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),(0,0,0,0,0,0),(0,0,b ,1,0,0))$ shortformdelta:={0, ss, 1, 0, ss, 0, ss, 0, 0, ss, 0, b}$ 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(3,0))$ Unknowns: {d(5,3),d(3,0)} Unknowns: {d(5,3),d(3,0)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(3,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,3) + d(4,1) - d(4, 0) + d(3,1)*b$ Unknowns: {d(6,3),d(4,1),d(4,0),d(3,1),b} Unknowns: {d(6,3),d(4,1),d(4,0),d(3,1),b} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(4,1) - d(4,0) + d(3,1)*b$ 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},6} qui est maintenant AA:=d(4,2) + d(3,2)*b + d(3,0 )$ Unknowns: {d(4,2),d(3,2),d(3,0),b} Unknowns: {d(4,2),d(3,2),d(3,0),b} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:= - (d(3,2)*b + d(3,0))$ on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,6)*b$ Unknowns: {d(0,6),b} Unknowns: {d(0,6),b} pas de selection possible de variable a coefficient independant des xi dans - d (0,6)*b on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,6)*b$ Unknowns: {d(1,6),b} Unknowns: {d(1,6),b} pas de selection possible de variable a coefficient independant des xi dans - d (1,6)*b on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,6)*b$ Unknowns: {d(2,6),b} Unknowns: {d(2,6),b} pas de selection possible de variable a coefficient independant des xi dans - d (2,6)*b on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,6)*b$ Unknowns: {d(3,6),b} Unknowns: {d(3,6),b} pas de selection possible de variable a coefficient independant des xi dans - d (3,6)*b on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,6)*b$ Unknowns: {d(4,6),b} Unknowns: {d(4,6),b} pas de selection possible de variable a coefficient independant des xi dans - d (4,6)*b on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6)*b$ Unknowns: {d(5,6),b} Unknowns: {d(5,6),b} pas de selection possible de variable a coefficient independant des xi dans - d (5,6)*b on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6)*b - 2*d(2,0) + d(1,1)*b + 2*d(0,0)*b$ Unknowns: {d(6,6),d(2,0),d(1,1),d(0,0),b} Unknowns: {d(6,6),d(2,0),d(1,1),d(0,0),b} bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=(b*( - d(6,6) + d(1,1) + 2*d(0,0)))/2$ 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)$ Unknown: d(5,6) Unknown: d(5,6) bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},6} qui est maintenant AA:= - d(6,6) + d(4,4) + d(3, 4)*b + d(0,0)$ Unknowns: {d(6,6),d(4,4),d(3,4),d(0,0),b} Unknowns: {d(6,6),d(4,4),d(3,4),d(0,0),b} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(4,4) + d(3,4)*b + d(0,0)$ on resout l'equation {{0,5},3} qui est maintenant AA:=d(1,5)$ Unknown: d(1,5) Unknown: d(1,5) bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},6} qui est maintenant AA:=d(4,5) + d(3,5)*b$ Unknowns: {d(4,5),d(3,5),b} Unknowns: {d(4,5),d(3,5),b} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:= - d(3,5)*b$ on resout l'equation {{0,6},6} qui est maintenant AA:=d(1,4)*b$ Unknowns: {d(1,4),b} Unknowns: {d(1,4),b} pas de selection possible de variable a coefficient independant des xi dans d(1, 4)*b 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},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 {{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(3,2)$ Unknowns: {d(5,4),d(3,2)} Unknowns: {d(5,4),d(3,2)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(3,2)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,4) - d(3,2)*b + d( 3,1) - d(3,0)$ Unknowns: {d(6,4),d(3,2),d(3,1),d(3,0),b} Unknowns: {d(6,4),d(3,2),d(3,1),d(3,0),b} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(3,2)*b + d(3,1) - d(3,0)$ on resout l'equation {{1,3},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 {{1,3},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 {{1,3},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 {{1,3},5} qui est maintenant AA:= - d(5,5) + 2*d(1,1) + d( 0,0)$ Unknowns: {d(5,5),d(1,1),d(0,0)} Unknowns: {d(5,5),d(1,1),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=2*d(1,1) + d(0,0)$ on resout l'equation {{1,3},6} qui est maintenant AA:=( - 2*d(6,5) + d(2,2)*b - 2*d(2,1) - d(0,2)*b**2 + 2*d(0,1)*b - d(0,0)*b)/2$ Unknowns: {d(6,5),d(2,2),d(2,1),d(0,2),d(0,1),d(0,0),b} Unknowns: {d(6,5),d(2,2),d(2,1),d(0,2),d(0,1),d(0,0),b} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=(d(2,2)*b - 2*d(2,1) - d(0,2)*b**2 + 2*d(0,1)*b - d(0,0) *b)/2$ on resout l'equation {{1,4},5} qui est maintenant AA:= - d(0,2)$ Unknown: d(0,2) Unknown: d(0,2) bonne inconnue W:=d(0,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,4},6} qui est maintenant AA:=d(1,1) + d(0,1) - d(0,0)$ Unknowns: {d(1,1),d(0,1),d(0,0)} Unknowns: {d(1,1),d(0,1),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:= - d(0,1) + d(0,0)$ 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},3},0}, {{{0,2},4},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},6},0}, {{{0,6},3},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},0},0}, {{{1,3},1},0}, {{{1,3},2},0}, {{{1,3},3},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},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},3},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},4},0}, {{{2,4},6},0}, {{{2,5},4},0}, {{{2,5},6},0}, {{{2,6},4},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},6},0}, {{{4,6},6},0}}$ il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(0,1),0,0,0,0,0),(0, - (d(0,1) - d(0,0)),0,0,0,0,0),(( - (d(2,2) - d(0,0))*b)/2,d(2,1),d(2,2),0,0,0,0),(d(3,0),d(3,1),d(3,2), - (d(0,1) - 2*d(0,0)) ,0,0,0),(d(4,0),d(4,1), - (d(3,2)*b + d(3,0)),((d(2,2) - d(0,0))*b)/2, - (d(0,1) - d(0,0) - d(2,2)),0,0),(d(5,0),d(5,1),d(5,2), - d(3,0),d(3,2), - (2*d(0,1) - 3 *d(0,0)),0),(d(6,0),d(6,1),d(6,2), - (d(4,0) - d(3,1)*b - d(4,1)),d(3,1) - d(3,0 ) - d(3,2)*b,((2*d(0,1) - d(0,0))*b - 2*d(2,1) + d(2,2)*b)/2, - (d(0,1) - 2*d(0, 0) - d(2,2))))$ 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 0 0 0 0] [ ] [0 0 b 1 0 0] pour shortformdelta:={0, ss, 1, 0, ss, 0, ss, 0, 0, ss, 0, b} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,2), d(2,1), d(0,1), d(0,0), b} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,2), d(2,1), d(0,1), d(0,0), b} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,2), d(2,1), d(0,1), d(0,0)}$ dim Der(gtildedelta):=15$ un element t1 d'un tore $ t1:=D(0,0)$ t1:= [ 1 0 0 0 0 0 0] [ ] [ 0 1 0 0 0 0 0] [ ] [ b ] [--- 0 0 0 0 0 0] [ 2 ] [ ] [ 0 0 0 2 0 0 0] [ ] [ - b ] [ 0 0 0 ------ 1 0 0] [ 2 ] [ ] [ 0 0 0 0 0 3 0] [ ] [ - b ] [ 0 0 0 0 0 ------ 2] [ 2 ] MATD:= mat((d(0,0),d(0,1),0,0,0,0,0), (0, - (d(0,1) - d(0,0)),0,0,0,0,0), - (d(2,2) - d(0,0))*b (------------------------,d(2,1),d(2,2),0,0,0,0), 2 (d(3,0),d(3,1),d(3,2), - (d(0,1) - 2*d(0,0)),0,0,0), (d(2,2) - d(0,0))*b (d(4,0),d(4,1), - (d(3,2)*b + d(3,0)),---------------------, 2 - (d(0,1) - d(0,0) - d(2,2)),0,0), (d(5,0),d(5,1),d(5,2), - d(3,0),d(3,2), - (2*d(0,1) - 3*d(0,0)),0), (d(6,0),d(6,1),d(6,2), - (d(4,0) - d(3,1)*b - d(4,1)), (2*d(0,1) - d(0,0))*b - 2*d(2,1) + d(2,2)*b d(3,1) - d(3,0) - d(3,2)*b,---------------------------------------------, 2 - (d(0,1) - 2*d(0,0) - d(2,2)))) {{x - 3, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - 2*arbcomplex(117) ] [----------------------] [ b ] [ ] [ arbcomplex(117) ] }, {x,1, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(118)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 2, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - 2*arbcomplex(119) ] [----------------------] [ b ] [ ] [ arbcomplex(119) ] [ ] [ 0 ] [ ] [ arbcomplex(120) ] }, {x - 1, 3, [ 2*arbcomplex(122) ] [-------------------] [ b ] [ ] [ arbcomplex(121) ] [ ] [ arbcomplex(122) ] [ ] [ 0 ] [ ] [ arbcomplex(123) ] [ ] [ 0 ] [ ] [ 0 ] }} Unknowns: {d(6,0),d(5,0),d(4,1),d(4,0),d(2,2),d(0,1),d(0,0),b} Unknowns: {d(6,0),d(5,0),d(4,1),d(4,0),d(2,2),d(0,1),d(0,0),b} commutant de t1 dans der(gtildedelta): mat((d(0,0),d(0,1),0,0,0,0,0), (0, - (d(0,1) - d(0,0)),0,0,0,0,0), - (d(2,2) - d(0,0))*b d(0,1)*b (------------------------,----------,d(2,2),0,0,0,0), 2 2 (0,0,0, - (d(0,1) - 2*d(0,0)),0,0,0), (d(2,2) - d(0,0))*b (d(4,0),d(4,1),0,---------------------, - (d(0,1) - d(0,0) - d(2,2)),0,0), 2 (d(5,0),0,0,0,0, - (2*d(0,1) - 3*d(0,0)),0), (d(0,1) - d(0,0) + d(2,2))*b (d(6,0),0,0,d(4,1) - d(4,0),0,------------------------------, 2 - (d(0,1) - 2*d(0,0) - d(2,2)))) on peut prendre comme element semi simple du commutant de t1 dans der(gtildede\ lta): Unknowns: {d(6,0),d(5,0),d(4,1),d(4,0),d(2,2),d(0,1),d(0,0),b} Unknowns: {d(6,0),d(5,0),d(4,1),d(4,0),d(2,2),d(0,1),d(0,0),b} t2:=D(0,1):= [0 1 0 0 0 0 0 ] [ ] [0 -1 0 0 0 0 0 ] [ ] [ b ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [0 0 0 -1 0 0 0 ] [ ] [0 0 0 0 -1 0 0 ] [ ] [0 