In the present case 2 we suppose A=((0,0),(0,0)).$ a:=1/2$ b:=1$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(1,0,0,0,0,0),(0,1/2,0,0,0,0),(0,0,0,0,0,0),(0,0 ,1,0,( - 1)/2,0))$ shortformdelta:={1,0,ss,0,1/2,ss,0,0,ss,1,0}$ 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:=( - 2*d(4,3) + d(2,1))/2$ Unknowns: {d(4,3),d(2,1)} Unknowns: {d(4,3),d(2,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(2,1)/2$ on resout l'equation {{0,1},5} qui est maintenant AA:= - (d(5,3) + d(2,0))$ Unknowns: {d(5,3),d(2,0)} Unknowns: {d(5,3),d(2,0)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:=( - 2*d(6,3) - d(5,1) - 2 *d(4,0) + 2*d(3,1))/2$ Unknowns: {d(6,3),d(5,1),d(4,0),d(3,1)} Unknowns: {d(6,3),d(5,1),d(4,0),d(3,1)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=( - d(5,1) - 2*d(4,0) + 2*d(3,1))/2$ on resout l'equation {{0,2},0} qui est maintenant AA:=( - d(0,4))/2$ Unknown: d(0,4) Unknown: d(0,4) bonne inconnue W:=d(0,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},1} qui est maintenant AA:=( - d(1,4))/2$ Unknown: d(1,4) Unknown: d(1,4) bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},2} qui est maintenant AA:=( - d(2,4))/2$ Unknown: d(2,4) Unknown: d(2,4) bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},3} qui est maintenant AA:=( - d(3,4) + 2*d(1,2))/2$ Unknowns: {d(3,4),d(1,2)} Unknowns: {d(3,4),d(1,2)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=2*d(1,2)$ on resout l'equation {{0,2},4} qui est maintenant AA:=( - d(4,4) + d(2,2) + d(0 ,0))/2$ Unknowns: {d(4,4),d(2,2),d(0,0)} Unknowns: {d(4,4),d(2,2),d(0,0)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(2,2) + d(0,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=( - d(5,4) + 2*d(1,0))/2$ Unknowns: {d(5,4),d(1,0)} Unknowns: {d(5,4),d(1,0)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=2*d(1,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:=( - d(6,4) - d(5,2) + 2*d (3,2) - 2*d(3,0))/2$ Unknowns: {d(6,4),d(5,2),d(3,2),d(3,0)} Unknowns: {d(6,4),d(5,2),d(3,2),d(3,0)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(5,2) + 2*d(3,2) - 2*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:=( - 2*d(6,6) + 3*d(2,0) + 2*d(1,1) + 4*d(0,0))/2$ Unknowns: {d(6,6),d(2,0),d(1,1),d(0,0)} Unknowns: {d(6,6),d(2,0),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=(3*d(2,0) + 2*d(1,1) + 4*d(0,0))/2$ on resout l'equation {{0,4},6} qui est maintenant AA:=2*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 {{0,5},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},4} qui est maintenant AA:=d(2,5)/2$ Unknown: d(2,5) Unknown: d(2,5) bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},6} qui est maintenant AA:=( - 2*d(5,5) + 4*d(3,5) + 3*d(2,0) + 2*d(1,1) + 2*d(0,0))/4$ Unknowns: {d(5,5),d(3,5),d(2,0),d(1,1),d(0,0)} Unknowns: {d(5,5),d(3,5),d(2,0),d(1,1),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=(4*d(3,5) + 3*d(2,0) + 2*d(1,1) + 2*d(0,0))/2$ 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},3} qui est maintenant AA:= - (d(3,5) + d(0,2))$ Unknowns: {d(3,5),d(0,2)} Unknowns: {d(3,5),d(0,2)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{1,2},4} qui est maintenant AA:=( - 2*d(4,5) + d(0,1))/2$ Unknowns: {d(4,5),d(0,1)} Unknowns: {d(4,5),d(0,1)} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=d(0,1)/2$ on resout l'equation {{1,2},5} qui est maintenant AA:=(2*d(2,2) - 3*d(2,0) + 4* d(0,2) - 2*d(0,0))/2$ Unknowns: {d(2,2),d(2,0),d(0,2),d(0,0)} Unknowns: {d(2,2),d(2,0),d(0,2),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=(3*d(2,0) - 4*d(0,2) + 2*d(0,0))/2$ 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},6} qui est