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),2*xi(1,1),0,0 ,0),(xi(4,1),xi(4,2), - xi(2,1),xi(2,2) + xi(1,1),0,0),(xi(5,1),xi(5,2),xi(5,3), xi(4,2) - xi(3,1),xi(2,2) + 2*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),0,0,xi(6 ,6)))$ $ [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 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx2 := [ ] [-1 0 0 0 0 0] [ ] [0 0 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] adx3 := [ ] [0 0 0 0 0 0] [ ] [0 -1 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=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 - xi(2,1) 0 0 0 ] [ ] [ 0 0 xi(5,3) - xi(3,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,3), xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,3},{5,6},{6,1},{6,2},{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,-1,0,0,0),(0,0,0,0,0,1),(0,a, 1,0,0,0))$ $ shortformdelta:={1, ss, 0, 0, ss, 0, 1, ss, 0, a, 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(3,1) + d(2 ,0))$ Unknowns: {d(4,2),d(3,1),d(2,0)} Unknowns: {d(4,2),d(3,1),d(2,0)} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:= - (d(3,1) + d(2,0))$ on resout l'equation {{0,1},5} qui est maintenant AA:=d(6,1) - d(5,2) - d(4,0)$ Unknowns: {d(6,1),d(5,2),d(4,0)} Unknowns: {d(6,1),d(5,2),d(4,0)} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(5,2) + d(4,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,2) + d(3,1) + d(2, 1)*a$ Unknowns: {d(6,2),d(3,1),d(2,1),a} Unknowns: {d(6,2),d(3,1),d(2,1),a} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=d(3,1) + d(2,1)*a$ on resout l'equation {{0,2},0} qui est maintenant AA:= - d(0,6)*a$ Unknowns: {d(0,6),a} Unknowns: {d(0,6),a} pas de selection possible de variable a coefficient numerique dans - d(0,6)*a on resout l'equation {{0,2},1} qui est maintenant AA:= - d(1,6)*a$ Unknowns: {d(1,6),a} Unknowns: {d(1,6),a} pas de selection possible de variable a coefficient numerique dans - d(1,6)*a on resout l'equation {{0,2},2} qui est maintenant AA:= - d(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 {{0,2},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 {{0,2},4} 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 {{0,2},5} qui est maintenant AA:= - d(5,6)*a + d(3,1) - d( 3,0) + d(2,1)*a$ Unknowns: {d(5,6),d(3,1),d(3,0),d(2,1),a} Unknowns: {d(5,6),d(3,1),d(3,0),d(2,1),a} bonne inconnue W:=d(3,1)$ sa valeur doit etre WW:=d(5,6)*a + d(3,0) - d(2,1)*a$ on resout l'equation {{0,2},6} qui est maintenant AA:=a*( - d(6,6) + d(1,1) + 2 *d(0,0))$ Unknowns: {d(6,6),d(1,1),d(0,0),a} Unknowns: {d(6,6),d(1,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(6,6) + d(1,1) + 2*d(0,0)) on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,6) + d(0,4)$ Unknowns: {d(0,6),d(0,4)} Unknowns: {d(0,6),d(0,4)} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:=d(0,4)$ on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,6) + d(1,4)$ Unknowns: {d(1,6),d(1,4)} Unknowns: {d(1,6),d(1,4)} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:=d(1,4)$ on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,6) + d(2,4) + d(1, 3)$ Unknowns: {d(2,6),d(2,4),d(1,3)} Unknowns: {d(2,6),d(2,4),d(1,3)} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:=d(2,4) + d(1,3)$ on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,6) + d(3,4)$ Unknowns: {d(3,6),d(3,4)} Unknowns: {d(3,6),d(3,4)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(3,4)$ on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,6) + d(4,4) - d(3, 3) - d(0,0)$ Unknowns: {d(4,6),d(4,4),d(3,3),d(0,0)} Unknowns: {d(4,6),d(4,4),d(3,3),d(0,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=d(4,4) - d(3,3) - d(0,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:=d(6,3) - d(5,6) + d(5,4) + d(2,0)$ Unknowns: {d(6,3),d(5,6),d(5,4),d(2,0)} Unknowns: {d(6,3),d(5,6),d(5,4),d(2,0)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,6) - d(5,4) - d(2,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6) + d(6,4) + d(3, 3) + d(2,3)*a + d(0,0)$ Unknowns: {d(6,6),d(6,4),d(3,3),d(2,3),d(0,0),a} Unknowns: {d(6,6),d(6,4),d(3,3),d(2,3),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(6,4) + d(3,3) + d(2,3)*a + d(0,0)$ on resout l'equation {{0,4},2} qui est maintenant AA:=d(1,4)$ Unknown: d(1,4) Unknown: d(1,4) bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},4} qui est maintenant AA:= - d(3,4)$ Unknown: d(3,4) Unknown: d(3,4) bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(6,4) + d(4,4)*a - d(3,3 )*a - d(0,0)*a$ Unknowns: {d(6,4),d(4,4),d(3,3),d(0,0),a} Unknowns: {d(6,4),d(4,4),d(3,3),d(0,0),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=a*( - d(4,4) + d(3,3) + d(0,0))$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(2,4)*a$ Unknowns: {d(2,4),a} Unknowns: {d(2,4),a} pas de selection possible de variable a coefficient numerique dans d(2,4)*a 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},4} 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},5} qui est maintenant AA:=d(6,5)$ Unknown: d(6,5) Unknown: d(6,5) bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},6} 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 {{0,6},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 {{0,6},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 {{0,6},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 {{0,6},5} qui est maintenant AA:= - d(5,5) - d(4,4)*a + d( 3,3)*a + d(3,3) + d(2,3)*a + d(0,0)*a + 2*d(0,0)$ Unknowns: {d(5,5),d(4,4),d(3,3),d(2,3),d(0,0),a} Unknowns: {d(5,5),d(4,4),d(3,3),d(2,3),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(4,4)*a + d(3,3)*a + d(3,3) + d(2,3)*a + d(0,0)*a + 2*d(0,0)$ on resout