a neq {-2,2}$ a:=a$ b:=0$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1,0,0,0),(0,0,0,0,0,1),(1,0,a ,0,0,0))$ shortformdelta:={0, ss, 0, 0, ss, 1, ss, 0, 0, ss, 1, a}$ phase 1 de la resolution des equations$ 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)$ Unknown: d(3,6) Unknown: d(3,6) bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},4} qui est maintenant AA:= - d(4,6) + d(3,1) - d(2, 0)$ Unknowns: {d(4,6),d(3,1),d(2,0)} Unknowns: {d(4,6),d(3,1),d(2,0)} bonne inconnue W:=d(4,6)$ 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(5,6)$ Unknowns: {d(6,1),d(5,6)} Unknowns: {d(6,1),d(5,6)} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(5,6)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,6) + d(3,1)*a + d( 1,1) + d(0,0)$ Unknowns: {d(6,6),d(3,1),d(1,1),d(0,0),a} Unknowns: {d(6,6),d(3,1),d(1,1),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(3,1)*a + d(1,1) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:=d(3,2) + d(1,0)$ Unknowns: {d(3,2),d(1,0)} Unknowns: {d(3,2),d(1,0)} bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:= - d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=d(6,2) - d(4,0)$ Unknowns: {d(6,2),d(4,0)} Unknowns: {d(6,2),d(4,0)} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=d(4,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(3,0) + d(1,2) - d(1, 0)*a$ Unknowns: {d(3,0),d(1,2),d(1,0),a} Unknowns: {d(3,0),d(1,2),d(1,0),a} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:=d(1,2) - d(1,0)*a$ on resout l'equation {{0,3},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 {{0,3},1} 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 {{0,3},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 {{0,3},3} 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 {{0,3},4} qui est maintenant AA:= - d(4,4) + d(3,3) - d(3, 1)*a + d(2,0)*a + d(0,0)$ Unknowns: {d(4,4),d(3,3),d(3,1),d(2,0),d(0,0),a} Unknowns: {d(4,4),d(3,3),d(3,1),d(2,0),d(0,0),a} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(3,3) - d(3,1)*a + d(2,0)*a + d(0,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:=d(6,3) - d(5,6)*a - d(5,4 )$ Unknowns: {d(6,3),d(5,6),d(5,4),a} Unknowns: {d(6,3),d(5,6),d(5,4),a} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,6)*a + d(5,4)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,4) + d(3,3)*a - d( 3,1)*a**2 + d(2,0) + d(1,3) - d(1,1)*a$ Unknowns: {d(6,4),d(3,3),d(3,1),d(2,0),d(1,3),d(1,1),a} Unknowns: {d(6,4),d(3,3),d(3,1),d(2,0),d(1,3),d(1,1),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(3,3)*a - d(3,1)*a**2 + d(2,0) + d(1,3) - d(1,1)*a$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(3,3)*a - d(3,1)*a**2 + 2*d(2,0) + d(1,3) - d(1,1)*a$ Unknowns: {d(3,3),d(3,1),d(2,0),d(1,3),d(1,1),a} Unknowns: {d(3,3),d(3,1),d(2,0),d(1,3),d(1,1),a} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=( - d(3,3)*a + d(3,1)*a**2 - d(1,3) + d(1,1)*a)/2$ 