a neq {-2,2}$ a:=a$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1,0,0,0),(0,0,a ,-1,0,0))$ shortformdelta:={0, ss, 1, 0, ss, 0, ss, 0, 1, ss, 0, a}$ phase 1 de la resolution des equations$ on resout l'equation {{0,1},0} qui est maintenant AA:= - d(0,3)$ Unknown: d(0,3) Unknown: d(0,3) bonne inconnue W:=d(0,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},1} qui est maintenant AA:= - d(1,3)$ Unknown: d(1,3) Unknown: d(1,3) bonne inconnue W:=d(1,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},2} qui est maintenant AA:= - d(2,3)$ Unknown: d(2,3) Unknown: d(2,3) bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},3} qui est maintenant AA:= - d(3,3) + d(1,1) + d(0, 0)$ Unknowns: {d(3,3),d(1,1),d(0,0)} Unknowns: {d(3,3),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(1,1) + d(0,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,3) + d(2,0))$ Unknowns: {d(4,3),d(2,0)} Unknowns: {d(4,3),d(2,0)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,3) + d(3,1)$ Unknowns: {d(5,3),d(3,1)} Unknowns: {d(5,3),d(3,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(3,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,3) - d(4,1) + d(3, 1)*a$ Unknowns: {d(6,3),d(4,1),d(3,1),a} Unknowns: {d(6,3),d(4,1),d(3,1),a} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - d(4,1) + d(3,1)*a$ on resout l'equation {{0,2},3} 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 {{0,2},4} qui est maintenant AA:=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 {{0,2},5} qui est maintenant AA:= - d(4,0) + d(3,2)$ Unknowns: {d(4,0),d(3,2)} Unknowns: {d(4,0),d(3,2)} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(3,2)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(4,2) + d(3,2)*a - d( 3,0)$ Unknowns: {d(4,2),d(3,2),d(3,0),a} Unknowns: {d(4,2),d(3,2),d(3,0),a} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:=d(3,2)*a - d(3,0)$ on resout l'equation {{0,3},0} qui est maintenant AA:= - (d(0,6)*a + d(0,5))$ Unknowns: {d(0,6),d(0,5),a} Unknowns: {d(0,6),d(0,5),a} bonne inconnue W:=d(0,5)$ sa valeur doit etre WW:= - d(0,6)*a$ on resout l'equation {{0,3},1} qui est maintenant AA:= - (d(1,6)*a + d(1,5))$ Unknowns: {d(1,6),d(1,5),a} Unknowns: {d(1,6),d(1,5),a} bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:= - d(1,6)*a$ on resout l'equation {{0,3},2} qui est maintenant AA:= - (d(2,6)*a + d(2,5))$ Unknowns: {d(2,6),d(2,5),a} Unknowns: {d(2,6),d(2,5),a} bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:= - d(2,6)*a$ on resout l'equation {{0,3},3} qui est maintenant AA:= - (d(3,6)*a + d(3,5))$ Unknowns: {d(3,6),d(3,5),a} Unknowns: {d(3,6),d(3,5),a} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(3,6)*a$ on resout l'equation {{0,3},4} qui est maintenant AA:= - (d(4,6)*a + d(4,5))$ Unknowns: {d(4,6),d(4,5),a} Unknowns: {d(4,6),d(4,5),a} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:= - d(4,6)*a$ on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6)*a - d(5,5) + d( 1,1) + 2*d(0,0)$ Unknowns: {d(5,6),d(5,5),d(1,1),d(0,0),a} Unknowns: {d(5,6),d(5,5),d(1,1),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(5,6)*a + d(1,1) + 2*d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6)*a - d(6,5) + 2* d(2,0) + d(1,1)*a + 2*d(0,0)*a$ Unknowns: {d(6,6),d(6,5),d(2,0),d(1,1),d(0,0),a} Unknowns: {d(6,6),d(6,5),d(2,0),d(1,1),d(0,0),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(6,6)*a + 2*d(2,0) + d(1,1)*a + 2*d(0,0)*a$ on resout l'equation {{0,4},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 {{0,4},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 {{0,4},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 {{0,4},3} qui est maintenant AA:=d(3,6) + d(1,4)$ Unknowns: {d(3,6),d(1,4)} Unknowns: {d(3,6),d(1,4)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,4},4} qui est maintenant AA:=d(4,6)$ Unknown: d(4,6) Unknown: d(4,6) bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(5,6) + d(3,4) + d(2,0)$ Unknowns: {d(5,6),d(3,4),d(2,0)} Unknowns: {d(5,6),d(3,4),d(2,0)} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - (d(3,4) + d(2,0))$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(6,6) - d(4,4) + d(3,4)* a - d(0,0)$ Unknowns: {d(6,6),d(4,4),d(3,4),d(0,0),a} Unknowns: {d(6,6),d(4,4),d(3,4),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(4,4) - d(3,4)*a + d(0,0)$ on resout l'equation {{0,5},5} qui est maintenant AA:=d(1,4)*a$ Unknowns: {d(1,4),a} Unknowns: {d(1,4),a} pas de selection possible de variable a coefficient numerique dans d(1,4)*a on resout l'equation {{0,5},6} qui est maintenant AA:=d(1,4)*a**2$ Unknowns: {d(1,4),a} Unknowns: {d(1,4),a} pas de selection possible de variable a coefficient numerique dans d(1,4)*a**2 on resout l'equation {{0,6},5} 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 {{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},3} qui est maintenant AA:= - (d(3,4) + d(0,2))$ Unknowns: {d(3,4),d(0,2)} Unknowns: {d(3,4),d(0,2)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,4) + d(2,2) + d(1, 1)$ Unknowns: {d(4,4),d(2,2),d(1,1)} Unknowns: {d(4,4),d(2,2),d(1,1)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - (d(5,4) + d(4,1))$ 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(6,4) + d(3,1))$ Unknowns: {d(6,4),d(3,1)} Unknowns: {d(6,4),d(3,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(3,1)$ on resout l'equation {{1,3},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 {{1,3},6} 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 {{2,4},5} qui est maintenant AA:=2*d(2,2) - d(2,0)*a + d(0 ,2)*a - 2*d(0,0)$ Unknowns: {d(2,2),d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,2),d(2,0),d(0,2),d(0,0),a} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=(d(2,0)*a - d(0,2)*a + 2*d(0,0))/2$ on resout l'equation {{2,4},6} qui est maintenant AA:=(d(2,0)*a**2 - 4*d(2,0) + d(0,2)*a**2 - 4*d(0,2))/2$ Unknowns: {d(2,0),d(0,2),a} Unknowns: {d(2,0),d(0,2),a} pas de selection possible de variable a coefficient numerique dans (d(2,0)*a**2 - 4*d(2,0) + d(0,2)*a**2 - 4*d(0,2))/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},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},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},3},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},3},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,6},3},0}, {{{1,6},4},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}, ((d(2,0) + d(0,2))*(a + 2)*(a - 2))/2}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},5},0}, {{{3,5},6},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}}$ Il y a une phase 2$ Si a neq conditionsura$ d(0,2) := - d(2,0)$ 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},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},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},3},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},3},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,6},3},0}, {{{1,6},4},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},5},0}, {{{3,4},6},0}, {{{3,5},5},0}, {{{3,5},6},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}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),0, - d(2,0),0,0,0,0),(0,d(1,1),0,0,0,0,0),(d(2,0),0,d(2,0)*a + d(0,0 ),0,0,0,0),(d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0),d(2,0),0,0),(d(3,2),d(4,1),d(3, 2)*a - d(3,0), - d(2,0),d(1,1) + d(0,0) + d(2,0)*a,0,0),(d(5,0),d(5,1),d(5,2),d( 3,1), - d(4,1),d(1,1) + 2*d(0,0) + 2*d(2,0)*a, - 2*d(2,0)),(d(6,0),d(6,1),d(6,2) , - (d(4,1) - d(3,1)*a), - d(3,1),2*d(2,0),d(1,1) + 2*d(0,0)))$ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 1 0 0 0] [ ] [0 0 a -1 0 0] pour shortformdelta:={0, ss, 1, 0, ss, 0, ss, 0, 1, ss, 0, a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(2,0), d(1,1), d(0,0), a} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(2,0), d(1,1), d(0,0), a} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(3,2), d(3,1), d(3,0), d(2,0), d(1,1), d(0,0)}$ dim Der(gtildedelta):=13$ un element t1 d'un tore $ t1:=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 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] MATD:= mat((d(0,0),0, - d(2,0),0,0,0,0), (0,d(1,1),0,0,0,0,0), (d(2,0),0,d(2,0)*a + d(0,0),0,0,0,0), (d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0),d(2,0),0,0), (d(3,2),d(4,1),d(3,2)*a - d(3,0), - d(2,0),d(1,1) + d(0,0) + d(2,0)*a,0,0), (d(5,0),d(5,1),d(5,2),d(3,1), - d(4,1),d(1,1) + 2*d(0,0) + 2*d(2,0)*a, - 2*d(2,0)), (d(6,0),d(6,1),d(6,2), - (d(4,1) - d(3,1)*a), - d(3,1),2*d(2,0), d(1,1) + 2*d(0,0))) Unknowns: {d(3,2),d(3,0),d(2,0),d(1,1),d(0,0),a} Unknowns: {d(3,2),d(3,0),d(2,0),d(1,1),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),0, - d(2,0),0,0,0,0), (0,d(1,1),0,0,0,0,0), (d(2,0),0,d(2,0)*a + d(0,0),0,0,0,0), (d(3,0),0,d(3,2),d(1,1) + d(0,0),d(2,0),0,0), (d(3,2),0,d(3,2)*a - d(3,0), - d(2,0),d(1,1) + d(0,0) + d(2,0)*a,0,0), (0,0,0,0,0,d(1,1) + 2*d(0,0) + 2*d(2,0)*a, - 2*d(2,0)), (0,0,0,0,0,2*d(2,0),d(1,1) + 2*d(0,0))) Unknowns: {d(3,2),d(3,0),d(2,0),d(1,1),d(0,0),a} Unknowns: {d(3,2),d(3,0),d(2,0),d(1,1),d(0,0),a} t2:=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] Unknowns: {d(2,0),d(1,1),d(0,0),a} Unknowns: {d(2,0),d(1,1),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),0, - d(2,0),0,0,0,0),(0,d(1,1),0,0,0,0,0),(d(2,0),0,d(2,0)*a + d(0,0 ),0,0,0,0),(0,0,0,d(1,1) + d(0,0),d(2,0),0,0),(0,0,0, - d(2,0),d(1,1) + d(0,0) + d(2,0)*a,0,0),(0,0,0,0,0,d(1,1) + 2*d(0,0) + 2*d(2,0)*a, - 2*d(2,0)),(0,0,0,0,0 ,2*d(2,0),d(1,1) + 2*d(0,0)))$ Unknowns: {d(2,0),d(1,1),d(0,0),a} Unknowns: {d(2,0),d(1,1),d(0,0),a} t3:=D(2,0):= [0 0 -1 0 0 0 0 ] [ ] [0 0 0 0 0 0 0 ] [ ] [1 0 a 0 0 0 0 ] [ ] [0 0 0 0 1 0 0 ] [ ] [0 0 0 -1 a 0 0 ] [ ] [0 0 0 0 0 2*a -2] [ ] [0 0 0 0 0 2 0 ] {{x,1, [ 0 ] [ ] [arbcomplex(147)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, 2 {x + 1 - a*x, 2, [ 3 2 ] [ - (a *x - a - 4*a*x + 3)*arbcomplex(148) ] [--------------------------------------------] [ 4 3 2 ] [ a *x - a - 5*a *x + 4*a + 3*x ] [ ] [ 0 ] [ ] [ arbcomplex(148) ] [ ] [ 3 2 ] [ (a *x - a - 4*a*x + 3)*arbcomplex(149) ] [ ----------------------------------------- ] [ 4 3 2 ] [ a *x - a - 5*a *x + 4*a + 3*x ] [ ] [ arbcomplex(149) ] [ ] [ 0 ] [ ] [ 0 ] }, 2 {x + 4 - 2*a*x, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 4 3 2 ] [ (4*a *x - 8*a - 16*a *x + 24*a + 9*x)*arbcomplex(150) ] [--------------------------------------------------------] [ 3 2 ] [ 2*(2*a *x - 4*a - 6*a*x + 9) ] [ ] [ arbcomplex(150) ] }} Unknowns: {d(2,0),d(1,1),d(0,0),a} Unknowns: {d(2,0),d(1,1),d(0,0),a} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),0, - d(2,0),0,0,0,0), (0,d(1,1),0,0,0,0,0), (d(2,0),0,d(2,0)*a + d(0,0),0,0,0,0), (0,0,0,d(1,1) + d(0,0),d(2,0),0,0), (0,0,0, - d(2,0),d(1,1) + d(0,0) + d(2,0)*a,0,0), (0,0,0,0,0,d(1,1) + 2*d(0,0) + 2*d(2,0)*a, - 2*d(2,0)), (0,0,0,0,0,2*d(2,0),d(1,1) + 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:= 2 - (sqrt(a - 4) + a) mat((1,0,-----------------------,0,0,0,0), 2 (0,1,0,0,0,0,0), 2 - (sqrt(a - 4) + a) (-----------------------,0,1,0,0,0,0), 2 2 sqrt(a - 4) + a (0,0,0,1,------------------,0,0), 2 2 sqrt(a - 4) + a (0,0,0,------------------,1,0,0), 2 2 sqrt(a - 4) + a (0,0,0,0,0,1,------------------), 2 2 sqrt(a - 4) + a (0,0,0,0,0,------------------,1)) 2 P**(-1)*t1*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 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] P**(-1)*t2*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)*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 sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a - 5*a + 4 0,0,0,0,0), (0,0,0,0,0,0,0), 2 2*sqrt(a - 4) (0,0,-------------------------,0,0,0,0), 2 2 sqrt(a - 4)*a + a - 4 (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), 