l'equation {{0,6},6} qui est maintenant AA:=a*(d(2,4) + d(1,3))$ Unknowns: {d(2,4),d(1,3),a} Unknowns: {d(2,4),d(1,3),a} pas de selection possible de variable a coefficient numerique dans a*(d(2,4) + d (1,3)) 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},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},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:= - 2*d(5,6)*a - d(5,4) - 2*d(3,0) + 2*d(2,1)*a - d(2,0)$ Unknowns: {d(5,6),d(5,4),d(3,0),d(2,1),d(2,0),a} Unknowns: {d(5,6),d(5,4),d(3,0),d(2,1),d(2,0),a} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - 2*d(5,6)*a - 2*d(3,0) + 2*d(2,1)*a - d(2,0)$ on resout l'equation {{1,2},6} qui est maintenant AA:=a*( - d(3,3) + 2*d(1,1) + d(0,1))$ Unknowns: {d(3,3),d(1,1),d(0,1),a} Unknowns: {d(3,3),d(1,1),d(0,1),a} pas de selection possible de variable a coefficient numerique dans a*( - d(3,3) + 2*d(1,1) + d(0,1)) 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) - d(0,1)$ Unknowns: {d(2,3),d(0,1)} Unknowns: {d(2,3),d(0,1)} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,3},5} qui est maintenant AA:=d(4,3) + d(2,1)$ Unknowns: {d(4,3),d(2,1)} Unknowns: {d(4,3),d(2,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(2,1)$ on resout l'equation {{1,3},6} qui est maintenant AA:=d(0,1)$ Unknown: d(0,1) Unknown: d(0,1) bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,4},5} qui est maintenant AA:= - d(3,3)*a - d(3,3) + 2* d(1,1)*a + 3*d(1,1) - d(0,0)$ Unknowns: {d(3,3),d(1,1),d(0,0),a} Unknowns: {d(3,3),d(1,1),d(0,0),a} bonne inconnue W:=d(0,0)$ sa valeur doit etre WW:= - d(3,3)*a - d(3,3) + 2*d(1,1)*a + 3*d(1,1)$ on resout l'equation {{1,6},4} 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 {{1,6},5} qui est maintenant AA:= - d(3,3) + 2*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:=2*d(1,1)$ Derivation equations to cancel (Reduce output) : \\{{{{0,1},0},0}, {{{0,1},1},0}, {{{0,1},2},0}, {{{0,1},3},0}, {{{0,1},4},0}, {{{0,1},5},0}, {{{0,1},6},0}, {{{0,2},0},0}, {{{0,2},1},0}, {{{0,2},2},0}, {{{0,2},3},0}, {{{0,2},4},0}, {{{0,2},5},0}, {{{0,2},6},0}, {{{0,3},0},0}, {{{0,3},1},0}, {{{0,3},2},0}, {{{0,3},3},0}, {{{0,3},4},0}, {{{0,3},5},0}, {{{0,3},6},0}, {{{0,4},2},0}, {{{0,4},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},2},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},0},0}, {{{0,6},1},0}, {{{0,6},2},0}, {{{0,6},3},0}, {{{0,6},4},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},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},2},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,6},2},0}, {{{1,6},4},0}, {{{1,6},5},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},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}, {{{4,6},5},0}, {{{5,6},5},0}}$ Il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((d(1,1),0,0,0,0,0,0),(0,d(1,1),0,0,0,0,0),(d(2,0),d(2,1),2*d(1,1),0,0,0,0),( d(3,0),d(5,6)*a + d(3,0) - d(2,1)*a,0,2*d(1,1),0,0,0),(d(4,0),d(4,1), - d(5,6)*a - d(3,0) + d(2,1)*a - d(2,0), - d(2,1),3*d(1,1),0,0),(d(5,0),d(5,1),d(5,2),d(5, 3), - 2*d(5,6)*a - 2*d(3,0) + 2*d(2,1)*a - d(2,0),4*d(1,1),d(5,6)),(d(6,0),d(5,2 ) + d(4,0),d(5,6)*a + d(3,0),2*d(5,6)*a + d(5,6) + 2*d(3,0) - 2*d(2,1)*a,0,0,3*d (1,1)))$ $ pour delta:= [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 0 0 0 1] [ ] [0 a 1 0 0 0] pour shortformdelta:={1, ss, 0, 0, ss, 0, 1, ss, 0, a, 1} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,0), d(2,1), d(2,0), d(1,1), a} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,0), d(2,1), d(2,0), d(1,1), a} listeparametresMATD{d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,0), d(2,1), d(2,0), d(1,1)}$ dim Der(gtildedelta):=12$ t1:=D(1,1):= [1 0 0 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 2 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 3 0 0] [ ] [0 0 0 0 0 4 0] [ ] [0 0 0 0 0 0 3] Unknown: d(1,1) Unknown: d(1,1) commutant de t1 dans der(gtildedelta): [d(1,1) 0 0 0 0 0 0 ] [ ] [ 0 d(1,1) 0 0 0 0 0 ] [ ] [ 0 0 2*d(1,1) 0 0 0 0 ] [ ] [ 0 0 0 2*d(1,1) 0 0 0 ] [ ] [ 0 0 0 0 3*d(1,1) 0 0 ] [ ] [ 0 0 0 0 0 4*d(1,1) 0 ] [ ] [ 0 0 0 0 0 0 3*d(1,1)] 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] [ ] [0 1 0 0 0 0 0] [ ] [0 0 2 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 3 0 0] [ ] [0 0 0 0 0 4 0] [ ] [0 0 0 0 0 0 3] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((d(1,1),0,0,0,0,0,0),(0,d(1,1),0,0,0,0,0),(d(2,0),d(2,1),2*d(1,1),0,0,0,0),( d(3,0),d(3,0) - d(2,1)*a,0,2*d(1,1),0,0,0),(d(4,0),d(4,1), - d(3,0) + d(2,1)*a - d(2,0), - d(2,1),3*d(1,1),0,0),(d(5,0),d(5,1),d(5,2),d(5,3), - 2*d(3,0) + 2*d(2 ,1)*a - d(2,0),4*d(1,1),d(5,6)),(d(6,0),d(5,2) + d(4,0),d(3,0),d(5,6) + 2*d(3,0) - 2*d(2,1)*a,0,0,3*d(1,1)))$ $ 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(1,1),0,0,0,0,0,0), (0,d(1,1),0,0,0,0,0), (d(2,0),d(2,1),2*d(1,1),0,0,0,0), (d(3,0),d(3,0) - d(2,1)*a,0,2*d(1,1),0,0,0), (d(4,0),d(4,1), - d(3,0) + d(2,1)*a - d(2,0), - d(2,1),3*d(1,1),0,0), (d(5,0),d(5,1),d(5,2),d(5,3), - 2*d(3,0) + 2*d(2,1)*a - d(2,0),4*d(1,1), d(5,6)), (d(6,0),d(5,2) + d(4,0),d(3,0),d(5,6) + 2*d(3,0) - 2*d(2,1)*a,0,0,3*d(1,1))) on voit apparaitre les poids sur la diagonale r(1) := d(1,1) r(2) := d(1,1) r(3) := 2*d(1,1) r(4) := 2*d(1,1) r(5) := 3*d(1,1) r(6) := 4*d(1,1) r(7) := 3*d(1,1) r(1) := gamma2 r(2) := gamma2 r(3) := 2*gamma2 r(4) := 2*gamma2 r(5) := 3*gamma2 r(6) := 4*gamma2 r(7) := 3*gamma2 