In the present case 2 we suppose A=((0,0),(0,0)).$ a neq {}$ a:=a$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(a,1,0,0,0,0),(0,0,0,0,0,0),(0,0,0 ,0,1,0))$ shortformdelta:={0, 1, ss, a, 1, ss, 0, 0, ss, 0, 0}$ phase 1 de la resolution des equations$ on resout l'equation {{0,1},0} qui est maintenant AA:= - d(0,4)*a$ Unknowns: {d(0,4),a} Unknowns: {d(0,4),a} pas de selection possible de variable a coefficient numerique dans - d(0,4)*a on resout l'equation {{0,1},1} 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 {{0,1},2} 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 {{0,1},3} qui est maintenant AA:= - d(3,4)*a + d(2,1)$ Unknowns: {d(3,4),d(2,1),a} Unknowns: {d(3,4),d(2,1),a} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=d(3,4)*a$ on resout l'equation {{0,1},4} qui est maintenant AA:=a*( - d(4,4) + d(3,4) + d (1,1) + d(0,0))$ Unknowns: {d(4,4),d(3,4),d(1,1),d(0,0),a} Unknowns: {d(4,4),d(3,4),d(1,1),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(4,4) + d(3,4) + d(1,1) + d(0,0)) on resout l'equation {{0,1},5} qui est maintenant AA:= - (d(5,4)*a + d(2,0))$ Unknowns: {d(5,4),d(2,0),a} Unknowns: {d(5,4),d(2,0),a} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:= - d(5,4)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,4)*a + d(5,1) - d( 4,0)$ Unknowns: {d(6,4),d(5,1),d(4,0),a} Unknowns: {d(6,4),d(5,1),d(4,0),a} bonne inconnue W:=d(5,1)$ sa valeur doit etre WW:=d(6,4)*a + d(4,0)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,4) + d(0,3))$ Unknowns: {d(0,4),d(0,3)} Unknowns: {d(0,4),d(0,3)} bonne inconnue W:=d(0,4)$ sa valeur doit etre WW:= - d(0,3)$ on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,4) + d(1,3))$ Unknowns: {d(1,4),d(1,3)} Unknowns: {d(1,4),d(1,3)} bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:= - d(1,3)$ on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,4) + d(2,3))$ Unknowns: {d(2,4),d(2,3)} Unknowns: {d(2,4),d(2,3)} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:= - d(2,3)$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,4) - d(3,3) + d(2, 2) + d(0,0)$ Unknowns: {d(3,4),d(3,3),d(2,2),d(0,0)} Unknowns: {d(3,4),d(3,3),d(2,2),d(0,0)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(3,3) + d(2,2) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,4) - d(4,3) + d(2, 2) + d(1,2)*a + d(0,0)$ Unknowns: {d(4,4),d(4,3),d(2,2),d(1,2),d(0,0),a} Unknowns: {d(4,4),d(4,3),d(2,2),d(1,2),d(0,0),a} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:= - d(4,3) + d(2,2) + d(1,2)*a + d(0,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,4) - d(5,3) + d(1, 0)$ Unknowns: {d(5,4),d(5,3),d(1,0)} Unknowns: {d(5,4),d(5,3),d(1,0)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(5,3) + d(1,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,4) - d(6,3) + d(5, 