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(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},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(3,1)*a + d( 1,1) + 2*d(0,0)$ Unknowns: {d(5,5),d(3,1),d(1,1),d(0,0),a} Unknowns: {d(5,5),d(3,1),d(1,1),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(3,1)*a + d(1,1) + 2*d(0,0)$ on resout l'equation {{1,2},4} qui est maintenant AA:=(d(3,3)*a**2 - 2*d(3,3) - d(3,1)*a**3 + 2*d(3,1)*a + 2*d(2,2) + d(1,3)*a - d(1,1)*a**2 + 2*d(1,1) - 2*d(0 ,0))/2$ Unknowns: {d(3,3),d(3,1),d(2,2),d(1,3),d(1,1),d(0,0),a} Unknowns: {d(3,3),d(3,1),d(2,2),d(1,3),d(1,1),d(0,0),a} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=( - d(3,3)*a**2 + 2*d(3,3) + d(3,1)*a**3 - 2*d(3,1)*a - d(1,3)*a + d(1,1)*a**2 - 2*d(1,1) + 2*d(0,0))/2$ on resout l'equation {{1,2},5} qui est maintenant AA:= - (d(5,4) + d(4,1))$ Unknowns: {d(5,4),d(4,1)} Unknowns: {d(5,4),d(4,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(4,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:=( - d(3,3)*a + d(3,1)*a** 2 - 2*d(3,1) - d(1,3) + d(1,1)*a - 2*d(0,2))/2$ Unknowns: {d(3,3),d(3,1),d(1,3),d(1,1),d(0,2),a} Unknowns: {d(3,3),d(3,1),d(1,3),d(1,1),d(0,2),a} bonne inconnue W:=d(1,3)$ sa valeur doit etre WW:= - d(3,3)*a + d(3,1)*a**2 - 2*d(3,1) + d(1,1)*a - 2*d(0, 2)$ 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},6} qui est maintenant AA:=d(2,1) - d(0,3) + d(0,1)* a$ Unknowns: {d(2,1),d(0,3),d(0,1),a} Unknowns: {d(2,1),d(0,3),d(0,1),a} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=d(0,3) - d(0,1)*a$ on resout l'equation {{1,4},5} qui est maintenant AA:=d(0,3) - d(0,1)*a$ Unknowns: {d(0,3),d(0,1),a} Unknowns: {d(0,3),d(0,1),a} bonne inconnue W:=d(0,3)$ sa valeur doit etre WW:=d(0,1)*a$ on resout l'equation {{1,6},5} 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 {{2,3},4} qui est maintenant AA:=d(3,3)*a - d(3,1)*a**2 + 2*d(3,1) - d(1,1)*a + 4*d(0,2)$ Unknowns: {d(3,3),d(3,1),d(1,1),d(0,2),a} Unknowns: {d(3,3),d(3,1),d(1,1),d(0,2),a} bonne inconnue W:=d(0,2)$ sa valeur doit etre WW:=( - d(3,3)*a + d(3,1)*a**2 - 2*d(3,1) + d(1,1)*a)/4$ on resout l'equation {{2,3},5} qui est maintenant AA:= - d(5,6) + d(4,3)$ Unknowns: {d(5,6),d(4,3)} Unknowns: {d(5,6),d(4,3)} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=d(4,3)$ on resout l'equation {{2,3},6} qui est maintenant AA:=( - d(3,3)*a**2 + 4*d(3,3 ) + d(3,1)*a**3 - 4*d(3,1)*a + d(1,1)*a**2 - 4*d(1,1))/2$ Unknowns: {d(3,3),d(3,1),d(1,1),a} Unknowns: {d(3,3),d(3,1),d(1,1),a} pas de selection possible de variable a coefficient numerique dans ( - d(3,3)*a **2 + 4*d(3,3) + d(3,1)*a**3 - 4*d(3,1)*a + d(1,1)*a**2 - 4*d(1,1))/2 on resout l'equation {{2,4},5} qui est maintenant AA:=( - d(3,3)*a**2 + 4*d(3,3 ) + d(3,1)*a**3 - 4*d(3,1)*a + d(1,1)*a**2 - 4*d(1,1))/2$ Unknowns: {d(3,3),d(3,1),d(1,1),a} Unknowns: {d(3,3),d(3,1),d(1,1),a} pas de selection