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(2)}, {{0,2},a*x(6)}, {{0,3},x(6) - x(4)}, {{0,4},0}, {{0,5},0}, {{0,6},x(5)}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},0}, {{1,6},0}, {{2,3},x(5)}, {{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(3)}, {{1,3},diay(7)*a}, {{1,4},diay(7) - diay(5)}, {{1,5},0}, {{1,6},0}, {{1,7},diay(6)}, {{2,3},diay(5)}, {{2,4},0}, {{2,5},diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},diay(6)}, {{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_{7,1.2}$ (iL)$ and that for a neq{0,-1}$ 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 isom computed for parama neq 0,-1 by calculisom6_53xCIII.red$ mat((1,( - sqrt(a + 1)*(sqrt(a**2 + a) + a))/sqrt(a**2 + a),0,0,0,0,0),(( - a*( sqrt(a**2 + a) + a + 1))/sqrt(a**2 + a),( - sqrt(a + 1)*(sqrt(a**2 + a) + a))/( sqrt(a**2 + a) + a + 1),0,0,0,0,0),(0,0,(2*a*( - 2*sqrt(a**2 + a)*a - 2*sqrt(a** 2 + a) - 2*a**2 - 3*a - 1))/sqrt(a**2 + a),(2*sqrt(a + 1)*( - 2*sqrt(a**2 + a)*a - sqrt(a**2 + a) - 2*a**2 - 2*a))/(sqrt(a**2 + a) + a + 1),0,0,0),(0,0,4*a*(2* sqrt(a**2 + a)*a + sqrt(a**2 + a) + 2*a**2 + 2*a),0,0,0,0),(0,0,0,0,(2*sqrt(a + 1)*a*(4*sqrt(a**2 + a)*a + 3*sqrt(a**2 + a) + 4*a**2 + 5*a + 1))/(sqrt(a**2 + a) + a + 1),2*a*(sqrt(a**2 + a) + a + 1),0),(0,0,0,0,0,0,(4*sqrt(a + 1)*a*( - 4* sqrt(a**2 + a)*a**2 - 5*sqrt(a**2 + a)*a - sqrt(a**2 + a) - 4*a**3 - 7*a**2 - 3* a))/(sqrt(a**2 + a) + a + 1)),(0,0,0,0,(2*sqrt(a + 1)*a*( - 2*sqrt(a**2 + a)*a - sqrt(a**2 + a) - 2*a**2 - 2*a))/(sqrt(a**2 + a) + a + 1),(2*a**2*(2*sqrt(a**2 + a)*a + 2*sqrt(a**2 + a) + 2*a**2 + 3*a + 1))/sqrt(a**2 + a),0))$ $ det(isom):= ( - 512*(8192*sqrt(a**2 + a)*a**6 + 24576*sqrt(a**2 + a)*a**5 + 28160*sqrt(a**2 + a)*a**4 + 15360*sqrt(a**2 + a)*a**3 + 4032*sqrt(a**2 + a)*a**2 + 448*sqrt(a**2 + a)*a + 14*sqrt(a**2 + a) + 8192*a**7 + 28672*a**6 + 39424*a** 5 + 26880*a**4 + 9408*a**3 + 1568*a**2 + 98*a + 1)*(a + 1)**4*a**7)/(16*sqrt(a** 2 + a)*a**2 + 12*sqrt(a**2 + a)*a + sqrt(a**2 + a) + 16*a**3 + 20*a**2 + 5*a)$ That the determinant is not zero doesn t appear too nicely on that expression.$ !We! refer! to! the! end! of! the! file! calculisom6_53x!C!I!I!I.red! and! its! output! rcalcul\ isom6_53x!C!I!I!I.r$ There it is written with b(1,1)=1 and b(2,1) the entry 2x1 of isom :$ Quote$ That the determinant is not zero doesn t appear too nicely on that expression.$ However keep in mind that it is simply :$ det(isom):=((b(2,1)**2! +! b(1,1)**2*parama)**7*(b(2,1)! +! b(1,1)*parama)*(l! - ! 1)\ ** 2*b(2,2)**4)/(b(1,1)**3*parama**4)$ Then that it is non zero results from b(2,1):=Q and b(1,1)=1.$ Indeed, Q**2 +a=0 would imply by E237: (Q+a)*Q**2*a=0 ,$ hence Q+a=0 since Q neq 0$ Now Q+a=0 reads L+1=L-1-2*a , that is a=-1 which is excluded.