2) - d(3,0)$ Unknowns: {d(6,4),d(6,3),d(5,2),d(3,0)} Unknowns: {d(6,4),d(6,3),d(5,2),d(3,0)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(6,3) + d(5,2) - d(3,0)$ on resout l'equation {{0,3},3} 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 {{0,3},4} qui est maintenant AA:=d(1,3)*a$ Unknowns: {d(1,3),a} Unknowns: {d(1,3),a} pas de selection possible de variable a coefficient numerique dans d(1,3)*a on resout l'equation {{0,3},6} qui est maintenant AA:=d(5,3)*a + d(5,3) - d(1,0 )*a$ Unknowns: {d(5,3),d(1,0),a} Unknowns: {d(5,3),d(1,0),a} pas de selection possible de variable a coefficient numerique dans d(5,3)*a + d( 5,3) - d(1,0)*a on resout l'equation {{0,4},4} qui est maintenant AA:= - d(1,3)*a$ Unknowns: {d(1,3),a} Unknowns: {d(1,3),a} pas de selection possible de variable a coefficient numerique dans - d(1,3)*a on resout l'equation {{0,4},6} qui est maintenant AA:= - d(5,3) + 2*d(1,0)$ Unknowns: {d(5,3),d(1,0)} Unknowns: {d(5,3),d(1,0)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=2*d(1,0)$ on resout l'equation {{0,5},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 {{0,5},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 {{0,5},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 {{0,5},3} qui est maintenant AA:= - d(3,6) + d(2,5)$ Unknowns: {d(3,6),d(2,5)} Unknowns: {d(3,6),d(2,5)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(2,5)$ on resout l'equation {{0,5},4} qui est maintenant AA:= - d(4,6) + d(2,5) + d(1, 5)*a$ Unknowns: {d(4,6),d(2,5),d(1,5),a} Unknowns: {d(4,6),d(2,5),d(1,5),a} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=d(2,5) + d(1,5)*a$ on resout l'equation {{0,5},5} qui est maintenant AA:= - d(5,6)$ Unknown: d(5,6) Unknown: d(5,6) bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},6} qui est maintenant AA:= - d(6,6) + d(5,5) + d(0, 0)$ Unknowns: {d(6,6),d(5,5),d(0,0)} Unknowns: {d(6,6),d(5,5),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(5,5) + d(0,0)$ on resout l'equation {{1,2},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 {{1,2},1} qui est maintenant AA:= - d(1,5)$ Unknown: d(1,5) Unknown: d(1,5) bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,2},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 {{1,2},3} qui est maintenant AA:= - d(3,5) + d(0,1)$ Unknowns: {d(3,5),d(0,1)} Unknowns: {d(3,5),d(0,1)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,5) - d(0,2)*a + d( 0,1)$ Unknowns: {d(4,5),d(0,2),d(0,1),a} Unknowns: {d(4,5),d(0,2),d(0,1),a} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:= - d(0,2)*a + d(0,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,5) + d(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) + d(1,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,5) + d(4,2) - d(3, 