= - d(2,5)*a$ on resout l'equation {{0,4},3} qui est maintenant AA:=d(3,5)*a + d(3,4) - d(1,0 )$ Unknowns: {d(3,5),d(3,4),d(1,0),a} Unknowns: {d(3,5),d(3,4),d(1,0),a} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(3,5)*a + d(1,0)$ on resout l'equation {{0,4},4} qui est maintenant AA:=d(4,5)*a + d(4,4) - d(2,5 )*a - d(2,2) - d(0,0)$ Unknowns: {d(4,5),d(4,4),d(2,5),d(2,2),d(0,0),a} Unknowns: {d(4,5),d(4,4),d(2,5),d(2,2),d(0,0),a} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:= - d(4,5)*a + d(2,5)*a + d(2,2) + d(0,0)$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(5,5)*a + d(5,4) - d(3,5 )*a - d(3,2) - d(2,5)*a**2 - d(2,2)*a + d(2,1) + 2*d(1,0) - d(0,0)*a$ Unknowns: {d(5,5),d(5,4),d(3,5),d(3,2),d(2,5),d(2,2),d(2,1),d(1,0),d(0,0),a} Unknowns: {d(5,5),d(5,4),d(3,5),d(3,2),d(2,5),d(2,2),d(2,1),d(1,0),d(0,0),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(5,5)*a + d(3,5)*a + d(3,2) + d(2,5)*a**2 + d(2,2)*a - d(2,1) - 2*d(1,0) + d(0,0)*a$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(6,5)*a + d(6,4) - d(4,5 )*a - d(4,2) - d(3,5)*a**2 - d(3,2)*a + d(3,1) + d(2,1)*a + d(1,0)*a + d(0,0)$ Unknowns: {d(6,5), d(6,4), d(4,5), d(4,2), d(3,5), d(3,2), d(3,1), d(2,1), d(1,0), d(0,0), a} Unknowns: {d(6,5), d(6,4), d(4,5), d(4,2), d(3,5), d(3,2), d(3,1), d(2,1), d(1,0), d(0,0), a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(6,5)*a + d(4,5)*a + d(4,2) + d(3,5)*a**2 + d(3,2)*a - d(3,1) - d(2,1)*a - d(1,0)*a - d(0,0)$ on resout l'equation {{0,5},4} 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 {{0,5},5} 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 {{0,5},6} qui est maintenant AA:=d(4,5) + d(1,0)$ Unknowns: {d(4,5),d(1,0)} Unknowns: {d(4,5),d(1,0)} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:= - d(1,0)$ 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:=2*d(1,0)$ Unknown: d(1,0) Unknown: d(1,0) bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,2},3} qui est maintenant AA:= - d(5,5) + d(2,2) + d(1, 1) + d(0,0)$ Unknowns: {d(5,5),d(2,2),d(1,1),d(0,0)} Unknowns: {d(5,5),d(2,2),d(1,1),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,2) + d(1,1) + d(0,0)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(6,5) + d(3,2) + d(1, 1)*a + d(0,1) - d(0,0)*a$ Unknowns: {d(6,5),d(3,2),d(1,1),d(0,1),d(0,0),a} Unknowns: {d(6,5),d(3,2),d(1,1),d(0,1),d(0,0),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(3,2) + d(1,1)*a + d(0,1) - d(0,0)*a$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,3) + d(4,2) - d(3, 1) + d(0,1)*a$ Unknowns: {d(5,3),d(4,2),d(3,1),d(0,1),a} Unknowns: {d(5,3),d(4,2),d(3,1),d(0,1),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(4,2) - d(3,1) + d(0,1)*a$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,3) + d(5,2) - d(4, 1) + d(0,1)$ Unknowns: {d(6,3),d(5,2),d(4,1),d(0,1)} Unknowns: {d(6,3),d(5,2),d(4,1),d(0,1)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,2) - d(4,1) + d(0,1)$ on resout l'equation {{1,3},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 {{1,3},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 {{1,3},4} 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 {{1,3},5} qui est maintenant AA:=(4*d(2,1) + 4*d(0,1) + d( 0,0)*a)/2$ Unknowns: {d(2,1),d(0,1),d(0,0),a} Unknowns: {d(2,1),d(0,1),d(0,0),a} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=( - 4*d(0,1) - d(0,0)*a)/4$ on resout l'equation {{1,3},6} qui est maintenant AA:=(8*d(0,1)*a - 3*d(0,0)*a **2 + 4*d(0,0))/4$ Unknowns: {d(0,1),d(0,0),a} Unknowns: {d(0,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (8*d(0,1)*a - 3*d(0,0)*a**2 + 4*d(0,0))/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 {{2,3},4} 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 {{2,3},5} qui est maintenant AA:=d(2,2) + d(0,2) - d(0,0)$ Unknowns: {d(2,2),d(0,2),d(0,0)} Unknowns: {d(2,2),d(0,2),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:= - d(0,2) + d(0,0)$ on resout l'equation {{2,3},6} qui est maintenant AA:=(a*(2*d(0,2) + d(0,0)))/2 $ Unknowns: {d(0,2),d(0,0),a} Unknowns: {d(0,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans (a*(2*d(0,2) + d(0,0)))/2 Derivation equations to cancel (Reduce output) : \\{{{{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},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},(8*d(0,1)*a - 3*d(0,0)*a**2 + 4*d(0,0))/4}, {{{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},(a*(2*d(0,2) + d(0,0)))/2}, {{{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},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},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},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},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},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},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((d(0,0),((3*a**2 - 4)*d(0,0))/(8*a),( - d(0,0))/2,0,0,0,0),(0,d(0,0)/2,0,0,0 ,0,0),(0,( - (5*a**2 - 4)*d(0,0))/(8*a),(3*d(0,0))/2,0,0,0,0),(( - (5*a**2 - 4)* d(0,0))/(8*a),d(3,1),d(3,2),2*d(0,0),0,0,0),(( - ((5*a**2 - 4)*d(0,0) - 8*d(3,1) ))/8,d(4,1),d(4,2),((3*a**2 - 4)*d(0,0) + 8*d(3,2)*a)/(8*a),(5*d(0,0))/2,0,0),(( - ((5*a**2 - 4)*d(0,0) - 8*d(3,1)*a**2 - 8*d(4,1)*a))/(8*a),d(5,1),d(5,2),((3*a **2 - 4)*d(0,0) - 8*d(3,1) + 8*d(4,2))/8,(8*d(3,2)*a + d(0,0)*a**2 - 4*d(0,0))/( 8*a),3*d(0,0),0),(d(6,0),d(6,1),d(6,2),((3*a**2 - 4)*d(0,0) - 8*d(4,1)*a + 8*d(5 ,2)*a)/(8*a),((3*a**2 - 4)*d(0,0) - 4*d(3,1) + 4*d(4,2))/4,( - ((a**2 + 4)*d(0,0 ) - 8*d(3,2)*a))/(8*a),(7*d(0,0))/2))$ $ 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] [ ] [0 a 1 0 0 0] [ ] [0 1 a 1 0 0] pour shortformdelta:={1,ss,a,ss,1} 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(0,0), 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(0,0), 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(0,0)}$ dim Der(gtildedelta):=10$ t1:=D(0,0):= 2 3*a - 4 - 1 mat((1,----------,------,0,0,0,0), 8*a 2 1 (0,---,0,0,0,0,0), 2 2 - (5*a - 4) 3 (0,---------------,---,0,0,0,0), 8*a 2 2 - (5*a - 4) (---------------,0,0,2,0,0,0), 8*a 2 2 - (5*a - 4) 3*a - 4 5 (---------------,0,0,----------,---,0,0), 8 8*a 2 2 2 - (5*a - 4) 3*a - 4 (a + 2)*(a - 2) (---------------,0,0,----------,-----------------,3,0), 8*a 8 8*a 2 2 2 3*a - 4 3*a - 4 - (a + 4) 7 (0,0,0,----------,----------,-------------,---)) 8*a 4 8*a 2 {{2*x - 1, 1, [ 4 ] [ - 92160*arbcomplex(393)*a ] [ --------------------------------------------- ] [ 6 4 2 2 ] [ (1137*a - 60*a - 3088*a - 64)*(5*a - 4) ] [ ] [ 5 ] [ 737280*arbcomplex(393)*a ] [-------------------------------------------------------------] [ 6 4 2 2 ] [ (1137*a - 60*a - 3088*a - 64)*(5*a - 4)*(a + 2)*(a - 2) ] [ ] [ 4 ] [ 92160*arbcomplex(393)*a ] [ -------------------------------------------------- ] [ 6 4 2 ] [ (1137*a - 60*a - 3088*a - 64)*(a + 2)*(a - 2) ] [ ] [ 3 ] [ - 7680*arbcomplex(393)*a ] [ -------------------------------- ] [ 6 4 2 ] [ 1137*a - 60*a - 3088*a - 64 ] [ ] [ 2 2 ] [ - 480*(9*a + 4)*arbcomplex(393)*a ] [ -------------------------------------- ] [ 6 4 2 ] [ 1137*a - 60*a - 3088*a - 64 ] [ ] [ 4 2 ] [ 24*(57*a - 288*a - 16)*arbcomplex(393)*a ] [ -------------------------------------------- ] [ 6 4 2 ] [ 1137*a - 60*a - 3088*a - 64 ] [ ] [ arbcomplex(393) ] }, {x - 3, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 4*arbcomplex(394)*a ] [---------------------] [ 2 ] [ a + 4 ] [ ] [ arbcomplex(394) ] }, {x - 2, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 3 ] [ 384*arbcomplex(395)*a ] [ ---------------------------------- ] [ 4 2 2 ] [ (45*a - 112*a - 16)*(3*a - 4) ] [ ] [ 2 ] [ - 96*arbcomplex(395)*a ] [ -------------------------- ] [ 4 2 ] [ 45*a - 112*a - 16 ] [ ] [ 2 ] [ - 12*(3*a + 4)*arbcomplex(395)*a ] [------------------------------------] [ 4 2 ] [ 45*a - 112*a - 16 ] [ ] [ arbcomplex(395) ] }, {x - 1, 1, [ 4 ] [ - 30720*arbcomplex(396)*a ] [ --------------------------------------------- ] [ 6 4 2 2 ] [ (521*a + 324*a - 1936*a - 64)*(5*a - 4) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 3 ] [ - 3840*arbcomplex(396)*a ] [ -------------------------------- ] [ 6 4 2 ] [ 521*a + 324*a - 1936*a - 64 ] [ ] [ 2 2 ] [ - 320*(5*a + 4)*arbcomplex(396)*a ] [ -------------------------------------- ] [ 6 4 2 ] [ 521*a + 324*a - 1936*a - 64 ] [ ] [ 2 ] [ 20*(41*a + 4)*(a + 2)*(a - 2)*arbcomplex(396)*a ] [--------------------------------------------------] [ 6 4 2 ] [ 521*a + 324*a - 1936*a - 64 ] [ ] [ arbcomplex(396) ] }, {2*x - 7,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(397)] }, {2*x - 5, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 2 ] [ - 32*arbcomplex(398)*a ] [ -------------------------- ] [ 4 2 ] [ 25*a - 32*a - 16 ] [ ] [ 8*(a + 2)*(a - 2)*arbcomplex(398)*a ] [-------------------------------------] [ 4 2 ] [ 25*a - 32*a - 16 ] [ ] [ arbcomplex(398) ] }, {2*x - 3, 1, [ 4 ] [ - 6144*arbcomplex(399)*a ] [--------------------------------------------] [ 6 4 2 2 ] [ (97*a + 516*a - 1040*a - 64)*(5*a - 4) ] [ ] [ 0 ] [ ] [ 4 ] [ 6144*arbcomplex(399)*a ] [--------------------------------------------] [ 6 4 2 2 ] [ (97*a + 516*a - 1040*a - 64)*(5*a - 4) ] [ ] [ 3 ] [ - 1536*arbcomplex(399)*a ] [ ------------------------------- ] [ 6 4 2 ] [ 97*a + 516*a - 1040*a - 64 ] [ ] [ 2 2 ] [ - 192*(a + 4)*arbcomplex(399)*a ] [ ------------------------------------ ] [ 6 4 2 ] [ 97*a + 516*a - 1040*a - 64 ] [ ] [ 4 2 ] [ 16*(25*a - 64*a - 16)*arbcomplex(399)*a ] [------------------------------------------- ] [ 6 4 2 ] [ 97*a + 516*a - 1040*a - 64 ] [ ] [ arbcomplex(399) ] }} Unknowns: {d(6,1),d(5,1),d(0,0),a} Unknowns: {d(6,1),d(5,1),d(0,0),a} commutant de t1 dans der(gtildedelta): 2 (3*a - 4)*d(0,0) - d(0,0) mat((d(0,0),-------------------,-----------,0,0,0,0), 8*a 2 d(0,0) (0,--------,0,0,0,0,0), 2 2 - (5*a - 4)*d(0,0) 3*d(0,0) (0,----------------------,----------,0,0,0,0), 8*a 2 2 - (5*a - 4)*d(0,0) (----------------------,0,0,2*d(0,0),0,0,0), 8*a 2 2 - (5*a - 4)*d(0,0) (3*a - 4)*d(0,0) 5*d(0,0) (----------------------,0,0,-------------------,----------,0,0), 8 8*a 2 2 2 - (5*a - 4)*d(0,0) (3*a - 4)*d(0,0) (----------------------,d(5,1),0,-------------------, 8*a 8 (a + 2)*(a - 2)*d(0,0) ------------------------,3*d(0,0),0), 8*a 2 2 2 (3*a - 4)*d(0,0) (3*a - 4)*d(0,0) - (a + 4)*d(0,0) (0,d(6,1),0,-------------------,-------------------,--------------------, 8*a 4 8*a 7*d(0,0) ----------)) 2 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:= - (a + 2)*(a - 2) mat((1,--------------------,-1,0,0,0,0), 8*a (0,1,0,0,0,0,0), 2 5*a - 4 (0,----------,1,0,0,0,0), 8*a 2 2 2 5*a - 4 - (5*a - 4)*(a + 2)*(a - 2) - (5*a - 4) (----------,-------------------------------,---------------,1,0,0,0), 8*a 2 4*a 96*a 2 2 2 2 (5*a + 4)*(5*a - 4) - (9*a + 4)*(5*a - 4)*(a + 2)*(a - 2) (-----------------------,------------------------------------------, 2 3 96*a 1536*a 2 2 2 - (5*a - 4)*(a + 4) - (3*a - 4) ------------------------,---------------,1,0,0), 2 4*a 32*a 2 2 - (41*a + 4)*(5*a - 4)*(a + 2)*(a - 2) (-------------------------------------------, 3 1536*a 4 2 2 (57*a - 288*a - 16)*(5*a - 4)*(a + 2)*(a - 