maintenant AA:=(3*d(2,1) + 2*d(0,1))/2$ 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:=( - 2*d(0,1))/3$ on resout l'equation {{1,4},6} qui est maintenant AA:= - 2*d(0,2)$ Unknown: d(0,2) Unknown: d(0,2) bonne inconnue W:=d(0,2)$ sa valeur doit etre WW:=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},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},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},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},3},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},5},0}, {{{1,5},6},0}, {{{1,6},3},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},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},6},0}, {{{3,5},6},0}, {{{3,6},6},0}, {{{4,5},6},0}, {{{4,6},6},0}, {{{5,6},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),(d(1,0),d(1,1),0,0,0,0,0),(d(2,0),( - 2*d(0,1))/3, (3*d(2,0) + 2*d(0,0))/2,0,0,0,0),(d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0),0,0,0),(d (4,0),d(4,1),d(4,2),( - d(0,1))/3,(3*d(2,0) + 4*d(0,0))/2,d(0,1)/2,0),(d(5,0),d( 5,1),d(5,2), - d(2,0),2*d(1,0),(2*(d(1,1) + d(0,0)) + 3*d(2,0))/2,0),(d(6,0),d(6 ,1),d(6,2),( - (2*(d(4,0) - d(3,1)) + d(5,1)))/2,2*(d(3,2) - d(3,0)) - d(5,2),d( 4,2) - d(3,1),(2*(d(1,1) + 2*d(0,0)) + 3*d(2,0))/2))$ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [1 0 0 0 0 0] [ ] [ 1 ] [0 --- 0 0 0 0] [ 2 ] [ ] [0 0 0 0 0 0] [ ] [ - 1 ] [0 0 1 0 ------ 0] [ 2 ] 1 pour shortformdelta:={1,0,ss,0,---,ss,0,0,ss,1,0} 2 Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(1,1), d(1,0), d(0,1), d(0,0)} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(1,1), d(1,0), d(0,1), d(0,0)} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(1,1), d(1,0), d(0,1), d(0,0)}$ dim Der(gtildedelta):=17$ 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 2 0 0] [ ] [0 0 0 0 0 1 0] [ ] [0 0 0 0 0 0 2] MATD:= mat((d(0,0),d(0,1),0,0,0,0,0), (d(1,0),d(1,1),0,0,0,0,0), - 2*d(0,1) 3*d(2,0) + 2*d(0,0) (d(2,0),-------------,---------------------,0,0,0,0), 3 2 (d(3,0),d(3,1),d(3,2),d(1,1) + d(0,0),0,0,0), - d(0,1) 3*d(2,0) + 4*d(0,0) d(0,1) (d(4,0),d(4,1),d(4,2),-----------,---------------------,--------,0), 3 2 2 2*(d(1,1) + d(0,0)) + 3*d(2,0) (d(5,0),d(5,1),d(5,2), - d(2,0),2*d(1,0),--------------------------------,0) 2 , - (2*(d(4,0) - d(3,1)) + d(5,1)) (d(6,0),d(6,1),d(6,2),-----------------------------------, 2 2*(d(3,2) - d(3,0)) - d(5,2),d(4,2) - d(3,1), 2*(d(1,1) + 2*d(0,0)) + 3*d(2,0) ----------------------------------)) 2 Unknowns: {d(5,2),d(5,0),d(3,2),d(3,0),d(2,0),d(1,1),d(0,0)} Unknowns: {d(5,2),d(5,0),d(3,2),d(3,0),d(2,0),d(1,1),d(0,0)} commutant de t1 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),0,0,0,0,0), 3*d(2,0) + 2*d(0,0) (d(2,0),0,---------------------,0,0,0,0), 2 (d(3,0),0,d(3,2),d(1,1) + d(0,0),0,0,0), 3*d(2,0) + 4*d(0,0) (0,0,0,0,---------------------,0,0), 2 2*(d(1,1) + d(0,0)) + 3*d(2,0) (d(5,0),0,d(5,2), - d(2,0),0,--------------------------------,0), 2 2*(d(1,1) + 2*d(0,0)) + 3*d(2,0) (0,0,0,0,2*(d(3,2) - d(3,0)) - d(5,2),0,----------------------------------)) 2 Unknowns: {d(5,2),d(5,0),d(3,2),d(3,0),d(2,0),d(1,1),d(0,0)} Unknowns: {d(5,2),d(5,0),d(3,2),d(3,0),d(2,0),d(1,1),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 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] Unknowns: {d(2,0),d(1,1),d(0,0)} Unknowns: {d(2,0),d(1,1),d(0,0)} commutant simultane de t1,t2 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),0,0,0,0,0), 3*d(2,0) + 2*d(0,0) (d(2,0),0,---------------------,0,0,0,0), 2 (0,0,0,d(1,1) + d(0,0),0,0,0), 3*d(2,0) + 4*d(0,0) (0,0,0,0,---------------------,0,0), 2 2*(d(1,1) + d(0,0)) + 