1)$ Unknowns: {d(6,5),d(4,2),d(3,1)} Unknowns: {d(6,5),d(4,2),d(3,1)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,3},4} qui est maintenant AA:= - d(0,3)*a$ Unknowns: {d(0,3),a} Unknowns: {d(0,3),a} pas de selection possible de variable a coefficient numerique dans - d(0,3)*a on resout l'equation {{1,3},6} qui est maintenant AA:=d(4,3) - d(3,3)*a + d(2,2 )*a + d(0,0)*a$ Unknowns: {d(4,3),d(3,3),d(2,2),d(0,0),a} Unknowns: {d(4,3),d(3,3),d(2,2),d(0,0),a} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=a*(d(3,3) - d(2,2) - d(0,0))$ on resout l'equation {{1,4},4} qui est maintenant AA:=d(0,3)*a$ Unknowns: {d(0,3),a} Unknowns: {d(0,3),a} pas de selection possible de variable a coefficient numerique dans d(0,3)*a on resout l'equation {{1,4},6} qui est maintenant AA:=a*( - d(3,3) + d(2,2) + d (1,2) + d(0,0))$ Unknowns: {d(3,3),d(2,2),d(1,2),d(0,0),a} Unknowns: {d(3,3),d(2,2),d(1,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans a*( - d(3,3) + d(2,2) + d(1,2) + d(0,0)) on resout l'equation {{1,5},6} qui est maintenant AA:= - d(0,2)*a + 2*d(0,1)$ Unknowns: {d(0,2),d(0,1),a} Unknowns: {d(0,2),d(0,1),a} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=(d(0,2)*a)/2$ on resout l'equation {{2,3},3} 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 {{2,3},5} 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 {{2,3},6} 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 {{2,4},6} qui est maintenant AA:=d(2,2) + d(1,2) - d(1,1)$ Unknowns: {d(2,2),d(1,2),d(1,1)} Unknowns: {d(2,2),d(1,2),d(1,1)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:= - d(1,2) + d(1,1)$ on resout l'equation {{2,5},6} qui est maintenant AA:=(d(0,2)*(a + 2))/2$ Unknowns: {d(0,2),a} Unknowns: {d(0,2),a} pas de selection possible de variable a coefficient numerique dans (d(0,2)*(a + 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},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},3},0}, {{{0,3},4},0}, {{{0,3},6},(a + 2)*d(1,0)}, {{{0,4},3},0}, {{{0,4},4},0}, {{{0,4},6},0}, {{{0,5},0},0}, {{{0,5},1},0}, {{{0,5},2},0}, {{{0,5},3},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},3},0}, {{{0,6},4},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}, {{{2,4},6},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},((a + 2)*d(0,2))/2}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},6},0}, {{{3,5},6},0}, {{{3,6},6},0}, {{{4,5},6},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ Il y a une phase 2 pour a neq -2$ 