2 2*sqrt(a - 4) (0,0,0,0,-------------------------,0,0), 2 2 sqrt(a - 4)*a + a - 4 2 4*sqrt(a - 4) (0,0,0,0,0,-------------------------,0), 2 2 sqrt(a - 4)*a + a - 4 2 4 2 2 2 5 (0,0,0,0,0,0,(2*(sqrt(a - 4)*a - 4*sqrt(a - 4)*a + 2*sqrt(a - 4) + a 3 2 3 2 4 - 6*a + 8*a))/(sqrt(a - 4)*a - 3*sqrt(a - 4)*a + a 2 - 5*a + 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(2,0) + (sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0,0,0),(0,d(1,1), 0,0,0,0,0),(0,0,((sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0) + 2*sqrt(a**2 - 4)*d(2,0)) /(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0),(( - ((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,2) - (sqrt(a**2 - 4)*a + a**2 - 4)*d(3,0)))/(sqrt(a**2 - 4)*a + a**2 - 4),(sqrt(a**2 - 4)*d(4,1) + d(4,1)*a - 2*d(3,1))/(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 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(2,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0 ,0,0),(0,((sqrt(a**2 - 4) + a)*d(3,1) - 2*d(4,1))/(sqrt(a**2 - 4)*a + a**2 - 4), ( - ((sqrt(a**2 - 4)*a + a**2 - 4)*d(3,0) - 2*sqrt(a**2 - 4)*d(3,2)))/(sqrt(a**2 - 4)*a + a**2 - 4),0,((d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*sqrt( a**2 - 4)*d(2,0))/(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) + (sqrt(a**2 - 4) + a)*d(6,0) - (a**2 - 2 + sqrt(a**2 - 4)*a) *d(6,2))/(sqrt(a**2 - 4)*a + a**2 - 4),(sqrt(a**2 - 4)*d(6,1) + d(6,1)*a - 2*d(5 ,1))/(sqrt(a**2 - 4)*a + a**2 - 4),((sqrt(a**2 - 4) + a)*d(5,0) - 2*d(5,2) - (a **2 - 2 + sqrt(a**2 - 4)*a)*d(6,0) + (sqrt(a**2 - 4) + a)*d(6,2))/(sqrt(a**2 - 4 )*a + a**2 - 4),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(4,1))/(sqrt(a**2 - 4)*a + a**2 - 4),((d(1,1) + 2*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 4*sqrt(a**2 - 4)*d(2,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0),(( - ((a**2 - 2 + sqrt(a**2 - 4)*a)*d(5,2) - (sqrt(a**2 - 4 ) + a)*d(5,0) + 2*d(6,0) - (sqrt(a**2 - 4) + a)*d(6,2)))/(sqrt(a**2 - 4)*a + a** 2 - 4),((sqrt(a**2 - 4) + a)*d(5,1) - 2*d(6,1))/(sqrt(a**2 - 4)*a + a**2 - 4),( - ((a**2 - 2 + sqrt(a**2 - 4)*a)*d(5,0) - (sqrt(a**2 - 4) + a)*d(5,2) - (sqrt(a **2 - 4) + a)*d(6,0) + 2*d(6,2)))/(sqrt(a**2 - 4)*a + a**2 - 4),( - (sqrt(a**2 - 4)*d(4,1)*a + d(4,1)*a**2 - 4*d(4,1) - 2*sqrt(a**2 - 4)*d(3,1)))/(sqrt(a**2 - 4 )*a + a**2 - 4),0,0,((d(1,1) + 2*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*(sqrt (a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(2,0))/(sqrt(a**2 - 4)*a + a** 2 - 4)))$ PP:= mat((1,0,( - (sqrt(a**2 - 4) + a))/2,0,0,0,0),(0,1,0,0,0,0,0),(( - (sqrt(a**2 - 4) + a))/2,0,1,0,0,0,0),(0,0,0,1,(sqrt(a**2 - 4) + a)/2,0,0),(0,0,0,(sqrt(a**2 - 4) + a)/2,1,0,0),(0,0,0,0,0,1,(sqrt(a**2 - 4) + a)/2),(0,0,0,0,0,(sqrt(a**2 - 4 ) + a)/2,1))$ avec PP:=P*Q:= mat((1,0,( - (sqrt(a**2 - 4) + a))/2,0,0,0,0),(0,1,0,0,0,0,0),(( - (sqrt(a**2 - 4) + a))/2,0,1,0,0,0,0),(0,0,0,1,(sqrt(a**2 - 4) + a)/2,0,0),(0,0,0,(sqrt(a**2 - 