a:=0$ b:=0$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1 ,0,0,0))$ shortformdelta:={0, ss, 0, 0, ss, 0, ss, 0, 0, ss, 0, 1}$ phase 1 de la resolution des equations$ on resout l'equation {{0,1},4} qui est maintenant AA:= - d(2,0)$ Unknown: d(2,0) Unknown: d(2,0) bonne inconnue W:=d(2,0)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},6} qui est maintenant AA:=d(3,1)$ Unknown: d(3,1) Unknown: d(3,1) bonne inconnue W:=d(3,1)$ 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)$ Unknown: d(4,0) Unknown: d(4,0) bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},6} qui est maintenant AA:=d(3,2) - d(3,0)$ Unknowns: {d(3,2),d(3,0)} Unknowns: {d(3,2),d(3,0)} bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:=d(3,0)$ on resout l'equation {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},6} qui est maintenant AA:= - d(6,6) + d(3,3) + d(0, 0)$ Unknowns: {d(6,6),d(3,3),d(0,0)} Unknowns: {d(6,6),d(3,3),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(3,3) + d(0,0)$ on resout l'equation {{0,4},6} 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 {{0,5},6} 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 {{1,2},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},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)$ Unknown: d(6,4) Unknown: d(6,4) bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,3},4} 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 {{1,3},6} qui est maintenant AA:=d(2,1) + d(0,1)$ Unknowns: {d(2,1),d(0,1)} Unknowns: {d(2,1),d(0,1)} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:= - d(0,1)$ on resout l'equation {{1,4},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 {{1,5},4} 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 {{2,3},4} 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},5} qui est maintenant AA:=d(4,3)$ Unknown: d(4,3) Unknown: d(4,3) bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{2,3},6} qui est maintenant AA:=d(2,2) + d(0,2) - d(0,0)$ Unknowns: {d(2,2),d(0,2),d(0,0)} Unknowns: {d(2,2),d(0,2),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:= - d(0,2) + d(0,0)$ on resout l'equation {{2,4},0} qui est maintenant AA:= - d(0,5)$ Unknown: d(0,5) Unknown: d(0,5) bonne inconnue W:=d(0,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{2,4},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 {{2,4},4} qui est maintenant AA:= - d(4,5)$ Unknown: d(4,5) Unknown: d(4,5) bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{2,4},5} qui est maintenant AA:= - d(5,5) + d(1,1) - 2*d( 0,2) + 2*d(0,0)$ Unknowns: {d(5,5),d(1,1),d(0,2),d(0,0)} Unknowns: {d(5,5),d(1,1),d(0,2),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(1,1) - 2*d(0,2) + 2*d(0,0)$ on resout l'equation {{2,4},6} qui est maintenant AA:= - d(6,5)$ Unknown: d(6,5) Unknown: d(6,5) bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=0$ Derivation equations to cancel (Reduce output) : \\{{{{0,1},4},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},5},0}, {{{0,4},6},0}, {{{0,5},6},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}, {{{1,5},4},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},6},0}, {{{3,6},6},0}, {{{4,5},5},0}, {{{4,6},5},0}}$ il n'y a pas de phase 2$ collect_eq:={{{{0,1},4},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},5},0}, {{{0,4},6},0}, {{{0,5},6},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}, {{{1,5},4},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},6},0}, {{{3,6},6},0}, {{{4,5},5},0}, {{{4,6},5},0}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),0,d(0,2),d(0,3),0,0,0),(0,d(1,1),d(1,2),0,0,0,0),(0,0, - (d(0,2) - d (0,0)),0,0,0,0),(d(3,0),0,d(3,0),d(3,3),0,0,0),(0,d(4,1),d(4,2),0, - (d(0,2) - d (0,0) - d(1,1)),0,0),(d(5,0),d(5,1),d(5,2),d(5,3), - d(4,1), - (2*(d(0,2) - d(0, 0)) - d(1,1)),0),(d(6,0),d(6,1),d(6,2),d(6,3),0,0,d(3,3) + d(0,0)))$ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 1 0 0 0] pour shortformdelta:={0, ss, 0, 0, ss, 0, ss, 0, 0, ss, 0, 1} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,3), d(3,0), d(1,2), d(1,1), d(0,3), d(0,2), d(0,0)} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,3), d(3,0), d(1,2), d(1,1), d(0,3), d(0,2), d(0,0)} listeparametresMATD{d(6,3), d(6,2), d(6,1), d(6,0), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,3), d(3,0), d(1,2), d(1,1), d(0,3), d(0,2), d(0,0)}$ dim Der(gtildedelta):=17$ 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 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] MATD:= mat((d(0,0),0,d(0,2),d(0,3),0,0,0), (0,d(1,1),d(1,2),0,0,0,0), (0,0, - (d(0,2) - d(0,0)),0,0,0,0), (d(3,0),0,d(3,0),d(3,3),0,0,0), (0,d(4,1),d(4,2),0, - (d(0,2) - d(0,0) - d(1,1)),0,0), (d(5,0),d(5,1),d(5,2),d(5,3), - d(4,1), - (2*(d(0,2) - d(0,0)) - d(1,1)),0), (d(6,0),d(6,1),d(6,2),d(6,3),0,0,d(3,3) + d(0,0))) Unknowns: {d(6,2),d(6,0),d(4,2),d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(6,2),d(6,0),d(4,2),d(3,3),d(1,1),d(0,2),d(0,0)} commutant de t1 dans der(gtildedelta): mat((d(0,0),0,d(0,2),0,0,0,0), (0,d(1,1),0,0,0,0,0), (0,0, - (d(0,2) - d(0,0)),0,0,0,0), (0,0,0,d(3,3),0,0,0), (0,0,d(4,2),0, - (d(0,2) - d(0,0) - d(1,1)),0,0), (0,0,0,0,0, - (2*(d(0,2) - d(0,0)) - d(1,1)),0), (d(6,0),0,d(6,2),0,0,0,d(3,3) + d(0,0))) Unknowns: {d(6,2),d(6,0),d(4,2),d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(6,2),d(6,0),d(4,2),d(3,3),d(1,1),d(0,2),d(0,0)} 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 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] {{x - 1, 3, [ 0 ] [ ] [arbcomplex(25)] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(26)] [ ] [arbcomplex(27)] [ ] [ 0 ] }, {x, 4, [arbcomplex(28)] [ ] [ 0 ] [ ] [arbcomplex(29)] [ ] [arbcomplex(30)] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(31)] }} Unknowns: {d(6,2),d(6,0),d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(6,2),d(6,0),d(3,3),d(1,1),d(0,2),d(0,0)} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),0,d(0,2),0,0,0,0),(0,d(1,1),0,0,0,0,0),(0,0, - (d(0,2) - d(0,0)),0,0 ,0,0),(0,0,0,d(3,3),0,0,0),(0,0,0,0, - (d(0,2) - d(0,0) - d(1,1)),0,0),(0,0,0,0, 0, - (2*(d(0,2) - d(0,0)) - d(1,1)),0),(d(6,0),0,d(6,2),0,0,0,d(3,3) + d(0,0)))$ Unknowns: {d(6,2),d(6,0),d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(6,2),d(6,0),d(3,3),d(1,1),d(0,2),d(0,0)} t3:=D(3,3):= [0 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 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 1] Unknowns: {d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(3,3),d(1,1),d(0,2),d(0,0)} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),0,d(0,2),0,0,0,0), (0,d(1,1),0,0,0,0,0), (0,0, - (d(0,2) - d(0,0)),0,0,0,0), (0,0,0,d(3,3),0,0,0), (0,0,0,0, - (d(0,2) - d(0,0) - d(1,1)),0,0), (0,0,0,0,0, - (2*(d(0,2) - d(0,0)) - d(1,1)),0), (0,0,0,0,0,0,d(3,3) + d(0,0))) Unknowns: {d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(3,3),d(1,1),d(0,2),d(0,0)} t4:=D(0,2):= [0 0 1 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 0] {{x + 1, 2, [ - arbcomplex(32)] [ ] [ 0 ] [ ] [ arbcomplex(32) ] [ ] [ 0 ] [ ] [ arbcomplex(33) ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 2,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(34)] [ ] [ 0 ] }, {x, 4, [arbcomplex(35)] [ ] [arbcomplex(36)] [ ] [ 0 ] [ ] [arbcomplex(37)] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(38)] }} Unknowns: {d(3,3),d(1,1),d(0,2),d(0,0)} Unknowns: {d(3,3),d(1,1),d(0,2),d(0,0)} commutant simultane de t1,t2,t3,t4 dans der(gtildedelta): mat((d(0,0),0,d(0,2),0,0,0,0), (0,d(1,1),0,0,0,0,0), (0,0, - (d(0,2) - d(0,0)),0,0,0,0), (0,0,0,d(3,3),0,0,0), (0,0,0,0, - (d(0,2) - d(0,0) - d(1,1)),0,0), (0,0,0,0,0, - (2*(d(0,2) - d(0,0)) - d(1,1)),0), (0,0,0,0,0,0,d(3,3) + d(0,0))) rank 4 with maximal torus t1,t2,t3,t4 4 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 -1 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 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)*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 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] P**(-1)*t3*P:= [0 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 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 1] P**(-1)*t4*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 -2 0] [ ] [0 0 0 0 0 0 0] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((d(0,0),0,0,d(0,3),0,0,0),(0,d(1,1),d(1,2),0,0,0,0),(0,0, - (d(0,2) - d(0,0) ),0,0,0,0),(d(3,0),0,0,d(3,3),0,0,0),(0,d(4,1),d(4,2),0, - (d(0,2) - d(0,0) - d( 1,1)),0,0),(d(5,0),d(5,1),d(5,2) - d(5,0),d(5,3), - d(4,1), - (2*(d(0,2) - d(0,0 )) - d(1,1)),0),(d(6,0),d(6,1),d(6,2) - d(6,0),d(6,3),0,0,d(3,3) + d(0,0)))$ PP:= mat((1,0,-1,0,0,0,0),(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))$ avec PP:=P*Q:= mat((1,0,-1,0,0,0,0),(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,d(0,3),0,0,0),(0,d(1,1),d(1,2),0,0,0,0),(0,0, - (d(0,2) - d(0,0) ),0,0,0,0),(d(3,0),0,0,d(3,3),0,0,0),(0,d(4,1),d(4,2),0, - (d(0,2) - d(0,0) - d( 1,1)),0,0),(d(5,0),d(5,1),d(5,2) - d(5,0),d(5,3), - d(4,1), - (2*(d(0,2) - d(0,0 )) - d(1,1)),0),(d(6,0),d(6,1),d(6,2) - d(6,0),d(6,3),0,0,d(3,3) + d(0,0)))$ on voit apparaitre les poids sur la diagonale$ r(1) := d(0,0)$ r(2) := d(1,1)$ r(3) := - (d(0,2) - d(0,0))$ r(4) := d(3,3)$ r(5) := - (d(0,2) - d(0,0) - d(1,1))$ r(6) := - (2*(d(0,2) - d(0,0)) - d(1,1))$ r(7) := d(3,3) + d(0,0)$ r(1) := gamma2$ r(2) := gamma3$ r(3) := gamma1$ r(4) := gamma4$ r(5) := gamma1 + gamma3$ r(6) := 2*gamma1 + gamma3$ r(7) := gamma2 + gamma4$ Le systeme de poids est le systeme 4.5$ calcul de relations de commutation de la base diaY(j) diagonalisant le tore$ listcommutateursdesx := {{{0,1},0}, {{0,2},0}, {{0,3},x(6)}, {{0,4},0}, {{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(0)$ diaY(2):=x(1)$ diaY(3):=x(2) - x(0)$ diaY(4):=x(3)$ diaY(5):=x(4)$ diaY(6):=x(5)$ diaY(7):=x(6)$ liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},0}, {{1,3},0}, {{1,4},diay(7)}, {{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},diay(6)}, {{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_{7,4.5}$ 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),(0,0,-1,0,0,0,0),(1,0,0,0,0,0,0),(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))$ det(isom):= -1$ ZZ(1):=diay(3)$ ZZ(2):=diay(1)$ ZZ(3):= - diay(2)$ ZZ(4):=diay(4)$ ZZ(5):=diay(5)$ ZZ(6):=diay(6)$ ZZ(7):=diay(7)$ listcommutateursdesZZ:=$ {{1,2},0}$ {{1,3},zz(5)}$ {{1,4},0}$ {{1,5},zz(6)}$ {{1,6},0}$ {{1,7},0}$ {{2,3},0}$ {{2,4},zz(7)}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4},0}$ {{3,5},0}$ {{3,6},0}$ {{3,7},0}$ {{4,5},0}$ {{4,6},0}$ {{4,7},0}$ {{5,6},0}$ {{5,7},0}$ {{6,7},0}$ On obtient donc les relations de commutations de $ g_{7,4.5}$ Et cela pour a:=0, b:=0.$ shortformdelta:={0, ss, 0, 0, ss, 0, ss, 0, 0, ss, 0, 1}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1 ,0,0,0))$