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},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},3},0}, {{{0,3},4},0}, {{{0,3},6},0}, {{{0,4},3},0}, {{{0,4},4},0}, {{{0,4},6},0}, {{{0,5},0},0}, {{{0,5},1},0}, {{{0,5},2},0}, {{{0,5},3},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},3},0}, {{{0,6},4},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}, {{{2,4},6},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},6},0}, {{{3,5},6},0}, {{{3,6},6},0}, {{{4,5},6},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),0,0,0,0,0,0),(0,d(1,1),d(1,2),0,0,0,0),(0, - d(1,2)*a, - (d(1,2) - d (1,1)),0,0,0,0),(d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0), - d(1,2),0,0),(d(4,0),d(4 ,1),d(4,2),d(1,2)*a,d(1,1) + d(0,0) - d(1,2),0,0),(d(5,0),d(4,0) - d(3,0)*a + d( 5,2)*a - d(6,3)*a,d(5,2),0,0, - (d(1,2) - 2*d(1,1)),0),(d(6,0),d(6,1),d(6,2),d(6 ,3),d(5,2) - d(3,0) - d(6,3),d(4,2) - d(3,1),2*d(1,1) + d(0,0) - d(1,2)))$ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] [ ] [a 1 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 1 0] pour shortformdelta:={0, 1, ss, a, 1, ss, 0, 0, ss, 0, 0} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(1,2), d(1,1), d(0,0), a} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(1,2), d(1,1), d(0,0), a} listeparametresMATD{d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(1,2), d(1,1), d(0,0)}$ dim Der(gtildedelta):=15$ 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 0 0 0 0 0] [ ] [0 0 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 1] MATD:= mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),d(1,2),0,0,0,0), (0, - d(1,2)*a, - (d(1,2) - d(1,1)),0,0,0,0), (d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0), - d(1,2),0,0), (d(4,0),d(4,1),d(4,2),d(1,2)*a,d(1,1) + d(0,0) - d(1,2),0,0), (d(5,0),d(4,0) - d(3,0)*a + d(5,2)*a - d(6,3)*a,d(5,2),0,0, - (d(1,2) - 2*d(1,1)),0), (d(6,0),d(6,1),d(6,2),d(6,3),d(5,2) - d(3,0) - d(6,3),d(4,2) - d(3,1), 2*d(1,1) + d(0,0) - d(1,2))) {{x, 3, [ 0 ] [ ] [arbcomplex(179)] [ ] [arbcomplex(180)] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(181)] [ ] [ 0 ] }, {x - 1, 4, [arbcomplex(182)] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(183)] [ ] [arbcomplex(184)] [ ] [ 0 ] [ ] [arbcomplex(185)] }} Unknowns: {d(6,3),d(6,0),d(5,2),d(4,0),d(3,0),d(1,2),d(1,1),d(0,0),a} Unknowns: {d(6,3),d(6,0),d(5,2),d(4,0),d(3,0),d(1,2),d(1,1),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),d(1,2),0,0,0,0), (0, - d(1,2)*a, - (d(1,2) - d(1,1)),0,0,0,0), (d(3,0),0,0,d(1,1) + d(0,0), - d(1,2),0,0), (d(4,0),0,0,d(1,2)*a,d(1,1) + d(0,0) - d(1,2),0,0), (0,d(4,0) - d(3,0)*a + d(5,2)*a - d(6,3)*a,d(5,2),0,0, - (d(1,2) - 2*d(1,1)) ,0), (d(6,0),0,0,d(6,3),d(5,2) - d(3,0) - d(6,3),0,2*d(1,1) + d(0,0) - d(1,2))) Unknowns: {d(6,3),d(6,0),d(5,2),d(4,0),d(3,0),d(1,2),d(1,1),d(0,0),a} Unknowns: {d(6,3),d(6,0),d(5,2),d(4,0),d(3,0),d(1,2),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 