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),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,0,0,0,0),(0,0,a,0,0,0),(1,0,0 ,0,0,0))$ shortformdelta:={0, ss, 0, 1, ss, a, 0, ss, 1, 0, 0}$ on resout l'equation {{0,1},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 {{0,1},1} 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 {{0,1},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 {{0,1},3} qui est maintenant AA:= - d(3,6) + d(2,1)$ Unknowns: {d(3,6),d(2,1)} Unknowns: {d(3,6),d(2,1)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,6) + d(2,0))$ Unknowns: {d(4,6),d(2,0)} Unknowns: {d(4,6),d(2,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,6) - d(4,0) + d(3, 1)*a$ Unknowns: {d(5,6),d(4,0),d(3,1),a} Unknowns: {d(5,6),d(4,0),d(3,1),a} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - d(4,0) + d(3,1)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,6) + d(1,1) + d(0, 0)$ Unknowns: {d(6,6),d(1,1),d(0,0)} Unknowns: {d(6,6),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(1,1) + d(0,0)$ on resout l'equation {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},3} qui est maintenant AA:= - d(3,3) + d(2,2) + d(0, 0)$ Unknowns: {d(3,3),d(2,2),d(0,0)} Unknowns: {d(3,3),d(2,2),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,2) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,3) + d(1,0)$ Unknowns: {d(4,3),d(1,0)} Unknowns: {d(4,3),d(1,0)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,3) + d(3,2)*a - d( 3,0)$ Unknowns: {d(5,3),d(3,2),d(3,0),a} Unknowns: {d(5,3),d(3,2),d(3,0),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(3,2)*a - d(3,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,3) + d(1,2)$ Unknowns: {d(6,3),d(1,2)} Unknowns: {d(6,3),d(1,2)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(1,2)$ on resout l'equation {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},5} qui est maintenant AA:= - d(5,5)*a + d(2,2)*a + d(2,0) + 2*d(0,0)*a$ Unknowns: {d(5,5),d(2,2),d(2,0),d(0,0),a} Unknowns: {d(5,5),d(2,2),d(2,0),d(0,0),a} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=a*(d(5,5) - d(2,2) - 2*d(0,0))$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,5)*a$ Unknowns: {d(6,5),a} Unknowns: {d(6,5),a} pas de selection possible de variable a coefficient numerique dans - d(6,5)*a on resout l'equation {{0,4},3} qui est maintenant AA:=d(2,4)$ Unknown: d(2,4) Unknown: d(2,4) bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(3,4)*a + d(1,0)$ Unknowns: {d(3,4),d(1,0),a} Unknowns: {d(3,4),d(1,0),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:= - d(3,4)*a$ on resout l'equation {{0,4},6} 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 {{0,5},3} 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)*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 {{0,5},6} 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 {{0,6},5} qui est maintenant AA:=d(2,1)*a$ Unknowns: {d(2,1),a} Unknowns: {d(2,1),a} pas de selection possible de variable a coefficient numerique dans d(2,1)*a 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},3} qui est maintenant