possible de variable a coefficient numerique dans ( - d(3,3)*a **2 + 4*d(3,3) + d(3,1)*a**3 - 4*d(3,1)*a + d(1,1)*a**2 - 4*d(1,1))/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},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},4},0}, {{{0,4},5},0}, {{{0,4},6},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},4},0}, {{{1,3},6},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},4},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}, ( - (d(3,3) - d(3,1)*a - d(1,1))*(a + 2)*(a - 2))/2}, {{{2,4},0},0}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5}, ( - (d(3,3) - d(3,1)*a - d(1,1))*(a + 2)*(a - 2))/2}, {{{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},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 y a une phase 2$ On suppose a neq {-2,2}. Alors $ d(3,3):=d(3,1)*a + d(1,1)$ 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},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},4},0}, {{{0,4},5},0}, {{{0,4},6},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},4},0}, {{{1,3},6},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},4},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},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},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},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}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),0,( - d(3,1))/2,0,0,0,0),(d(1,0),d(1,1),d(1,2), - d(3,1),0,0,0),(d(3 ,1)/2,0,(d(3,1)*a + 2*d(0,0))/2,0,0,0,0),(d(1,2) - d(1,0)*a,d(3,1), - d(1,0),d(3 ,1)*a + d(1,1),0,0,0),(d(4,0),d(4,1),d(4,2),d(4,3),(2*(d(1,1) + d(0,0)) + d(3,1) *a)/2,0,d(3,1)/2),(d(5,0),d(5,1),d(5,2),d(5,3), - d(4,1),d(1,1) + 2*d(0,0) + d(3 ,1)*a,d(4,3)),(d(6,0),d(4,3),d(4,0),d(4,3)*a - d(4,1),( - d(3,1))/2,0,d(1,1) + d (0,0) + d(3,1)*a))$ pour delta:= [0 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 0 0 0 1] [ ] [1 0 a 0 0 0] pour shortformdelta:={0, ss, 0, 0, ss, 1, ss, 0, 0, ss, 1, a} Unknowns: {d(6,0), d(5,3), d(5,2), d(5,1), d(5,0), d(4,3), d(4,2), d(4,1), d(4,0), d(3,1), d(1,2), d(1,1), d(1,0), d(0,0), a} Unknowns: {d(6,0), d(5,3), d(5,2), d(5,1), d(5,0), d(4,3), d(4,2), d(4,1), d(4,0), d(3,1), d(1,2), d(1,1), d(1,0), d(0,0), a} listeparametresMATD{d(6,0), d(5,3), d(5,2), d(5,1), d(5,0), d(4,3), d(4,2), d(4,1), d(4,0), d(3,1), d(1,2), d(1,1), d(1,0), d(0,0)}$ dim Der(gtildedelta):=14$ un element t1 d'un tore $ t1:=D(1,1):= [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 0] [ ] [0 0 0 0 0 0 1] MATD:= - d(3,1) mat((d(0,0),0,-----------,0,0,0,0), 2 (d(1,0),d(1,1),d(1,2), - d(3,1),0,0,0), d(3,1) d(3,1)*a + 2*d(0,0) (--------,0,---------------------,0,0,0,0), 2 2 (d(1,2) - d(1,0)*a,d(3,1), - d(1,0),d(3,1)*a + d(1,1),0,0,0), 2*(d(1,1) + d(0,0)) + d(3,1)*a d(3,1) (d(4,0),d(4,1),d(4,2),d(4,3),--------------------------------,0,--------), 2 2 (d(5,0),d(5,1),d(5,2),d(5,3), - d(4,1),d(1,1) + 