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] Unknowns: {d(1,2),d(1,1),d(0,0),a} Unknowns: {d(1,2),d(1,1),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),d(1,2),0,0,0,0), (0, - d(1,2)*a, - (d(1,2) - d(1,1)),0,0,0,0), (0,0,0,d(1,1) + d(0,0), - d(1,2),0,0), (0,0,0,d(1,2)*a,d(1,1) + d(0,0) - d(1,2),0,0), (0,0,0,0,0, - (d(1,2) - 2*d(1,1)),0), (0,0,0,0,0,0,2*d(1,1) + d(0,0) - d(1,2))) Unknowns: {d(1,2),d(1,1),d(0,0),a} Unknowns: {d(1,2),d(1,1),d(0,0),a} t3:=D(1,2):= [0 0 0 0 0 0 0 ] [ ] [0 0 1 0 0 0 0 ] [ ] [0 - a -1 0 0 0 0 ] [ ] [0 0 0 0 -1 0 0 ] [ ] [0 0 0 a -1 0 0 ] [ ] [0 0 0 0 0 -1 0 ] [ ] [0 0 0 0 0 0 -1] {{x,1, [arbcomplex(194)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 1, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(195)] [ ] [arbcomplex(196)] }, {(x + 1)*x + a, 2, [ 0 ] [ ] [ arbcomplex(197) ] [ ----------------- ] [ x ] [ ] [ arbcomplex(197) ] [ ] [ - arbcomplex(198) ] [--------------------] [ x ] [ ] [ arbcomplex(198) ] [ ] [ 0 ] [ ] [ 0 ] }} Unknowns: {d(1,2),d(1,1),d(0,0),a} Unknowns: {d(1,2),d(1,1),d(0,0),a} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),d(1,2),0,0,0,0), (0, - d(1,2)*a, - (d(1,2) - d(1,1)),0,0,0,0), (0,0,0,d(1,1) + d(0,0), - d(1,2),0,0), (0,0,0,d(1,2)*a,d(1,1) + d(0,0) - d(1,2),0,0), (0,0,0,0,0, - (d(1,2) - 2*d(1,1)),0), (0,0,0,0,0,0,2*d(1,1) + d(0,0) - d(1,2))) For a:=1/4, t3 is not semisimple 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:= mat((1,0,0,0,0,0,0), (0,1,1,0,0,0,0), sqrt( - 4*a + 1) - 1 - (sqrt( - 4*a + 1) + 1) (0,----------------------,---------------------------,0,0,0,0), 2 2 (0,0,0,1,1,0,0), - (sqrt( - 4*a + 1) - 1) sqrt( - 4*a + 1) + 1 (0,0,0,---------------------------,----------------------,0,0), 2 2 (0,0,0,0,0,1,0), (0,0,0,0,0,0,1)) detP:=4*a - 1 P**(-1)*t1*P:= [1 0 0 0 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 1 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 1] P**(-1)*t2*P:= [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 0 0] [ ] [0 0 0 0 0 2 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t3*P:= mat((0,0,0,0,0,0,0), - (sqrt( - 4*a + 1) + 4*a - 1) (0,---------------------------------,0,0,0,0,0), 2*sqrt( - 4*a + 1) - (sqrt( - 4*a + 1) - 4*a + 1) (0,0,---------------------------------,0,0,0,0), 2*sqrt( - 4*a + 1) - (sqrt( - 4*a + 1) + 4*a - 1) (0,0,0,---------------------------------,0,0,0), 2*sqrt( - 4*a + 1) - (sqrt( - 4*a + 1) - 4*a + 1) (0,0,0,0,---------------------------------,0,0), 2*sqrt( - 4*a + 1) (0,0,0,0,0,-1,0), (0,0,0,0,0,0,-1)) matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((d(0,0),0,0,0,0,0,0),(0,( - (sqrt( - 4*a + 1)*d(1,2) + 4*d(1,2)*a - d(1,2) - 2*sqrt( - 4*a + 1)*d(1,1)))/(2*sqrt( - 