AA:= - d(3,4) + d(0,1)$ Unknowns: {d(3,4),d(0,1)} Unknowns: {d(3,4),d(0,1)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=d(0,1)$ 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(3, 1)$ Unknowns: {d(5,4),d(4,2),d(3,1)} Unknowns: {d(5,4),d(4,2),d(3,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - (d(6,4) + d(0,2))$ Unknowns: {d(6,4),d(0,2)} Unknowns: {d(6,4),d(0,2)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{1,3},5} qui est maintenant AA:=d(2,1)$ Unknown: d(2,1) Unknown: d(2,1) bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,4},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},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)$ Unknown: d(6,5) Unknown: d(6,5) bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,6},5} qui est maintenant AA:=2*a*( - d(1,1) + d(0,0))$ Unknowns: {d(1,1),d(0,0),a} Unknowns: {d(1,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans 2*a*( - d(1,1 ) + d(0,0)) on resout l'equation {{2,3},5} qui est maintenant AA:=d(2,2) - 2*d(1,1) + d(0,2 )*a + d(0,0)$ Unknowns: {d(2,2),d(1,1),d(0,2),d(0,0),a} Unknowns: {d(2,2),d(1,1),d(0,2),d(0,0),a} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=2*d(1,1) - d(0,2)*a - d(0,0)$ on resout l'equation {{2,4},5} qui est maintenant AA:=d(1,2) + d(0,1)$ Unknowns: {d(1,2),d(0,1)} Unknowns: {d(1,2),d(0,1)} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:= - d(0,1)$ 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},3},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},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},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},4},0}, {{{1,6},5},2*a*( - d(1,1) + d(0,0))}, {{{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},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{3,4},5},0}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}}$ Il y a une phase 2$ a neq {0}$ 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},3},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},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},4},0}, {{{1,5},5},0}, {{{1,5},6},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},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{3,4},5},0}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(0,1),d(0,2),0,0,0,0),( - d(0,1)*a,d(0,0), - d(0,1),0,0,0,0),(0,0, - d(0,2)*a + d(0,0),0,0,0,0),(d(3,0),d(3,1),d(3,2), - d(0,2)*a + 2*d(0,0),d(0,1) ,0,0),(d(4,0),d(4,1),d(4,2), - d(0,1)*a, - d(0,2)*a + 2*d(0,0),0,0),(d(5,0),d(5, 1),d(5,2),d(3,2)*a - d(3,0),d(4,2) - d(3,1), - d(0,2)*a + 3*d(0,0), - d(4,0) + d (3,1)*a),(d(6,0),d(6,1),d(6,2), - d(0,1), - d(0,2),0,2*d(0,0)))$ pour delta:= [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 a 0 0 0] [ ] [1 0 0 0 0 0] pour shortformdelta:={0, ss, 0, 1, ss, a, 0, ss, 1, 0, 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(4,0), d(3,2), d(3,1), d(3,0), d(0,2), d(0,1), d(0,0), a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(0,2), d(0,1), d(0,0), a} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(0,2), d(0,1), d(0,0)}$ dim Der(gtildedelta):=15$ t1:=D(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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] Unknowns: {d(0,2),d(0,1),d(0,0),a} Unknowns: {d(0,2),d(0,1),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),d(0,1),d(0,2),0,0,0,0), ( - d(0,1)*a,d(0,0), - d(0,1),0,0,0,0), (0,0, - d(0,2)*a + d(0,0),0,0,0,0), (0,0,0, - d(0,2)*a + 2*d(0,0),d(0,1),0,0), (0,0,0, - d(0,1)*a, - d(0,2)*a + 2*d(0,0),0,0), (0,0,0,0,0, - d(0,2)*a + 3*d(0,0),0), (0,0,0, - d(0,1), - d(0,2),0,2*d(0,0))) Unknowns: {d(0,2),d(0,1),d(0,0),a} Unknowns: {d(0,2),d(0,1),d(0,0),a} t2:=D(0,1):= [ 0 1 0 0 0 0 0] [ ] [ - a 0 -1 0 0 0 0] [ ] [ 0 0 0 0 0 0 0] [ ] [ 0 0 0 0 1 0 0] [ ] [ 0 0 0 - a 0 0 0] [ ] [ 0 0 0 0 0 0 0] [ ] [ 0 0 0 -1 0 0 0] 2 {{a + x , 2, [ - arbcomplex(121)*x ] [----------------------] [ a ] [ ] [ arbcomplex(121) ] [ ] [ 0 ] [ ] [ - arbcomplex(122)*x ] [ ] [ arbcomplex(122)*a ] [ ] [ 0 ] [ ] [ arbcomplex(122) ] }, {x, 3, [ - arbcomplex(123) ] [--------------------] [ a ] [ ] [ 0 ] [ ] [ arbcomplex(123) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(124) ] [ ] [ arbcomplex(125) ] }} Unknowns: {d(0,2),d(0,1),d(0,0),a} Unknowns: {d(0,2),d(0,1),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),d(0,1),d(0,2),0,0,0,0),( - d(0,1)*a,d(0,0), - d(0,1),0,0,0,0),(0,0, - d(0,2)*a + d(0,0),0,0,0,0),(0,0,0, - d(0,2)*a + 2*d(0,0),d(0,1),0,0),(0,0,0, - d(0,1)*a, - d(0,2)*a + 2*d(0,0),0,0),(0,0,0,0,0, - d(0,2)*a + 3*d(0,0),0),(0,0, 0, - d(0,1), - d(0,2),0,2*d(0,0)))$ Unknowns: {d(0,2),d(0,1),d(0,0),a} Unknowns: {d(0,2),d(0,1),d(0,0),a} t3:=D(0,2):= [0 0 1 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 - a 0 0 0 0] [ ] [0 0 0 - a 0 0 0] [ ] [0 0 0 0 - a 0 0] [ ] [0 0 0 0 0 - a 0] [ ] [0 0 0 0 -1 0 0] Unknowns: {d(0,2),d(0,1),d(0,0),a} Unknowns: {d(0,2),d(0,1),d(0,0),a} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),d(0,1),d(0,2),0,0,0,0), ( - d(0,1)*a,d(0,0), - d(0,1),0,0,0,0), (0,0, - d(0,2)*a + d(0,0),0,0,0,0), (0,0,0, - d(0,2)*a + 2*d(0,0),d(0,1),0,0), (0,0,0, - d(0,1)*a, - d(0,2)*a + 2*d(0,0),0,0), (0,0,0,0,0, - d(0,2)*a + 3*d(0,0),0), (0,0,0, - d(0,1), - d(0,2),0,2*d(0,0))) rank 3 with maximal torus t1,t2,t3 3 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:= [ - i - 1 ] [ 1 --------- ------ 0 0 0 0] [ sqrt(a) a ] [ ] [ - i*sqrt(a) 1 0 0 0 0 0] [ ] [ 0 0 1 0 0 0 0] [ ] [ i ] [ 0 0 0 1 --------- 0 0] [ sqrt(a) ] [ ] [ i*a ] [ 0 0 0 --------- 1 0 0] [ sqrt(a) ] [ ] [ 0 0 0 0 0 1 0] [ ] [ i 1 ] [ 0 0 0 --------- --- 0 1] [ sqrt(a) a ] P**(-1)*t1*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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t2*P:= [ - i*a ] [--------- 0 0 0 0 0 0] [ sqrt(a) ] [ ] [ i*a ] [ 0 --------- 0 0 0 0 0] [ sqrt(a) ] [ ] [ 0 0 0 0 0 0 0] [ ] [ 0 0 0 i*sqrt(a) 0 0 0] [ ] [ 0 0 0 0 - i*sqrt(a) 0 0] [ ] [ 0 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0 0] P**(-1)*t3*P:= [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 - a 0 0 0 0] [ ] [0 0 0 - a 0 0 0] [ ] [0 0 0 0 - a 0 0] [ ] [0 0 0 0 0 - a 0] [ ] [0 0 0 0 0 0 0] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((( - i*d(0,1)*a + sqrt(a)*d(0,0))/sqrt(a),0,0,0,0,0,0),(0,(i*d(0,1)*a + sqrt (a)*d(0,0))/sqrt(a),0,0,0,0,0),(0,0, - d(0,2)*a + d(0,0),0,0,0,0),(( - d(4,1)*a - i*sqrt(a)*d(4,0) - i*sqrt(a)*d(3,1)*a + d(3,0)*a)/(2*a),( - i*sqrt(a)*d(4,1) - d(4,0) + d(3,1)*a - i*sqrt(a)*d(3,0))/(2*a),( - i*sqrt(a)*d(4,2)*a + i*sqrt(a)* d(4,0) + d(3,2)*a**2 - d(3,0)*a)/(2*a**2), - d(0,2)*a + i*sqrt(a)*d(0,1) + 2*d(0 ,0),0,0,0),(( - i*sqrt(a)*d(4,1) + d(4,0) - d(3,1)*a - i*sqrt(a)*d(3,0))/2,(d(4, 1)*a - i*sqrt(a)*d(4,0) - i*sqrt(a)*d(3,1)*a - d(3,0)*a)/(2*a),(d(4,2)*a - d(4,0 ) - i*sqrt(a)*d(3,2)*a + i*sqrt(a)*d(3,0))/(2*a),0, - d(0,2)*a - i*sqrt(a)*d(0,1 ) + 2*d(0,0),0,0),( - i*sqrt(a)*d(5,1) + d(5,0),(sqrt(a)*d(5,1) - i*d(5,0))/sqrt (a),(d(5,2)*a - d(5,0))/a,(i*d(4,2)*a - i*d(4,0) + sqrt(a)*d(3,2)*a - sqrt(a)*d( 3,0))/sqrt(a),(sqrt(a)*d(4,2)*a - sqrt(a)*d(4,0) + i*d(3,2)*a**2 - i*d(3,0)*a)/( sqrt(a)*a), - d(0,2)*a + 3*d(0,0), - d(4,0) + d(3,1)*a),(( - i*sqrt(a)*d(6,1)*a + d(6,0)*a + i*sqrt(a)*d(4,1) - d(4,0))/a,(sqrt(a)*d(6,1)*a - i*d(6,0)*a - sqrt( a)*d(4,1) + i*d(4,0))/(sqrt(a)*a),(d(6,2)*a**2 - d(6,0)*a - d(4,2)*a + d(4,0))/a **2,0,0,0,2*d(0,0)))$ PP:= [ - i - 1 ] [ 1 --------- ------ 0 0 0 0] [ sqrt(a) a ] [ ] [ - i*sqrt(a) 1 0 0 0 0 0] [ ] [ 0 0 1 0 0 0 0] [ ] [ i ] [ 0 0 0 1 --------- 0 0] [ sqrt(a) ] [ ] [ i*a ] [ 0 0 0 --------- 1 0 0] [ sqrt(a) ] [ ] [ 0 0 0 0 0 1 0] [ ] [ i 1 ] [ 0 0 0 --------- --- 0 1] [ sqrt(a) a ] avec PP:=P*Q:= [ - i - 1 ] [ 1 --------- ------ 0 0 0 0] [ sqrt(a) a ] [ ] [ - i*sqrt(a) 1 0 0 0 0 0] [ ] [ 0 0 1 0 0 0 0] [ ] [ i ] [ 0 0 0 1 --------- 0 0] [ sqrt(a) ] [ ] [ i*a ] [ 0 0 0 --------- 1 0 0] [ sqrt(a) ] [ ] [ 0 0 0 0 0 1 0] [ ] [ i 1 ] [ 0 0 0 --------- --- 0 1] [ sqrt(a) a ] MATDDIAGONALISE:= - i*d(0,1)*a + sqrt(a)*d(0,0) mat((--------------------------------,0,0,0,0,0,0), sqrt(a) i*d(0,1)*a + sqrt(a)*d(0,0) (0,-----------------------------,0,0,0,0,0), sqrt(a) (0,0, - d(0,2)*a + d(0,0),0,0,0,0), - d(4,1)*a - i*sqrt(a)*d(4,0) - i*sqrt(a)*d(3,1)*a + d(3,0)*a (----------------------------------------------------------------, 2*a - i*sqrt(a)*d(4,1) - d(4,0) + d(3,1)*a - i*sqrt(a)*d(3,0) ------------------------------------------------------------, 2*a 2 - i*sqrt(a)*d(4,2)*a + i*sqrt(a)*d(4,0) + d(3,2)*a - d(3,0)*a -----------------------------------------------------------------, 2 2*a - d(0,2)*a + i*sqrt(a)*d(0,1) + 2*d(0,0),0,0,0), - i*sqrt(a)*d(4,1) + d(4,0) - d(3,1)*a - i*sqrt(a)*d(3,0) (------------------------------------------------------------, 2 d(4,1)*a - i*sqrt(a)*d(4,0) - i*sqrt(a)*d(3,1)*a - d(3,0)*a -------------------------------------------------------------, 2*a d(4,2)*a - d(4,0) - i*sqrt(a)*d(3,2)*a + i*sqrt(a)*d(3,0) -----------------------------------------------------------,0, 2*a - d(0,2)*a - i*sqrt(a)*d(0,1) + 2*d(0,0),0,0), sqrt(a)*d(5,1) - i*d(5,0) ( - i*sqrt(a)*d(5,1) + d(5,0),---------------------------, sqrt(a) d(5,2)*a - d(5,0) -------------------, a i*d(4,2)*a - i*d(4,0) + sqrt(a)*d(3,2)*a - sqrt(a)*d(3,0) -----------------------------------------------------------, sqrt(a) 2 sqrt(a)*d(4,2)*a - sqrt(a)*d(4,0) + i*d(3,2)*a - i*d(3,0)*a --------------------------------------------------------------, sqrt(a)*a - d(0,2)*a + 3*d(0,0), - d(4,0) + d(3,1)*a), - i*sqrt(a)*d(6,1)*a + d(6,0)*a + i*sqrt(a)*d(4,1) - d(4,0) (--------------------------------------------------------------, a sqrt(a)*d(6,1)*a - i*d(6,0)*a - sqrt(a)*d(4,1) + i*d(4,0) -----------------------------------------------------------, sqrt(a)*a 2 d(6,2)*a - d(6,0)*a - d(4,2)*a + d(4,0) ------------------------------------------,0,0,0,2*d(0,0))) 2 a on voit apparaitre les poids sur la diagonale - i*d(0,1)*a + sqrt(a)*d(0,0) r(1) := -------------------------------- sqrt(a) i*d(0,1)*a + sqrt(a)*d(0,0) r(2) := ----------------------------- sqrt(a) r(3) := - d(0,2)*a + d(0,0) r(4) := - d(0,2)*a + i*sqrt(a)*d(0,1) + 2*d(0,0) r(5) := - d(0,2)*a - i*sqrt(a)*d(0,1) + 2*d(0,0) r(6) := - d(0,2)*a + 3*d(0,0) r(7) := 2*d(0,0) r(4)-(r(2)+ r(3)) :=0 r(5)-(r(1)+ r(3)) :=0 r(1) := gamma2 r(2) := gamma1 r(3) := gamma3 gamma2 + 2*gamma3 + gamma1 + 2*i*sqrt(a)*d(0,1) r(4) := ------------------------------------------------- 2 gamma2 + 2*gamma3 + gamma1 - 2*i*sqrt(a)*d(0,1) r(5) := ------------------------------------------------- 2 r(6) := gamma2 + gamma3 + gamma1 r(7) := gamma1 + gamma2 Note that 2*i*d(0,1) is simply equal to gamma1-gamma2, hence r(4) gamma1+ gamma3 and r(5)= gamma2+ gamma3. Le systeme de poids est le systeme 3.1 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(6)}, {{0,2},x(3)}, {{0,3},a*x(5)}, {{0,4},0}, {{0,5},0}, {{0,6},0}, {{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):= - i*sqrt(a)*x(1) + x(0) sqrt(a)*x(1) - i*x(0) diaY(2):=----------------------- sqrt(a) x(2)*a - x(0) diaY(3):=--------------- a i*x(6) + i*x(4)*a + sqrt(a)*x(3) diaY(4):=---------------------------------- sqrt(a) sqrt(a)*x(6) + sqrt(a)*x(4)*a + i*x(3)*a diaY(5):=------------------------------------------ sqrt(a)*a diaY(6):=x(5) diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},2*diay(7)}, {{1,3}, - i*sqrt(a)*diay(5)}, {{1,4},2*diay(6)*a}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},( - i*diay(4))/sqrt(a)}, {{2,4},0}, {{2,5},2*diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},0}, {{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_{7,3.1}$ (iL)$ 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),(0,0,1,0,0,0,0),(0,0,0,0,( - i)/sqrt(a),0,0) ,(0,0,0,0,0, - i*sqrt(a),0),(0,0,0,0,0,0, - 2*i*sqrt(a)),(0,0,0,-2,0,0,0))$ det(isom):= ( - 4*i*a)/sqrt(a)$ ZZ(1):=diay(2)$ ZZ(2):=diay(1)$ ZZ(3):=diay(3)$ ZZ(4):= - 2*diay(7)$ ZZ(5):=( - i*diay(4))/sqrt(a)$ ZZ(6):= - i*sqrt(a)*diay(5)$ ZZ(7):= - 2*i*sqrt(a)*diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(5)}$ {{1,4},0}$ {{1,5},0}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(6)}$ {{2,4},0}$ {{2,5},zz(7)}$ {{2,6},0}$ {{2,7},0}$ {{3,4},0}$ {{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_{7,3.1}$ (iL)$ with L:=1$ Et cela pour a:=a$ and that for a neq {0}$ shortformdelta:={0, ss, 0, 1, ss, a, 0, ss, 1, 0, 0}$ delta:= mat((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,a,0,0,0),(1,0,0 ,0,0,0))$