2*d(0,0) + d(3,1)*a,d(4,3)), - d(3,1) (d(6,0),d(4,3),d(4,0),d(4,3)*a - d(4,1),-----------,0, 2 d(1,1) + d(0,0) + d(3,1)*a)) Unknowns: {d(5,3),d(5,1),d(4,3),d(4,1),d(3,1),d(1,1),d(0,0),a} Unknowns: {d(5,3),d(5,1),d(4,3),d(4,1),d(3,1),d(1,1),d(0,0),a} commutant de t1 dans der(gtildedelta): - d(3,1) mat((d(0,0),0,-----------,0,0,0,0), 2 (0,d(1,1),0, - d(3,1),0,0,0), d(3,1) d(3,1)*a + 2*d(0,0) (--------,0,---------------------,0,0,0,0), 2 2 (0,d(3,1),0,d(3,1)*a + d(1,1),0,0,0), 2*(d(1,1) + d(0,0)) + d(3,1)*a d(3,1) (0,d(4,1),0,d(4,3),--------------------------------,0,--------), 2 2 (0,d(5,1),0,d(5,3), - d(4,1),d(1,1) + 2*d(0,0) + d(3,1)*a,d(4,3)), - d(3,1) (0,d(4,3),0,d(4,3)*a - d(4,1),-----------,0,d(1,1) + d(0,0) + d(3,1)*a)) 2 Unknowns: {d(5,3),d(5,1),d(4,3),d(4,1),d(3,1),d(1,1),d(0,0),a} Unknowns: {d(5,3),d(5,1),d(4,3),d(4,1),d(3,1),d(1,1),d(0,0),a} t2:=D(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 0 0 0 0] [ ] [0 0 0 0 1 0 0] [ ] [0 0 0 0 0 2 0] [ ] [0 0 0 0 0 0 1] Unknowns: {d(3,1),d(1,1),d(0,0),a} Unknowns: {d(3,1),d(1,1),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),0,( - d(3,1))/2,0,0,0,0),(0,d(1,1),0, - d(3,1),0,0,0),(d(3,1)/2,0,(d (3,1)*a + 2*d(0,0))/2,0,0,0,0),(0,d(3,1),0,d(3,1)*a + d(1,1),0,0,0),(0,0,0,0,(2* (d(1,1) + d(0,0)) + d(3,1)*a)/2,0,d(3,1)/2),(0,0,0,0,0,d(1,1) + 2*d(0,0) + d(3,1 )*a,0),(0,0,0,0,( - d(3,1))/2,0,d(1,1) + d(0,0) + d(3,1)*a))$ Unknowns: {d(3,1),d(1,1),d(0,0),a} Unknowns: {d(3,1),d(1,1),d(0,0),a} t3:=D(3,1):= [ - 1 ] [ 0 0 ------ 0 0 0 0 ] [ 2 ] [ ] [ 0 0 0 -1 0 0 0 ] [ ] [ 1 a ] [--- 0 --- 0 0 0 0 ] [ 2 2 ] [ ] [ 0 1 0 a 0 0 0 ] [ ] [ a 1 ] [ 0 0 0 0 --- 0 ---] [ 2 2 ] [ ] [ 0 0 0 0 0 a 0 ] [ ] [ - 1 ] [ 0 0 0 0 ------ 0 a ] [ 2 ] ((2*a**2 - 6*a*x + 4*x**2 + 1)*(2*a*x - 4*x**2 - 1)*(a*x - x**2 - 1)*(a - x))/16 $ {{ - (a - x), 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(185)] [ ] [ 0 ] }, 2 { - (4*x + 1 - 2*a*x), 1, [ 2 ] [ (4*a - 10*a*x - 3)*arbcomplex(186) ] [-------------------------------------] [ 2 ] [ 2*a *x - 5*a + 6*x ] [ ] [ 0 ] [ ] [ arbcomplex(186) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, 2 { - (x + 1 - a*x), 1, [ 0 ] [ ] [ - (a - x)*arbcomplex(187)] [ ] [ 0 ] [ ] [ arbcomplex(187) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, 2 {2*(a - 3*x)*a + 4*x + 1, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 2 ] [ (2*a - 2*a*x - 3)*arbcomplex(188) ] [------------------------------------] [ 2*(2*a - 3*x) ] [ ] [ 0 ] [ ] [ arbcomplex(188) ] }} {{ - (a - x), 1, mat((0),(0),(0),(0),(0),(arbcomplex(189)),(0))$ }, { - (4*x**2 + 1 - 2*a*x), 1, mat((((4*a**2 - 10*a*x - 3)*arbcomplex(190))/(2*a**2*x - 5*a + 6*x)),(0),( arbcomplex(190)),(0),(0),(0),(0))$ }, { - (x**2 + 1 - a*x), 1, mat((0),( - (a - x)*arbcomplex(191)),(0),(arbcomplex(191)),(0),(0),(0))$ }, {2*(a - 3*x)*a + 4*x**2 + 1, 1, mat((0),(0),(0),(0),(((2*a**2 - 2*a*x - 3)*arbcomplex(192))/(2*(2*a - 