$ Another possibility is to let Reduce compute the roots of the numerator $ num:=num(det(isom)):=512*a**7*( - 8192*sqrt(a**2 + a)*a**14 - 90112*sqrt(a**2 + a)*a**13 - 454144*sqrt(a**2 + a)*a**12 - 1387520*sqrt(a**2 + a)*a**11 - 2865088* sqrt(a**2 + a)*a**10 - 4218816*sqrt(a**2 + a)*a**9 - 4553486*sqrt(a**2 + a)*a**8 - 3644272*sqrt(a**2 + a)*a**7 - 2161160*sqrt(a**2 + a)*a**6 - 937872*sqrt(a**2 + a)*a**5 - 290004*sqrt(a**2 + a)*a**4 - 60944*sqrt(a**2 + a)*a**3 - 8008*sqrt(a **2 + a)*a**2 - 560*sqrt(a**2 + a)*a - 14*sqrt(a**2 + a) - 8192*a**15 - 94208*a **14 - 498176*a**13 - 1603840*a**12 - 3507392*a**11 - 5503008*a**10 - 6376034*a **9 - 5529233*a**8 - 3595840*a**7 - 1738828*a**6 - 612612*a**5 - 151606*a**4 - 24752*a**3 - 2380*a**2 - 106*a - 1)$ den:=den(det(isom)):=16*sqrt(a**2 + a)*a**6 + 76*sqrt(a**2 + a)*a**5 + 145*sqrt( a**2 + a)*a**4 + 140*sqrt(a**2 + a)*a**3 + 70*sqrt(a**2 + a)*a**2 + 16*sqrt(a**2 + a)*a + sqrt(a**2 + a) + 16*a**7 + 84*a**6 + 181*a**5 + 204*a**4 + 126*a**3 + 40*a**2 + 5*a$ solve(num=0,parama):={}$ solve(den=0,parama):={a=-1,a=0}$ End Quote$ ZZ(1):=( - ((a + 1 + sqrt(a**2 + a))*diay(2)*a - sqrt(a**2 + a)*diay(1)))/sqrt(a **2 + a)$ ZZ(2):=( - sqrt(a + 1)*((a + 1 + sqrt(a**2 + a))*diay(1) + sqrt(a**2 + a)*diay(2 ))*(sqrt(a**2 + a) + a))/(sqrt(a**2 + a)*(a + 1 + sqrt(a**2 + a)))$ ZZ(3):=(2*(2*diay(4)*a - diay(3))*(2*sqrt(a**2 + a) + 2*a + 1)*(a + 1)*a)/sqrt(a **2 + a)$ ZZ(4):=( - 2*sqrt(a + 1)*(sqrt(a**2 + a)*(2*a + 1) + 2*(a + 1)*a)*diay(3))/(a + 1 + sqrt(a**2 + a))$ ZZ(5):=(2*sqrt(a + 1)*((sqrt(a**2 + a)*(4*a + 3) + (4*a + 1)*(a + 1))*diay(5) - (sqrt(a**2 + a)*(2*a + 1) + 2*(a + 1)*a)*diay(7))*a)/(a + 1 + sqrt(a**2 + a))$ ZZ(6):=(2*(2*sqrt(a**2 + a)*diay(7)*a + 2*diay(7)*a**2 + diay(7)*a + sqrt(a**2 + a)*diay(5) + diay(5)*a)*(a + 1)*a)/sqrt(a**2 + a)$ ZZ(7):=( - 4*sqrt(a + 1)*((4*a + 3)*a + sqrt(a**2 + a)*(4*a + 1))*(a + 1)*diay(6 )*a)/(a + 1 + sqrt(a**2 + a))$ 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}, (sqrt(a**2 + a)*zz(5)*(64*a**3 + 112*a**2 + 56*a + 7) + zz(5)*(64*a**3 + 80*a**2 + 24*a + 1)*(a + 1))/(sqrt(a**2 + a)*(16*a**2 + 20*a + 5) + (16*a**2 + 12*a + 1 )*(a + 1))}$ {{2,4},zz(6)}$ {{2,5},0}$ {{2,6},zz(7)}$ {{2,7},0}$ {{3,4}, ( - 2*sqrt(a**2 + a)*zz(7)*(16*a**2 + 20*a + 5)*a - 2*zz(7)*(16*a**2 + 12*a + 1) *(a + 1)*a)/(sqrt(a**2 + a)*(8*a**2 + 8*a + 1) + 4*(2*a + 1)*(a + 1)*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_{7,1.2}$ (iL)$ with L:=2*a + 1 + 2*sqrt(a**2 + a)$ In fact let :$ h:=((64*sqrt(a**2 + a)*a**3 + 112*sqrt(a**2 + a)*a**2 + 56*sqrt(a**2 + a)*a + 7* sqrt(a**2 + a) + 64*a**4 + 144*a**3 + 104*a**2 + 25*a + 1)*zz(5))/(sqrt(a**2 + a )*(16*a**2 + 20*a + 5) + (16*a**2 + 12*a + 1)*(a + 1))$ k:=( - 2*(16*sqrt(a**2 + a)*a**2 + 20*sqrt(a**2 + a)*a + 5*sqrt(a**2 + a) + 16*a **3 + 28*a**2 + 13*a + 1)*a*zz(7))/(sqrt(a**2 + a)*(8*a**2 + 8*a + 1) + 4*(2*a + 1)*(a + 1)*a)$ Then one has simply :$ h-L*ZZ(5):=0$ k-(1-L)*ZZ(7):=0$ Et cela pour a:=a$ and that for a neq {0,-1}$ shortformdelta:={1, ss, 0, 0, ss, 0, 1, ss, 0, a, 1}$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,-1,0,0,0),(0,0,0,0,0,1),(0,a, 1,0,0,0))$ $