4*a + 1)),0,0,0,0,0),(0,0,( - (sqrt( - 4 *a + 1)*d(1,2) - 4*d(1,2)*a + d(1,2) - 2*sqrt( - 4*a + 1)*d(1,1)))/(2*sqrt( - 4* a + 1)),0,0,0,0),(((sqrt( - 4*a + 1) + 1)*d(3,0) - 2*d(4,0))/(2*sqrt( - 4*a + 1) ),((sqrt( - 4*a + 1) + 1)*d(3,1) - 2*d(3,2)*a - 2*d(4,1) - (sqrt( - 4*a + 1) - 1 )*d(4,2))/(2*sqrt( - 4*a + 1)),((2*a - 1 - sqrt( - 4*a + 1))*d(3,2) + (sqrt( - 4 *a + 1) + 1)*d(3,1) - 2*d(4,1) + (sqrt( - 4*a + 1) + 1)*d(4,2))/(2*sqrt( - 4*a + 1)),(2*sqrt( - 4*a + 1)*(d(1,1) + d(0,0)) - (sqrt( - 4*a + 1) + 4*a - 1)*d(1,2) )/(2*sqrt( - 4*a + 1)),0,0,0),(((sqrt( - 4*a + 1) - 1)*d(3,0) + 2*d(4,0))/(2* sqrt( - 4*a + 1)),( - ((2*a - 1 + sqrt( - 4*a + 1))*d(3,2) - (sqrt( - 4*a + 1) - 1)*d(3,1) - 2*d(4,1) - (sqrt( - 4*a + 1) - 1)*d(4,2)))/(2*sqrt( - 4*a + 1)),(( sqrt( - 4*a + 1) - 1)*d(3,1) + 2*d(3,2)*a + 2*d(4,1) - (sqrt( - 4*a + 1) + 1)*d( 4,2))/(2*sqrt( - 4*a + 1)),0,(2*sqrt( - 4*a + 1)*(d(1,1) + d(0,0)) - (sqrt( - 4* a + 1) - 4*a + 1)*d(1,2))/(2*sqrt( - 4*a + 1)),0,0),(d(5,0),(2*(d(4,0) - d(3,0)* a) + (sqrt( - 4*a + 1) - 1)*d(5,2))/2,(2*(d(4,0) - d(3,0)*a) - (sqrt( - 4*a + 1) + 1)*d(5,2))/2,0,0, - (d(1,2) - 2*d(1,1)),0),(d(6,0),(sqrt( - 4*a + 1)*d(6,2) - d(6,2) + 2*d(6,1))/2,( - (sqrt( - 4*a + 1)*d(6,2) + d(6,2) - 2*d(6,1)))/2,( - ( (d(5,2) - d(3,0))*(sqrt( - 4*a + 1) - 1) - (sqrt( - 4*a + 1) + 1)*d(6,3)))/2,((d (5,2) - d(3,0))*(sqrt( - 4*a + 1) + 1) - (sqrt( - 4*a + 1) - 1)*d(6,3))/2,d(4,2) - d(3,1),2*d(1,1) + d(0,0) - d(1,2)))$ PP:= mat((1,0,0,0,0,0,0), (0,1,1,0,0,0,0), sqrt( - 4*a + 1) - 1 - (sqrt( - 4*a + 1) + 1) (0,----------------------,---------------------------,0,0,0,0), 2 2 (0,0,0,1,1,0,0), - (sqrt( - 4*a + 1) - 1) sqrt( - 4*a + 1) + 1 (0,0,0,---------------------------,----------------------,0,0), 2 2 (0,0,0,0,0,1,0), (0,0,0,0,0,0,1)) PP**(-1)*t1*PP:= [1 0 0 0 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 1 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 1] PP**(-1)*t2*PP:= [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 0 0] [ ] [0 0 0 0 0 2 0] [ ] [0 0 0 0 0 0 2] PP**(-1)*t3*PP:= mat((0,0,0,0,0,0,0), - (sqrt( - 4*a + 1) + 4*a - 1) (0,---------------------------------,0,0,0,0,0), 2*sqrt( - 4*a + 1) - (sqrt( - 4*a + 1) - 4*a + 1) (0,0,---------------------------------,0,0,0,0), 2*sqrt( - 4*a + 1) - (sqrt( - 4*a + 1) + 4*a - 1) (0,0,0,---------------------------------,0,0,0), 2*sqrt( - 4*a + 1) - (sqrt( - 4*a + 1) - 4*a + 1) (0,0,0,0,---------------------------------,0,0), 2*sqrt( - 4*a + 1) (0,0,0,0,0,-1,0), (0,0,0,0,0,0,-1)) avec PP:=P*Q:= mat((1,0,0,0,0,0,0), (0,1,1,0,0,0,0), sqrt( - 4*a + 1) - 1 - (sqrt( - 4*a + 1) + 1) (0,----------------------,---------------------------,0,0,0,0), 2 2 (0,0,0,1,1,0,0), - (sqrt( - 4*a + 1) - 1) sqrt( - 4*a + 1) + 