3*x))),(0) ,(arbcomplex(192)))$ }}$ Unknowns: {d(3,1),d(1,1),d(0,0),a} Unknowns: {d(3,1),d(1,1),d(0,0),a} commutant simultane de t1,t2,t3 dans der(gtildedelta):$ mat((d(0,0),0,( - d(3,1))/2,0,0,0,0),(0,d(1,1),0, - d(3,1),0,0,0),(d(3,1)/2,0,(d (3,1)*a + 2*d(0,0))/2,0,0,0,0),(0,d(3,1),0,d(3,1)*a + d(1,1),0,0,0),(0,0,0,0,(2* (d(1,1) + d(0,0)) + d(3,1)*a)/2,0,d(3,1)/2),(0,0,0,0,0,d(1,1) + 2*d(0,0) + d(3,1 )*a,0),(0,0,0,0,( - d(3,1))/2,0,d(1,1) + d(0,0) + d(3,1)*a))$ le calcul du commutant de t1 et t2 et t3 dans der(gtildedelta) montre que t1,\ t2,t3 est un tore maximal. 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:= 2 - (sqrt(a - 4) + a) mat((1,0,-----------------------,0,0,0,0), 2 2 - (sqrt(a - 4) + a) (0,1,0,-----------------------,0,0,0), 2 2 - (sqrt(a - 4) + a) (-----------------------,0,1,0,0,0,0), 2 2 - (sqrt(a - 4) + a) (0,-----------------------,0,1,0,0,0), 2 2 sqrt(a - 4) + a (0,0,0,0,1,0,------------------), 2 (0,0,0,0,0,1,0), 2 sqrt(a - 4) + a (0,0,0,0,------------------,0,1)) 2 P**(-1)*t1*P:= [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 0] [ ] [0 0 0 0 0 0 1] P**(-1)*t2*P:= [1 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 2 0] [ ] [0 0 0 0 0 0 1] P**(-1)*t3*P:= 2 4 2 2 2 5 3 sqrt(a - 4)*a - 4*sqrt(a - 4)*a + 2*sqrt(a - 4) + a - 6*a + 8*a mat((------------------------------------------------------------------------,0, 2 3 2 4 2 2*(sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a - 5*a + 4) 0,0,0,0,0), 2 4 2 2 2 5 3 sqrt(a - 4)*a - 4*sqrt(a - 4)*a + 2*sqrt(a - 4) + a - 6*a + 8*a (0,------------------------------------------------------------------------, 2 3 2 4 2 sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a - 5*a + 4 0,0,0,0,0), 2 sqrt(a - 4) (0,0,-------------------------,0,0,0,0), 2 2 sqrt(a - 4)*a + a - 4 2 2*sqrt(a - 4) (0,0,0,-------------------------,0,0,0), 2 2 sqrt(a - 4)*a + a - 4 2 4 2 2 2 5 (0,0,0,0,(2*sqrt(a - 4)*a - 7*sqrt(a - 4)*a + 2*sqrt(a - 4) + 2*a 3 - 11*a + 12*a)/(2 2 3 2 4 2 *(sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a - 5*a + 4)),0,0), (0,0,0,0,0,a,0), (0,0,0,0,0,0, 2 4 2 2 2 5 3 sqrt(a - 4)*a - 2*sqrt(a - 4)*a - 2*sqrt(a - 4) + a - 4*a ------------------------------------------------------------------)) 2 3 2 4 2 2*(sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a - 5*a + 4) Here is to be underlined that the denominators do not vanish fo a neq 2,-2 since: 2 3 2 4 2 u := sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a - 5*a + 4 2 2 u := sqrt(a - 4)*(a - 3)*a + (a + 2)*(a + 1)*(a - 1)*(a - 2) 2 Let w1:=(a - 3)*a 2 and w2:= - sqrt(a - 4)*(a + 1)*(a - 1) u=0 reads w1+w2=0 hence w1**2 = w2**2 Now, w1**2-w2**2:=4 matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1) + 2*(sqrt(a** 2 - 4)*a + a**2 - 4)*d(0,0))/(2*(sqrt(a**2 - 4)*a + a**2 - 4)),0,0,0,0,0,0),(0,( (sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1) + (sqrt(a**2 - 