2) --------------------------------------------------, 4 30720*a 4 2 2 2 2 (25*a - 64*a - 16)*(5*a - 4) - (3*a + 4)*(3*a - 4) ---------------------------------,--------------------------, 3 2 384*a 32*a - (a + 2)*(a - 2) --------------------,1,0), 4*a 6 4 2 2 - (521*a + 324*a - 1936*a - 64)*(5*a - 4) (------------------------------------------------, 4 30720*a 6 4 2 2 (1137*a - 60*a - 3088*a - 64)*(5*a - 4)*(a + 2)*(a - 2) -------------------------------------------------------------, 5 737280*a 6 4 2 2 (97*a + 516*a - 1040*a - 64)*(5*a - 4) --------------------------------------------, 4 6144*a 4 2 2 4 2 2 (45*a - 112*a - 16)*(3*a - 4) - (25*a - 32*a - 16) a + 4 ----------------------------------,-------------------------,--------,1)) 3 2 4*a 384*a 32*a P**(-1)*t1*P:= [1 0 0 0 0 0 0 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 --- 0 0 0 0 ] [ 2 ] [ ] [0 0 0 2 0 0 0 ] [ ] [ 5 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 3 0 ] [ ] [ 7 ] [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),0,0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,0,(3*d(0,0))/2,0,0,0,0),(0,( - d(3,1))/2,d(3,2),2*d(0,0),0,0,0),(( - d(3,1))/2,( - (8*d(4,1)*a + 3*d(3,1)*a** 2 - 4*d(3,1)))/(8*a),(4*d(4,2) - 16*d(4,1)*a - 2*d(3,2)*a + d(3,1)*a**2)/(5*a**2 ),d(3,2),(5*d(0,0))/2,0,0),(( - (8*d(4,1)*a + 3*d(3,1)*a**2 - 4*d(3,1)))/(8*a),( - (576*d(5,1)*a**2 + 80*d(4,1)*a**3 - 320*d(4,1)*a + 91*d(3,1)*a**4 - 192*d(3,1 )*a**2 + 80*d(3,1)))/(384*a**2),((55*a**2 + 64)*d(4,1)*a - 16*d(4,2) - 100*d(5,1 )*a**2 + 20*d(5,2)*a)/(25*a**3),(4*d(4,2) - 16*d(4,1)*a - 2*d(3,2)*a + d(3,1)*a **2)/(5*a**2),d(3,2),3*d(0,0),0),((384*d(6,0)*a**2 - 960*d(5,1)*a**2 - 464*d(4,1 )*a**3 + 320*d(4,1)*a - 91*d(3,1)*a**4 - 192*d(3,1)*a**2 - 80*d(3,1))/(384*a**2) ,( - (15360*d(6,1)*a**3 + 960*d(6,0)*a**4 - 3840*d(6,0)*a**2 - 5440*d(5,1)*a**4 - 8960*d(5,1)*a**2 + 4480*d(4,1)*a**5 - 3200*d(4,1)*a**3 + 2560*d(4,1)*a + 391*d (3,1)*a**6 - 2148*d(3,1)*a**4 + 5008*d(3,1)*a**2 - 448*d(3,1)))/(7680*a**3),( - ((1300*d(4,1)*a**4 + 640*d(4,1)*a**2 + 512*d(4,1) - 375*d(3,1)*a**5 - 500*d(3,1) *a**3)*a - 128*d(4,2) - 40*(31*a**2 + 4)*(5*a**2 + 4)*d(5,1) + 960*d(5,2)*a + 1000*d(6,0)*a**4 + 960*(5*a**2 + 4)*d(6,1)*a - 640*d(6,2)))/(1000*a**4),(40*d(5, 2)*a - 200*d(5,1)*a**2 - 32*d(4,2) + 110*d(4,1)*a**3 + 128*d(4,1)*a - 25*d(3,1)* a**4)/(50*a**3),(4*d(4,2) - 16*d(4,1)*a - 2*d(3,2)*a + d(3,1)*a**2)/(5*a**2),d(3 ,2),(7*d(0,0))/2))$ $ PP:= - (a + 2)*(a - 2) mat((1,--------------------,-1,0,0,0,0), 8*a (0,1,0,0,0,0,0), 2 5*a - 4 (0,----------,1,0,0,0,0), 8*a 2 2 2 5*a - 4 - (5*a - 4)*(a + 2)*(a - 2) - (5*a - 4) (----------,-------------------------------,---------------,1,0,0,0), 8*a 2 4*a 96*a 2 2 2 2 (5*a + 4)*(5*a - 4) - (9*a + 4)*(5*a - 4)*(a + 2)*(a - 