4) + a)/2,1,0,0),(0,0,0,0,0,1,(sqrt(a**2 - 4) + a)/2),(0,0,0,0,0,(sqrt(a**2 - 4 ) + a)/2,1))$ MATDDIAGONALISE:= mat((((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(2,0) + (sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0,0,0),(0,d(1,1), 0,0,0,0,0),(0,0,((sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0) + 2*sqrt(a**2 - 4)*d(2,0)) /(sqrt(a**2 - 4)*a + a**2 - 4),0,0,0,0),(( - ((sqrt(a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(3,2) - (sqrt(a**2 - 4)*a + a**2 - 4)*d(3,0)))/(sqrt(a**2 - 4)*a + a**2 - 4),(sqrt(a**2 - 4)*d(4,1) + d(4,1)*a - 2*d(3,1))/(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 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(2,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0 ,0,0),(0,((sqrt(a**2 - 4) + a)*d(3,1) - 2*d(4,1))/(sqrt(a**2 - 4)*a + a**2 - 4), ( - ((sqrt(a**2 - 4)*a + a**2 - 4)*d(3,0) - 2*sqrt(a**2 - 4)*d(3,2)))/(sqrt(a**2 - 4)*a + a**2 - 4),0,((d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*sqrt( a**2 - 4)*d(2,0))/(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) + (sqrt(a**2 - 4) + a)*d(6,0) - (a**2 - 2 + sqrt(a**2 - 4)*a) *d(6,2))/(sqrt(a**2 - 4)*a + a**2 - 4),(sqrt(a**2 - 4)*d(6,1) + d(6,1)*a - 2*d(5 ,1))/(sqrt(a**2 - 4)*a + a**2 - 4),((sqrt(a**2 - 4) + a)*d(5,0) - 2*d(5,2) - (a **2 - 2 + sqrt(a**2 - 4)*a)*d(6,0) + (sqrt(a**2 - 4) + a)*d(6,2))/(sqrt(a**2 - 4 )*a + a**2 - 4),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(4,1))/(sqrt(a**2 - 4)*a + a**2 - 4),((d(1,1) + 2*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 4*sqrt(a**2 - 4)*d(2,0))/(sqrt(a**2 - 4)*a + a**2 - 4),0),(( - ((a**2 - 2 + sqrt(a**2 - 4)*a)*d(5,2) - (sqrt(a**2 - 4 ) + a)*d(5,0) + 2*d(6,0) - (sqrt(a**2 - 4) + a)*d(6,2)))/(sqrt(a**2 - 4)*a + a** 2 - 4),((sqrt(a**2 - 4) + a)*d(5,1) - 2*d(6,1))/(sqrt(a**2 - 4)*a + a**2 - 4),( - ((a**2 - 2 + sqrt(a**2 - 4)*a)*d(5,0) - (sqrt(a**2 - 4) + a)*d(5,2) - (sqrt(a **2 - 4) + a)*d(6,0) + 2*d(6,2)))/(sqrt(a**2 - 4)*a + a**2 - 4),( - (sqrt(a**2 - 4)*d(4,1)*a + d(4,1)*a**2 - 4*d(4,1) - 2*sqrt(a**2 - 4)*d(3,1)))/(sqrt(a**2 - 4 )*a + a**2 - 4),0,0,((d(1,1) + 2*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*(sqrt (a**2 - 4)*a**2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(2,0))/(sqrt(a**2 - 4)*a + a** 2 - 4)))$ 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(2,0) + (sqrt(a **2 - 4)*a + a**2 - 4)*d(0,0))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(2) := d(1,1)$ r(3) := ((sqrt(a**2 - 4)*a + a**2 - 4)*d(0,0) + 2*sqrt(a**2 - 4)*d(2,0))/(sqrt(a **2 - 4)*a + a**2 - 4)$ r(4) := ((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(2,0))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(5) := ((d(1,1) + d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*sqrt(a**2 - 4)*d(2, 0))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(6) := ((d(1,1) + 2*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 4*sqrt(a**2 - 4)*d( 