0 0 0 0 -2 0 ] [ ] [ b ] [0 0 0 0 0 --- -1] [ 2 ] {{x + 2, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - 2*arbcomplex(128) ] [----------------------] [ b ] [ ] [ arbcomplex(128) ] }, {x, 2, [arbcomplex(129)] [ ] [ 0 ] [ ] [arbcomplex(130)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 1, 4, [ 2*arbcomplex(131) ] [ ------------------- ] [ b ] [ ] [ - 2*arbcomplex(131) ] [----------------------] [ b ] [ ] [ arbcomplex(131) ] [ ] [ arbcomplex(132) ] [ ] [ arbcomplex(133) ] [ ] [ 0 ] [ ] [ arbcomplex(134) ] }} Unknowns: {d(4,1),d(2,2),d(0,1),d(0,0),b} Unknowns: {d(4,1),d(2,2),d(0,1),d(0,0),b} commutant simultane de t1,t2 dans der(gtildedelta): mat((d(0,0),d(0,1),0,0,0,0,0), (0, - (d(0,1) - d(0,0)),0,0,0,0,0), - (d(2,2) - d(0,0))*b d(0,1)*b (------------------------,----------,d(2,2),0,0,0,0), 2 2 (0,0,0, - (d(0,1) - 2*d(0,0)),0,0,0), (d(2,2) - d(0,0))*b (0,d(4,1),0,---------------------, - (d(0,1) - d(0,0) - d(2,2)),0,0), 2 (0,0,0,0,0, - (2*d(0,1) - 3*d(0,0)),0), (d(0,1) - d(0,0) + d(2,2))*b (0,0,0,d(4,1),0,------------------------------, 2 - (d(0,1) - 2*d(0,0) - d(2,2)))) Unknowns: {d(4,1),d(2,2),d(0,1),d(0,0),b} Unknowns: {d(4,1),d(2,2),d(0,1),d(0,0),b} t3:=D(2,2):= [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 0 0 0] [ ] [0 0 0 0 1 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 1] Unknowns: {d(2,2),d(0,1),d(0,0)} Unknowns: {d(2,2),d(0,1),d(0,0)} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),d(0,1),0,0,0,0,0), (0, - (d(0,1) - d(0,0)),0,0,0,0,0), (0,0,d(2,2),0,0,0,0), (0,0,0, - (d(0,1) - 2*d(0,0)),0,0,0), (0,0,0,0, - (d(0,1) - d(0,0) - d(2,2)),0,0), (0,0,0,0,0, - (2*d(0,1) - 3*d(0,0)),0), (0,0,0,0,0,0, - (d(0,1) - 2*d(0,0) - d(2,2)))) 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:= [1 0 0 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 0 0 0] [ ] [0 0 0 0 1 0 0] [ ] [0 0 0 0 0 1 0] [ ] [0 0 0 0 0 0 1] P**(-1)*t1*P:= [1 0 0 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 1 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t2*P:= [0 1 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 -2 0 ] [ ] [0 0 0 0 0 0 -1] P**(-1)*t3*P:= [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 0 0 0] [ ] [0 0 0 0 1 0 0] [ ] [0 0 0 0 0 0 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),d(0,1),0,0,0,0,0),(0, - (d(0,1) - d(0,0)),0,0,0,0,0),(0,d(2,1),d(2,2 ),0,0,0,0),(d(3,0),d(3,1),d(3,2), - (d(0,1) - 2*d(0,0)),0,0,0),(d(4,0),d(4,1), - d(3,0),0, - (d(0,1) - d(0,0) - d(2,2)),0,0),(d(5,0),d(5,1),d(5,2), - d(3,0),d(3 ,2), - (2*d(0,1) - 3*d(0,0)),0),(d(6,0),d(6,1),d(6,2),d(4,1) - d(4,0),d(3,1) - d (3,0), - d(2,1), - (d(0,1) - 2*d(0,0) - d(2,2))))$ PP:= mat((1,-1,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,1,0,0,0),(0,0,0,0,1, 0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1))$ PP**(-1)*t1*PP:= mat((1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,2,0,0,0),(0,0,0,0,1,0 ,0),(0,0,0,0,0,3,0),(0,0,0,0,0,0,2))$ PP**(-1)*t2*PP:= mat((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,-2,0),(0,0,0,0,0,0,-1))$ PP**(-1)*t3*PP:= mat((0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,1,0 ,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,1))$ PP**(-1)*MATD*PP:= mat((d(0,0),0,0,0,0,0,0),(0, - (d(0,1) - d(0,0)),0,0,0,0,0),(0,d(2,1),d(2,2),0,0 ,0,0),(d(3,0),d(3,1) - d(3,0),d(3,2), - (d(0,1) - 2*d(0,0)),0,0,0),(d(4,0),d(4,1 ) - d(4,0), - d(3,0),0, - (d(0,1) - d(0,0) - d(2,2)),0,0),(d(5,0),d(5,1) - d(5,0 ),d(5,2), - d(3,0),d(3,2), - (2*d(0,1) - 3*d(0,0)),0),(d(6,0),d(6,1) - d(6,0),d( 6,2),d(4,1) - d(4,0),d(3,1) - d(3,0), - d(2,1), - (d(0,1) - 2*d(0,0) - d(2,2)))) $ avec PP:=P*Q:= mat((1,-1,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,1,0,0,0),(0,0,0,0,1, 0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1))$ MATDDIAGONALISE:= mat((d(0,0),0,0,0,0,0,0),(0, - (d(0,1) - d(0,0)),0,0,0,0,0),(0,d(2,1),d(2,2),0,0 ,0,0),(d(3,0),d(3,1) - d(3,0),d(3,2), - (d(0,1) - 2*d(0,0)),0,0,0),(d(4,0),d(4,1 ) - d(4,0), - d(3,0),0, - (d(0,1) - d(0,0) - d(2,2)),0,0),(d(5,0),d(5,1) - d(5,0 ),d(5,2), - d(3,0),d(3,2), - (2*d(0,1) - 3*d(0,0)),0),(d(6,0),d(6,1) - d(6,0),d( 6,2),d(4,1) - d(4,0),d(3,1) - d(3,0), - d(2,1), - (d(0,1) - 2*d(0,0) - d(2,2)))) $ on voit apparaitre les poids sur la diagonale$ ladiag := {{1,d(0,0)}, {2, - (d(0,1) - d(0,0))}, {3,d(2,2)}, {4, - (d(0,1) - 2*d(0,0))}, {5, - (d(0,1) - d(0,0) - d(2,2))}, {6, - (2*d(0,1) - 3*d(0,0))}, {7, - (d(0,1) - 2*d(0,0) - d(2,2))}}$ calcul de relations de commutation de la base diaY(j) diagonalisant le tore$ listcommutateursdesx := {{{0,1},x(3)}, {{0,2},0}, {{0,3},0}, {{0,4},x(6)}, {{0,5},0}, {{0,6},0}, {{1,2},x(4)}, {{1,3},x(5)}, {{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)$ diaY(2):=x(1) - x(0)$ diaY(3):=x(2)$ diaY(4):=x(3)$ diaY(5):=x(4)$ diaY(6):=x(5)$ diaY(7):=x(6)$ liste des commutateurs des diaY(i) := (diadiaY=diaY$ {{{1,2},diadiay(4)}, {{1,3},0}, {{1,4},0}, {{1,5},diadiay(7)}, {{1,6},0}, {{1,7},0}, {{2,3},diadiay(5)}, {{2,4},diadiay(6)}, {{2,5},0}, {{2,6},0}, {{2,7},0}, {{3,4}, - diadiay(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}}$ on pose :$ avec comme matrice de changement de base :$ [0 0 1 0 0 0 0 ] [ ] [1 0 0 0 0 0 0 ] [ ] [0 1 0 0 0 0 0 ] [ ] [0 0 0 0 -1 0 0 ] [ ] [0 0 0 1 0 0 0 ] [ ] [0 0 0 0 0 0 -1] [ ] [0 0 0 0 0 1 0 ] det(isom):= 1 *** zz declared operator ZZ(1):=diay(2) ZZ(2):=diay(3) ZZ(3):=diay(1) ZZ(4):=diay(5) ZZ(5):= - diay(4) ZZ(6):=diay(7) ZZ(7):= - diay(6) *** zzz declared operator liste des commutateurs des ZZ(i) (ZZZ=ZZ:= {{{1,2},zzz(4)}, {{1,3},zzz(5)}, {{1,4},0}, {{1,5},zzz(7)}, {{1,6},0}, {{1,7},0}, {{2,3},0}, {{2,4},0}, {{2,5},zzz(6)}, {{2,6},0}, {{2,7},0}, {{3,4},zzz(6)}, {{3,5},0}, {{3,6},0}, {{3,7},0}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}} On obtient donc les relations de commutation de g_{7,3.22}. Cela pour a = 0 et quel que soit b.