2) (-----------------------,------------------------------------------, 2 3 96*a 1536*a 2 2 2 - (5*a - 4)*(a + 4) - (3*a - 4) ------------------------,---------------,1,0,0), 2 4*a 32*a 2 2 - (41*a + 4)*(5*a - 4)*(a + 2)*(a - 2) (-------------------------------------------, 3 1536*a 4 2 2 (57*a - 288*a - 16)*(5*a - 4)*(a + 2)*(a - 2) --------------------------------------------------, 4 30720*a 4 2 2 2 2 (25*a - 64*a - 16)*(5*a - 4) - (3*a + 4)*(3*a - 4) ---------------------------------,--------------------------, 3 2 384*a 32*a - (a + 2)*(a - 2) --------------------,1,0), 4*a 6 4 2 2 - (521*a + 324*a - 1936*a - 64)*(5*a - 4) (------------------------------------------------, 4 30720*a 6 4 2 2 (1137*a - 60*a - 3088*a - 64)*(5*a - 4)*(a + 2)*(a - 2) -------------------------------------------------------------, 5 737280*a 6 4 2 2 (97*a + 516*a - 1040*a - 64)*(5*a - 4) --------------------------------------------, 4 6144*a 4 2 2 4 2 2 (45*a - 112*a - 16)*(3*a - 4) - (25*a - 32*a - 16) a + 4 ----------------------------------,-------------------------,--------,1)) 3 2 4*a 384*a 32*a avec PP:=P*Q:= - (a + 2)*(a - 2) mat((1,--------------------,-1,0,0,0,0), 8*a (0,1,0,0,0,0,0), 2 5*a - 4 (0,----------,1,0,0,0,0), 8*a 2 2 2 5*a - 4 - (5*a - 4)*(a + 2)*(a - 2) - (5*a - 4) (----------,-------------------------------,---------------,1,0,0,0), 8*a 2 4*a 96*a 2 2 2 2 (5*a + 4)*(5*a - 4) - (9*a + 4)*(5*a - 4)*(a + 2)*(a - 2) (-----------------------,------------------------------------------, 2 3 96*a 1536*a 2 2 2 - (5*a - 4)*(a + 4) - (3*a - 4) ------------------------,---------------,1,0,0), 2 4*a 32*a 2 2 - (41*a + 4)*(5*a - 4)*(a + 2)*(a - 2) (-------------------------------------------, 3 1536*a 4 2 2 (57*a - 288*a - 16)*(5*a - 4)*(a + 2)*(a - 2) --------------------------------------------------, 4 30720*a 4 2 2 2 2 (25*a - 64*a - 16)*(5*a - 4) - (3*a + 4)*(3*a - 4) ---------------------------------,--------------------------, 3 2 384*a 32*a - (a + 2)*(a - 2) --------------------,1,0), 4*a 6 4 2 2 - (521*a + 324*a - 1936*a - 64)*(5*a - 4) (------------------------------------------------, 4 30720*a 6 4 2 2 (1137*a - 60*a - 3088*a - 64)*(5*a - 4)*(a + 2)*(a - 2) -------------------------------------------------------------, 5 737280*a 6 4 2 2 (97*a + 516*a - 1040*a - 64)*(5*a - 4) --------------------------------------------, 4 6144*a 4 2 2 4 2 2 (45*a - 112*a - 16)*(3*a - 4) - (25*a - 32*a - 16) a + 4 ----------------------------------,-------------------------,--------,1)) 3 2 4*a 384*a 32*a MATDDIAGONALISE:= mat((d(0,0),0,0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 3*d(0,0) (0,0,----------,0,0,0,0), 2 - d(3,1) (0,-----------,d(3,2),2*d(0,0),0,0,0), 2 2 - d(3,1) - (8*d(4,1)*a + 3*d(3,1)*a - 4*d(3,1)) (-----------,------------------------------------------, 