4)*a + a**2 - 4)*d(1,1))/(sqrt(a**2 - 4)*a + a**2 - 4),( - ((sqrt(a**2 - 4)*a**2 - 2 *sqrt(a**2 - 4) + a**3 - 4*a)*d(1,0) - (sqrt(a**2 - 4)*a + a**2 - 4)*d(1,2)))/( sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0),(0,0,((sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0) + sqrt(a**2 - 4)*d(3,1))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0),((sqrt(a**2 - 4 )*d(1,2)*a + d(1,2)*a**2 - 4*d(1,2) - 2*sqrt(a**2 - 4)*d(1,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,((sqrt(a**2 - 4)*a + a**2 - 4)*d(1,1) + 2*sqrt(a**2 - 4)*d(3,1) )/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0),(((d(4,2) - d(4,0)*a + d(6,0))*(sqrt(a**2 - 4) + a))/(sqrt(a**2 - 4)*a + a**2 - 4),( - ((sqrt(a**2 - 4)*a**2 - 2*sqrt(a** 2 - 4) + a**3 - 4*a)*d(4,3) - (sqrt(a**2 - 4)*a + a**2 - 4)*d(4,1)))/(sqrt(a**2 - 4)*a + a**2 - 4),(2*((sqrt(a**2 - 4) + a)*d(4,0) - d(4,2)) - (a**2 - 2 + sqrt( a**2 - 4)*a)*d(6,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0,((d(1,1) + d(0,0))*(sqrt(a **2 - 4)*a + a**2 - 4) + (sqrt(a**2 - 4)*a**2 - sqrt(a**2 - 4) + a**3 - 4*a)*d(3 ,1))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0),(( - (sqrt(a**2 - 4)*d(5,2) + d(5,2)*a - 2*d(5,0)))/2,( - (sqrt(a**2 - 4)*d(5,3) + d(5,3)*a - 2*d(5,1)))/2,( - ((sqrt(a **2 - 4) + a)*d(5,0) - 2*d(5,2)))/2,( - ((sqrt(a**2 - 4) + a)*d(5,1) - 2*d(5,3)) )/2,(sqrt(a**2 - 4)*d(4,3) + d(4,3)*a - 2*d(4,1))/2,d(1,1) + 2*d(0,0) + d(3,1)*a ,( - ((sqrt(a**2 - 4) + a)*d(4,1) - 2*d(4,3)))/2),(( - ((a**2 - 2 + sqrt(a**2 - 4)*a)*d(4,2) - 2*(sqrt(a**2 - 4) + a)*d(4,0) + 2*d(6,0)))/(sqrt(a**2 - 4)*a + a **2 - 4),0,((d(4,2) - d(4,0)*a + d(6,0))*(sqrt(a**2 - 4) + a))/(sqrt(a**2 - 4)*a + a**2 - 4),( - ((sqrt(a**2 - 4)*a + a**2 - 4)*d(4,1) - 2*sqrt(a**2 - 4)*d(4,3) ))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,(2*(d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a **2 - 4) + (sqrt(a**2 - 4)*a**2 + 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1))/(2*( sqrt(a**2 - 4)*a + a**2 - 4))))$ PP:= mat((1,0,( - (sqrt(a**2 - 4) + a))/2,0,0,0,0),(0,( - (sqrt(a**2 - 4) + a))/2,0,1 ,0,0,0),(( - (sqrt(a**2 - 4) + a))/2,0,1,0,0,0,0),(0,1,0,( - (sqrt(a**2 - 4) + a ))/2,0,0,0),(0,0,0,0,(sqrt(a**2 - 4) + a)/2,1,0),(0,0,0,0,0,0,1),(0,0,0,0,1,( sqrt(a**2 - 4) + a)/2,0))$ avec PP:=P*Q:= mat((1,0,( - (sqrt(a**2 - 4) + a))/2,0,0,0,0),(0,( - (sqrt(a**2 - 4) + a))/2,0,1 ,0,0,0),(( - (sqrt(a**2 - 4) + a))/2,0,1,0,0,0,0),(0,1,0,( - (sqrt(a**2 - 4) + a ))/2,0,0,0),(0,0,0,0,(sqrt(a**2 - 4) + a)/2,1,0),(0,0,0,0,0,0,1),(0,0,0,0,1,( sqrt(a**2 - 4) + a)/2,0))$ MATDDIAGONALISE:= mat((((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1) + 2*(sqrt(a** 2 - 4)*a + a**2 - 4)*d(0,0))/(2*(sqrt(a**2 - 4)*a + a**2 - 4)),0,0,0,0,0,0),(( sqrt(a**2 - 4)*d(1,2)*a + d(1,2)*a**2 - 4*d(1,2) - 2*sqrt(a**2 - 4)*d(1,0))/( sqrt(a**2 - 4)*a + a**2 - 4),((sqrt(a**2 - 4)*a + a**2 - 4)*d(1,1) + 2*sqrt(a**2 - 