2,0))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(7) := ((d(1,1) + 2*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*(sqrt(a**2 - 4)*a **2 - 2*sqrt(a**2 - 4) + a**3 - 4*a)*d(2,0))/(sqrt(a**2 - 4)*a + a**2 - 4)$ r(4)-(r(1)+r(2)):=0$ r(7)-(r(2)+2*r(1)):=0$ r(1) := (((a**2 - 2)*gamma3 - (a + 2)*(a - 2)*d(0,0))*(sqrt(a**2 - 4)*a + a**2 - 4) + 2*(a + 2)*(a - 2)*d(2,0)*a)/(2*(sqrt(a**2 - 4)*a + a**2 - 4))$ r(2) := gamma1$ r(3) := gamma3$ r(4) := ( - ((d(0,0)*a**2 - 4*d(0,0) - a**2*gamma3 - 2*gamma1 + 2*gamma3)*(sqrt( a**2 - 4)*a + a**2 - 4) - 2*(a + 2)*(a - 2)*d(2,0)*a))/(2*(sqrt(a**2 - 4)*a + a **2 - 4))$ r(5) := gamma1 + gamma3$ r(6) := gamma1 + 2*gamma3$ r(7) := ( - ((d(0,0)*a**2 - 4*d(0,0) - a**2*gamma3 - gamma1 + 2*gamma3)*(sqrt(a **2 - 4)*a + a**2 - 4) - 2*(a + 2)*(a - 2)*d(2,0)*a))/(sqrt(a**2 - 4)*a + a**2 - 4)$ Le systeme de poids est le systeme 3.4$ calcul de relations de commutation de la base diaY(j) diagonalisant le tore$ listcommutateursdesx := {{{0,1},x(3)}, {{0,2},0}, {{0,3},x(6)*a + x(5)}, {{0,4}, - x(6)}, {{0,5},0}, {{0,6},0}, {{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):=x(1)$ diaY(3):=(2*x(2) - x(0)*(sqrt(a**2 - 4) + a))/2$ diaY(4):=(x(4)*(sqrt(a**2 - 4) + a) + 2*x(3))/2$ diaY(5):=(2*x(4) + x(3)*(sqrt(a**2 - 4) + a))/2$ diaY(6):=(x(6)*(sqrt(a**2 - 4) + a) + 2*x(5))/2$ diaY(7):=(2*x(6) + x(5)*(sqrt(a**2 - 4) + a))/2$ liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},diay(4)}, {{1,3},0}, {{1,4}, ( - (sqrt(a**2 - 4) + a)*(a + 2)*(a - 2)*diay(7))/(sqrt(a**2 - 4)*a + a**2 - 4)} , {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},diay(5)}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{2,7},0}, {{3,4},0}, {{3,5}, ( - (sqrt(a**2 - 4)*a + a**2 - 2)*(a + 2)*(a - 2)*diay(6))/(sqrt(a**2 - 4)*a + a **2 - 4)}, {{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_{7,3.4}$ 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,-1,0,0,0),(0,0,0,0,1, 0,0),(0,0,0,0,0,0,( - (a**2 - 2 + sqrt(a**2 - 4)*a)*(a + 2)*(a - 2))/(sqrt(a**2 - 4)*a + a**2 - 4)),(0,0,0,0,0,((sqrt(a**2 - 4) + a)*(a + 2)*(a - 2))/(sqrt(a**2 - 4)*a + a**2 - 4),0))$ det(isom):= ((sqrt(a**2 - 4)*a**2 - sqrt(a**2 - 4) + a**3 - 3*a)*(a + 2)*(a - 2) )/(sqrt(a**2 - 4)*a + a**2 - 2)$ ZZ(1):=diay(2)$ ZZ(2):=diay(1)$ ZZ(3):=diay(3)$ ZZ(4):= - diay(4)$ ZZ(5):=diay(5)$ ZZ(6):=((sqrt(a**2 - 4) + a)*(a + 2)*(a - 2)*diay(7))/(sqrt(a**2 - 4)*a + a**2 - 4)$ ZZ(7):=( - (a**2 - 2 + sqrt(a**2 - 4)*a)*(a + 2)*(a - 2)*diay(6))/(sqrt(a**2 - 4 )*a + a**2 - 4)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(5)}$ {{1,4},0}$ {{1,5},0}$ {{1,6},0}$ {{1,7},0}$ {{2,3},0}$ {{2,4},zz(6)}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4},0}$ {{3,5},zz(7)}$ {{3,6},0}$ {{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_{7,3.4}$ Et cela pour a different de {-2,2}.$ shortformdelta:={0, ss, 1, 0, ss, 0, ss, 0, 1, ss, 0, a}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1,0,0,0),(0,0,a ,-1,0,0))$