2 8*a - ((2*d(3,2) - d(3,1)*a + 16*d(4,1))*a - 4*d(4,2)) 5*d(0,0) -----------------------------------------------------,d(3,2),----------,0,0 2 2 5*a ), 2 - (8*d(4,1)*a + 3*d(3,1)*a - 4*d(3,1)) 2 (------------------------------------------,( - (576*d(5,1)*a 8*a 3 4 2 + 80*d(4,1)*a - 320*d(4,1)*a + 91*d(3,1)*a - 192*d(3,1)*a 2 + 80*d(3,1)))/(384*a ), 2 2 (55*a + 64)*d(4,1)*a - 16*d(4,2) - 100*d(5,1)*a + 20*d(5,2)*a -----------------------------------------------------------------, 3 25*a - ((2*d(3,2) - d(3,1)*a + 16*d(4,1))*a - 4*d(4,2)) -----------------------------------------------------,d(3,2),3*d(0,0),0), 2 5*a 2 2 2 (( - (16*(29*a - 20)*d(4,1)*a + (13*a + 20)*(7*a + 4)*d(3,1) 2 2 2 3 + 960*d(5,1)*a - 384*d(6,0)*a ))/(384*a ),( - (15360*d(6,1)*a 4 2 4 2 + 960*d(6,0)*a - 3840*d(6,0)*a - 5440*d(5,1)*a - 8960*d(5,1)*a 5 3 6 + 4480*d(4,1)*a - 3200*d(4,1)*a + 2560*d(4,1)*a + 391*d(3,1)*a 4 2 3 - 2148*d(3,1)*a + 5008*d(3,1)*a - 448*d(3,1)))/(7680*a ),( - (( 4 2 5 1300*d(4,1)*a + 640*d(4,1)*a + 512*d(4,1) - 375*d(3,1)*a 3 - 500*d(3,1)*a )*a - 128*d(4,2) 2 2 4 - 40*(31*a + 4)*(5*a + 4)*d(5,1) + 960*d(5,2)*a + 1000*d(6,0)*a 2 4 + 960*(5*a + 4)*d(6,1)*a - 640*d(6,2)))/(1000*a ),( 2 3 (110*d(4,1)*a + 128*d(4,1) - 25*d(3,1)*a )*a - 32*d(4,2) 2 3 - 200*d(5,1)*a + 40*d(5,2)*a)/(50*a ), - ((2*d(3,2) - d(3,1)*a + 16*d(4,1))*a - 4*d(4,2)) 7*d(0,0) -----------------------------------------------------,d(3,2),----------)) 2 2 5*a on voit apparaitre les poids sur la diagonale *** r declared operator r(1) := d(0,0) d(0,0) r(2) := -------- 2 3*d(0,0) r(3) := ---------- 2 r(4) := 2*d(0,0) 5*d(0,0) r(5) := ---------- 2 r(6) := 3*d(0,0) 7*d(0,0) r(7) := ---------- 2 r(1) := 2*gamma1 r(2) := gamma1 r(3) := 3*gamma1 r(4) := 4*gamma1 r(5) := 5*gamma1 r(6) := 6*gamma1 r(7) := 7*gamma1 Le systeme de poids est le systeme 1.1 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},0}, {{0,2},x(6) + x(5)*a + x(4)}, {{0,3},x(6)*a + x(5)}, {{0,4},x(6)}, {{0,5},0}, {{0,6},0}, {{1,2},x(3)}, {{1,3},x(4)}, {{1,4},x(5)}, {{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}} 8 6 4 2 diaY(1):=( - 2605*x(6)*a + 464*x(6)*a + 10976*x(6)*a - 7424*x(6)*a 7 5 3 - 256*x(6) - 4100*x(5)*a + 19280*x(5)*a - 11200*x(5)*a 6 2 5 - 1280*x(5)*a + 8000*x(4)*a - 5120*x(4)*a + 19200*x(3)*a 3 4 4 - 15360*x(3)*a + 30720*x(0)*a )/(30720*a ) 10 8 6 4 diaY(2):=(5685*x(6)*a - 27588*x(6)*a + 4192*x(6)*a + 72832*x(6)*a 2 9 7 - 47872*x(6)*a - 1024*x(6) + 6840*x(5)*a - 67392*x(5)*a 5 3 8 + 185856*x(5)*a - 101376*x(5)*a - 6144*x(5)*a - 21600*x(4)*a 6 4 2 7 + 94080*x(4)*a - 23040*x(4)*a - 30720*x(4)*a - 38400*x(3)*a 5 3 6 4 + 184320*x(3)*a - 122880*x(3)*a + 460800*x(2)*a - 368640*x(2)*a 5 6 4 5 + 737280*x(1)*a - 92160*x(0)*a + 368640*x(0)*a )/(737280*a ) 8 6 4 2 diaY(3):=(485*x(6)*a + 2192*x(6)*a - 7264*x(6)*a + 3840*x(6)*a + 256*x(6) 7 5 3 + 2000*x(5)*a - 6720*x(5)*a + 2816*x(5)*a + 1024*x(5)*a 6 4 2 5 - 960*x(4)*a - 3072*x(4)*a + 3072*x(4)*a - 7680*x(3)*a 3 4 4 4 + 6144*x(3)*a + 6144*x(2)*a - 6144*x(0)*a )/(6144*a ) 6 4 2 5 diaY(4):=(135*x(6)*a - 516*x(6)*a + 400*x(6)*a + 64*x(6) - 108*x(5)*a 4 2 3 3 + 192*x(5)*a - 288*x(4)*a + 384*x(4)*a + 384*x(3)*a )/(384*a ) 4 2 3 diaY(5):=( - 25*x(6)*a + 32*x(6)*a + 16*x(6) - 8*x(5)*a + 32*x(5)*a 2 2 + 32*x(4)*a )/(32*a ) 2 x(6)*a + 4*x(6) + 4*x(5)*a diaY(6):=----------------------------- 4*a diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},0}, {{1,3},diay(5)}, {{1,4},diay(6)}, {{1,5},diay(7)}, {{1,6},0}, {{1,7},0}, {{2,3},diay(4)}, {{2,4},diay(5)}, {{2,5},diay(6)}, {{2,6},diay(7)}, {{2,7},0}, {{3,4}, - diay(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}}$ Now we make explicit the isomorphism with an algebra of the book:$ Namely g_{7,1.1}$ (v)$ 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((0,1,0,0,0,0,0),(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,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)$ ZZ(3):=diay(3)/a$ ZZ(4):=diay(4)/a$ ZZ(5):=diay(5)/a$ ZZ(6):=diay(6)/a$ ZZ(7):=diay(7)/a$ listcommutateursdesZZ:=$ {{1,2},0}$ {{1,3},zz(4)}$ {{1,4},zz(5)}$ {{1,5},zz(6)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(5)}$ {{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_{7,1.1}$ (v)$ Et cela pour a:=a$ and that for a neq {0}$ shortformdelta:={1,ss,a,ss,1}$ 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),(0,a,1,0,0,0),(0,1,a ,1,0,0))$ $ The isomorphism from g_{7,1.1} to gtildedelta$ was constructed in 2 steps and is given by$ the product matrix P*isom:= mat((( - (a + 2)*(a - 2))/(8*a),1,( - 1)/a,0,0,0,0),(1,0,0,0,0,0,0),((5*a**2 - 4 )/(8*a),0,1/a,0,0,0,0),(( - (5*a**2 - 4)*(a + 2)*(a - 2))/(96*a**2),(5*a**2 - 4) /(8*a),( - (5*a**2 - 4))/(4*a**2),1/a,0,0,0),(( - (9*a**2 + 4)*(5*a**2 - 4)*(a + 2)*(a - 2))/(1536*a**3),((5*a**2 + 4)*(5*a**2 - 4))/(96*a**2),( - (5*a**2 - 4)* (a**2 + 4))/(32*a**3),( - (3*a**2 - 4))/(4*a**2),1/a,0,0),(((57*a**4 - 288*a**2 - 16)*(5*a**2 - 4)*(a + 2)*(a - 2))/(30720*a**4),( - (41*a**2 + 4)*(5*a**2 - 4)* (a + 2)*(a - 2))/(1536*a**3),((25*a**4 - 64*a**2 - 16)*(5*a**2 - 4))/(384*a**4), ( - (3*a**2 + 4)*(3*a**2 - 4))/(32*a**3),( - (a + 2)*(a - 2))/(4*a**2),1/a,0),(( (1137*a**6 - 60*a**4 - 3088*a**2 - 64)*(5*a**2 - 4)*(a + 2)*(a - 2))/(737280*a** 5),( - (521*a**6 + 324*a**4 - 1936*a**2 - 64)*(5*a**2 - 4))/(30720*a**4),((97*a **6 + 516*a**4 - 1040*a**2 - 64)*(5*a**2 - 4))/(6144*a**5),((45*a**4 - 112*a**2 - 16)*(3*a**2 - 4))/(384*a**4),( - (25*a**4 - 32*a**2 - 16))/(32*a**3),(a**2 + 4 )/(4*a**2),1/a))$ $ which we record here under the name PSI$