4)*d(3,1))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0,0),(0,0,((sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0) + sqrt(a**2 - 4)*d(3,1))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0 ),(0,0,( - ((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(1,0) - (sqrt (a**2 - 4)*a + a**2 - 4)*d(1,2)))/(sqrt(a**2 - 4)*a + a**2 - 4),((sqrt(a**2 - 4) *a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1) + (sqrt(a**2 - 4)*a + a**2 - 4)*d( 1,1))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0),(( - ((a**2 - 2 + sqrt(a**2 - 4)*a)*d (4,2) - 2*(sqrt(a**2 - 4) + a)*d(4,0) + 2*d(6,0)))/(sqrt(a**2 - 4)*a + a**2 - 4) ,( - ((sqrt(a**2 - 4)*a + a**2 - 4)*d(4,1) - 2*sqrt(a**2 - 4)*d(4,3)))/(sqrt(a** 2 - 4)*a + a**2 - 4),((d(4,2) - d(4,0)*a + d(6,0))*(sqrt(a**2 - 4) + a))/(sqrt(a **2 - 4)*a + a**2 - 4),0,(2*(d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + ( sqrt(a**2 - 4)*a**2 + 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1))/(2*(sqrt(a**2 - 4)* a + a**2 - 4)),0,0),(((d(4,2) - d(4,0)*a + d(6,0))*(sqrt(a**2 - 4) + a))/(sqrt(a **2 - 4)*a + a**2 - 4),0,(2*((sqrt(a**2 - 4) + a)*d(4,0) - d(4,2)) - (a**2 - 2 + sqrt(a**2 - 4)*a)*d(6,0))/(sqrt(a**2 - 4)*a + a**2 - 4),( - ((sqrt(a**2 - 4)*a **2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(4,3) - (sqrt(a**2 - 4)*a + a**2 - 4)*d(4, 1)))/(sqrt(a**2 - 4)*a + a**2 - 4),0,((d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + (sqrt(a**2 - 4)*a**2 - sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1))/(sqrt(a**2 - 4)*a + a**2 - 4),0),(( - (sqrt(a**2 - 4)*d(5,2) + d(5,2)*a - 2*d(5,0)))/2,( - ( (sqrt(a**2 - 4) + a)*d(5,1) - 2*d(5,3)))/2,( - ((sqrt(a**2 - 4) + a)*d(5,0) - 2* d(5,2)))/2,( - (sqrt(a**2 - 4)*d(5,3) + d(5,3)*a - 2*d(5,1)))/2,( - ((sqrt(a**2 - 4) + a)*d(4,1) - 2*d(4,3)))/2,(sqrt(a**2 - 4)*d(4,3) + d(4,3)*a - 2*d(4,1))/2, d(1,1) + 2*d(0,0) + d(3,1)*a))$ on voit apparaitre les poids sur la diagonale$ r(1) := ((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1) + 2*(sqrt( a**2 - 4)*a + a**2 - 4)*d(0,0))/(2*(sqrt(a**2 - 4)*a + a**2 - 4))$ r(2) := ((sqrt(a**2 - 4)*a + a**2 - 4)*d(1,1) + 2*sqrt(a**2 - 4)*d(3,1))/(sqrt(a **2 - 4)*a + a**2 - 4)$ r(3) := ((sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0) + sqrt(a**2 - 4)*d(3,1))/(sqrt(a** 2 - 4)*a + a**2 - 4)$ r(4) := ((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1) + (sqrt(a **2 - 4)*a + a**2 - 4)*d(1,1))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(5) := (2*(d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + (sqrt(a**2 - 4)*a** 2 + 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1))/(2*(sqrt(a**2 - 4)*a + a**2 - 4))$ r(6) := ((d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + (sqrt(a**2 - 4)*a**2 - sqrt(a**2 - 4) + a**3 - 4*a)*d(3,1))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(7) := d(1,1) + 2*d(0,0) + d(3,1)*a$ R(4)-(2*r(1)+r(2)-2*r(3)):= 0$ R(7)-(r(1)+r(5)):= 0$ R(7)-(2*r(1)+r(2)):= 0$ R(6)-(r(3)+r(4)):= 0$ R(6)-(2*r(1)+r(2)-r(3)):= 0$ R(7)-(r(3)+r(6)):= 0$ R(5)-(r(1)+r(2)):= 0$ R(7)-(2*r(1)+r(2)):= 0$ R(4)-(2*r(1)+r(2)-2*r(3)):= 0$ Le systeme de poids est le systeme 3.16$ calcul de relations de commutation de la base diaY(j) diagonalisant le tore$ listcommutateursdesx := {{{0,1},x(6)}, {{0,2},0}, {{0,3},x(6)*a + x(4)}, {{0,4},0}, {{0,5},0}, {{0,6},x(5)}, {{1,2},x(4)}, {{1,3},0}, {{1,4},0}, {{1,5},0}, {{1,6},0}, {{2,3},x(6)}, {{2,4},x(5)}, {{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(2)*(sqrt(a**2 - 4) + a) - 2*x(0)))/2$ diaY(2):=(2*x(3) - x(1)*(sqrt(a**2 - 4) + a))/2$ diaY(3):=(2*x(2) - x(0)*(sqrt(a**2 - 4) + a))/2$ diaY(4):=( - (x(3)*(sqrt(a**2 - 4) + a) - 2*x(1)))/2$ diaY(5):=(2*x(6) + x(4)*(sqrt(a**2 - 4) + a))/2$ diaY(6):=(x(6)*(sqrt(a**2 - 4) + a) + 2*x(4))/2$ diaY(7):=x(5)$ liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2}, ( - (sqrt(a**2 - 4) + a)*(a + 2)*(a - 2)*diay(5))/(sqrt(a**2 - 4)*a + a**2 - 4)} , {{1,3},0}, {{1,4},0}, {{1,5}, ( - (sqrt(a**2 - 4)*a + a**2 - 4)*diay(7))/2}, {{1,6},0}, {{1,7},0}, {{2,3},0}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{2,7},0}, {{3,4}, ((sqrt(a**2 - 4)*a + a**2 - 2)*(a + 2)*(a - 2)*diay(6))/(sqrt(a**2 - 4)*a + a**2 - 4)}, {{3,5},0}, {{3,6}, ( - (sqrt(a**2 - 4)*a + a**2 - 4)*diay(7))/2}, {{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_{7,3.16}$ 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((1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,( - (sqrt(a**2 - 4)*( a**2 - 2) + (a + 2)*(a - 2)*a))/(sqrt(a**2 - 4)*(a**2 - 3)*a + (a + 2)*(a + 1)*( a - 1)*(a - 2)),0,0,0),(0,0,0,0,( - (sqrt(a**2 - 4) + a)*(a + 2)*(a - 2))/(sqrt( a**2 - 4)*a + a**2 - 4),0,0),(0,0,0,0,0,( - (sqrt(a**2 - 4) + a)*(a + 2)*(a - 2) )/(sqrt(a**2 - 4)*a + a**2 - 4),0),(0,0,0,0,0,0,((sqrt(a**2 - 4) + a)*(a + 2)*(a - 2))/2))$ det(isom):= - (a + 2)**2*(a - 2)**2$ ZZ(1):=diay(1)$ ZZ(2):=diay(2)$ ZZ(3):=diay(3)$ ZZ(4):=( - (sqrt(a**2 - 4)*(a**2 - 2) + (a + 2)*(a - 2)*a)*diay(4))/(sqrt(a**2 - 4)*(a**2 - 3)*a + (a + 2)*(a + 1)*(a - 1)*(a - 2))$ ZZ(5):=( - (sqrt(a**2 - 4) + a)*(a + 2)*(a - 2)*diay(5))/(sqrt(a**2 - 4)*a + a** 2 - 4)$ ZZ(6):=( - (sqrt(a**2 - 4) + a)*(a + 2)*(a - 2)*diay(6))/(sqrt(a**2 - 4)*a + a** 2 - 4)$ ZZ(7):=((sqrt(a**2 - 4) + a)*(a + 2)*(a - 2)*diay(7))/2$ listcommutateursdesZZ:=$ {{1,2},zz(5)}$ {{1,3},0}$ {{1,4},0}$ {{1,5},zz(7)}$ {{1,6},0}$ {{1,7},0}$ {{2,3},0}$ {{2,4},0}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4},zz(6)}$ {{3,5},0}$ {{3,6},zz(7)}$ {{3,7},0}$ {{4,5},0}$ {{4,6},0}$ {{4,7},0}$ {{5,6},0}$ {{5,7},0}$ {{6,7},0}$ On obtient donc les relations de commutations de $ g_{7,3.16}$ Et cela pour a:=a, b:=0.$ Et cela pour a different de {-2,2}.$ shortformdelta:={0, ss, 0, 0, ss, 1, ss, 0, 0, ss, 1, a}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1,0,0,0),(0,0,0,0,0,1),(1,0,a ,0,0,0))$