1 (0,0,0,---------------------------,----------------------,0,0), 2 2 (0,0,0,0,0,1,0), (0,0,0,0,0,0,1)) MATDDIAGONALISE:= mat((d(0,0),0,0,0,0,0,0), (0,( - (sqrt( - 4*a + 1)*d(1,2) + 4*d(1,2)*a - d(1,2) - 2*sqrt( - 4*a + 1)*d(1,1)))/(2*sqrt( - 4*a + 1)),0,0,0,0,0), (0,0,( - (sqrt( - 4*a + 1)*d(1,2) - 4*d(1,2)*a + d(1,2) - 2*sqrt( - 4*a + 1)*d(1,1)))/(2*sqrt( - 4*a + 1)),0,0,0,0), (sqrt( - 4*a + 1) + 1)*d(3,0) - 2*d(4,0) (------------------------------------------,((sqrt( - 4*a + 1) + 1)*d(3,1) 2*sqrt( - 4*a + 1) - 2*d(3,2)*a - 2*d(4,1) - (sqrt( - 4*a + 1) - 1)*d(4,2))/(2 *sqrt( - 4*a + 1)),((2*a - 1 - sqrt( - 4*a + 1))*d(3,2) + (sqrt( - 4*a + 1) + 1)*d(3,1) - 2*d(4,1) + (sqrt( - 4*a + 1) + 1)*d(4,2))/(2*sqrt( - 4*a + 1)),( 2*sqrt( - 4*a + 1)*(d(1,1) + d(0,0)) - (sqrt( - 4*a + 1) + 4*a - 1)*d(1,2))/(2*sqrt( - 4*a + 1)),0,0,0), (sqrt( - 4*a + 1) - 1)*d(3,0) + 2*d(4,0) (------------------------------------------,( - ( 2*sqrt( - 4*a + 1) (2*a - 1 + sqrt( - 4*a + 1))*d(3,2) - (sqrt( - 4*a + 1) - 1)*d(3,1) - 2*d(4,1) - (sqrt( - 4*a + 1) - 1)*d(4,2)))/(2*sqrt( - 4*a + 1)),( (sqrt( - 4*a + 1) - 1)*d(3,1) + 2*d(3,2)*a + 2*d(4,1) - (sqrt( - 4*a + 1) + 1)*d(4,2))/(2*sqrt( - 4*a + 1)),0,( 2*sqrt( - 4*a + 1)*(d(1,1) + d(0,0)) - (sqrt( - 4*a + 1) - 4*a + 1)*d(1,2))/(2*sqrt( - 4*a + 1)),0,0), 2*(d(4,0) - d(3,0)*a) + (sqrt( - 4*a + 1) - 1)*d(5,2) (d(5,0),-------------------------------------------------------, 2 2*(d(4,0) - d(3,0)*a) - (sqrt( - 4*a + 1) + 1)*d(5,2) -------------------------------------------------------,0,0, 2 - (d(1,2) - 2*d(1,1)),0), sqrt( - 4*a + 1)*d(6,2) - d(6,2) + 2*d(6,1) (d(6,0),---------------------------------------------, 2 - (sqrt( - 4*a + 1)*d(6,2) + d(6,2) - 2*d(6,1)) --------------------------------------------------,( - ( 2 (d(5,2) - d(3,0))*(sqrt( - 4*a + 1) - 1) - (sqrt( - 4*a + 1) + 1)*d(6,3)))/2, (d(5,2) - d(3,0))*(sqrt( - 4*a + 1) + 1) - (sqrt( - 4*a + 1) - 1)*d(6,3) --------------------------------------------------------------------------, 2 d(4,2) - d(3,1),2*d(1,1) + d(0,0) - d(1,2))) on voit apparaitre les poids sur la diagonale r(1) := d(0,0) r(2) := - (sqrt( - 4*a + 1)*d(1,2) + 4*d(1,2)*a - d(1,2) - 2*sqrt( - 4*a + 1)*d(1,1)) -------------------------------------------------------------------------------- 2*sqrt( - 4*a + 1) r(3) := - (sqrt( - 4*a + 1)*d(1,2) - 4*d(1,2)*a + d(1,2) - 2*sqrt( - 4*a + 1)*d(1,1)) -------------------------------------------------------------------------------- 2*sqrt( - 4*a + 1) r(4) := 2*sqrt( - 4*a + 1)*(d(1,1) + d(0,0)) - (sqrt( - 4*a + 1) + 4*a - 1)*d(1,2) ---------------------------------------------------------------------------- 2*sqrt( - 4*a + 1) r(5) := 2*sqrt( - 4*a + 1)*(d(1,1) + d(0,0)) - (sqrt( - 4*a + 1) - 4*a + 1)*d(1,2) ---------------------------------------------------------------------------- 2*sqrt( - 4*a + 1) r(6) := - (d(1,2) - 2*d(1,1)) r(7) := 2*d(1,1) + d(0,0) - d(1,2) r(2)+r(3):= - (d(1,2) - 2*d(1,1)) - (4*a - 1)*d(1,2) r(2)-r(3):=--------------------- sqrt( - 4*a + 1) r(4)-(r(1)+r(2)):=0 r(5)-(r(1)+r(3)):=0 r(6)-(r(2)+r(3)):=0 r(7)-(r(1)+r(2)+r(3)):=0 r(1) := gamma1 r(2) := gamma2 r(3) := gamma3 r(4) := gamma1 + gamma2 r(5) := gamma1 + gamma3 r(6) := - (d(1,2) - 2*d(1,1)) r(7) := 2*d(1,1) + gamma1 - d(1,2) 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},a*x(4)}, {{0,2},x(4) + x(3)}, {{0,3},0}, {{0,4},0}, {{0,5},x(6)}, {{0,6},0}, {{1,2},x(5)}, {{1,3},0}, {{1,4},x(6)}, {{1,5},0}, {{1,6},0}, {{2,3},x(6)}, {{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) x(2)*(sqrt( - 4*a + 1) - 1) + 2*x(1) diaY(2):=-------------------------------------- 2 - (x(2)*(sqrt( - 4*a + 1) + 1) - 2*x(1)) diaY(3):=------------------------------------------- 2 - (x(4)*(sqrt( - 4*a + 1) - 1) - 2*x(3)) diaY(4):=------------------------------------------- 2 x(4)*(sqrt( - 4*a + 1) + 1) + 2*x(3) diaY(5):=-------------------------------------- 2 diaY(6):=x(5) diaY(7):=x(6) liste des commutateurs des diaY(i) : listcommutateurdesdiaY:={{{1,2}, ( - (sqrt( - 4*a + 1) + 4*a - 1)*diay(4))/(2*sqrt( - 4*a + 1))}, {{1,3}, ( - (sqrt( - 4*a + 1) - 4*a + 1)*diay(5))/(2*sqrt( - 4*a + 1))}, {{1,4},0}, {{1,5},0}, {{1,6},diay(7)}, {{1,7},0}, {{2,3}, - sqrt( - 4*a + 1)*diay(6)}, {{2,4},0}, {{2,5},sqrt( - 4*a + 1)*diay(7)}, {{2,6},0}, {{2,7},0}, {{3,4}, - sqrt( - 4*a + 1)*diay(7)}, {{3,5},0}, {{3,6},0}, {{3,7},0}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ Now we make explicit the isomorphism with an algebra of the book:$ Namely g_{7,3.1}$ (iL),$ for a neq 0$ pour a neq{0,1/4,-2}$ 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( - 4*a + 1) - 1 )/2,0,0,0),(0,0,0,0,( - (sqrt( - 4*a + 1) + 1))/2,0,0),(0,0,0,0,0, - sqrt( - 4*a + 1),0),(0,0,0,0,0,0, - sqrt( - 4*a + 1)))$ det(isom):= - (4*a - 1)*a$ ZZ(1):=diay(1)$ ZZ(2):=diay(2)$ ZZ(3):=diay(3)$ ZZ(4):=((sqrt( - 4*a + 1) - 1)*diay(4))/2$ ZZ(5):=( - (sqrt( - 4*a + 1) + 1)*diay(5))/2$ ZZ(6):= - sqrt( - 4*a + 1)*diay(6)$ ZZ(7):= - sqrt( - 4*a + 1)*diay(7)$ 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},(sqrt( - 4*a + 1)*zz(7) + zz(7))/2}$ {{2,6},0}$ {{2,7},0}$ {{3,4},(sqrt( - 4*a + 1)*zz(7) - zz(7))/2}$ {{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}$ On obtient donc les relations de commutations de $ g_{7,3.1}$ (iL)$ avec L:=(sqrt( - 4*a + 1) + 1)/2$ Et cela pour a different de {0,1/4,-2}.$ shortformdelta:={0, 1, ss, a, 1, ss, 0, 0, ss, 0, 0}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(a,1,0,0,0,0),(0,0,0,0,0,0),(0,0,0 ,0,1,0))$