3*d(2,0) (0,0,0, - d(2,0),0,--------------------------------,0), 2 2*(d(1,1) + 2*d(0,0)) + 3*d(2,0) (0,0,0,0,0,0,----------------------------------)) 2 Unknowns: {d(2,0),d(1,1),d(0,0)} Unknowns: {d(2,0),d(1,1),d(0,0)} t3:=D(2,0):= [0 0 0 0 0 0 0 ] [ ] [0 0 0 0 0 0 0 ] [ ] [ 3 ] [1 0 --- 0 0 0 0 ] [ 2 ] [ ] [0 0 0 0 0 0 0 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 -1 0 --- 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 0 0 ---] [ 2 ] {{x, 3, [ - 3*arbcomplex(60) ] [---------------------] [ 2 ] [ ] [ arbcomplex(59) ] [ ] [ arbcomplex(60) ] [ ] [ 3*arbcomplex(61) ] [ ------------------ ] [ 2 ] [ ] [ 0 ] [ ] [ arbcomplex(61) ] [ ] [ 0 ] }, {2*x - 3, 4, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(62)] [ ] [ 0 ] [ ] [arbcomplex(63)] [ ] [arbcomplex(64)] [ ] [arbcomplex(65)] }} Unknowns: {d(2,0),d(1,1),d(0,0)} Unknowns: {d(2,0),d(1,1),d(0,0)} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),0,0,0,0,0,0), (0,d(1,1),0,0,0,0,0), 3*d(2,0) + 2*d(0,0) (d(2,0),0,---------------------,0,0,0,0), 2 (0,0,0,d(1,1) + d(0,0),0,0,0), 3*d(2,0) + 4*d(0,0) (0,0,0,0,---------------------,0,0), 2 2*(d(1,1) + d(0,0)) + 3*d(2,0) (0,0,0, - d(2,0),0,--------------------------------,0), 2 2*(d(1,1) + 2*d(0,0)) + 3*d(2,0) (0,0,0,0,0,0,----------------------------------)) 2 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:= [ 1 0 0 0 0 0 0] [ ] [ 0 1 0 0 0 0 0] [ ] [ - 2 ] [------ 0 1 0 0 0 0] [ 3 ] [ ] [ 0 0 0 1 0 0 0] [ ] [ 0 0 0 0 1 0 0] [ ] [ 2 ] [ 0 0 0 --- 0 1 0] [ 3 ] [ ] [ 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 1 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 1 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 0 0 0] [ ] [0 0 0 0 0 1 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 ] [ ] [ 3 ] [0 0 --- 0 0 0 0 ] [ 2 ] [ ] [0 0 0 0 0 0 0 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 0 --- 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 0 0 ---] [ 2 ] 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),(d(1,0),d(1,1),0,0,0,0,0),(0,0,(3*d(2,0) + 2*d(0,0 ))/2,0,0,0,0),(( - (2*d(3,2) - 3*d(3,0)))/3,d(3,1),d(3,2),d(1,1) + d(0,0),0,0,0) ,(( - (2*d(4,2) - 3*d(4,0)))/3,d(4,1),d(4,2),0,(3*d(2,0) + 4*d(0,0))/2,d(0,1)/2, 0),((9*d(5,0) + 4*d(3,2) - 6*d(3,0) - 6*d(5,2))/9,(3*d(5,1) - 2*d(3,1))/3,(3*d(5 ,2) - 2*d(3,2))/3,0,2*d(1,0),(2*(d(1,1) + d(0,0)) + 3*d(2,0))/2,0),(( - (2*d(6,2 ) - 3*d(6,0)))/3,d(6,1),d(6,2),( - (2*(3*d(4,0) - d(3,1) - 2*d(4,2)) + 3*d(5,1)) )/6,2*(d(3,2) - d(3,0)) - d(5,2),d(4,2) - d(3,1),(2*(d(1,1) + 2*d(0,0)) + 3*d(2, 0))/2))$ PP:= [ 1 0 0 0 0 0 0] [ ] [ 0 1 0 0 0 0 0] [ ] [ - 2 ] [------ 0 1 0 0 0 0] [ 3 ] [ ] [ 0 0 0 1 0 0 0] [ ] [ 0 0 0 0 1 0 0] [ ] [ 2 ] [ 0 0 0 --- 0 1 0] [ 3 ] [ ] [ 0 0 0 0 0 0 1] avec PP:=P*Q:= [ 1 0 0 0 0 0 0] [ ] [ 0 1 0 0 0 0 0] [ ] [ - 2 ] [------ 0 1 0 0 0 0] [ 3 ] [ ] [ 0 0 0 1 0 0 0] [ ] [ 0 0 0 0 1 0 0] [ ] [ 2 ] [ 0 0 0 --- 0 1 0] [ 3 ] [ ] [ 0 0 0 0 0 0 1] MATDDIAGONALISE:= mat((d(0,0),d(0,1),0,0,0,0,0), (d(1,0),d(1,1),0,0,0,0,0), 3*d(2,0) + 2*d(0,0) (0,0,---------------------,0,0,0,0), 2 - (2*d(3,2) - 3*d(3,0)) (--------------------------,d(3,1),d(3,2),d(1,1) + d(0,0),0,0,0), 3 - (2*d(4,2) - 3*d(4,0)) 3*d(2,0) + 4*d(0,0) d(0,1) (--------------------------,d(4,1),d(4,2),0,---------------------,--------,0 3 2 2 ), 9*d(5,0) + 4*d(3,2) - 6*d(3,0) - 6*d(5,2) 3*d(5,1) - 2*d(3,1) (-------------------------------------------,---------------------, 9 3 3*d(5,2) - 2*d(3,2) 2*(d(1,1) + d(0,0)) + 3*d(2,0) ---------------------,0,2*d(1,0),--------------------------------,0), 3 2 - (2*d(6,2) - 3*d(6,0)) (--------------------------,d(6,1),d(6,2), 3 - (2*(3*d(4,0) - d(3,1) - 2*d(4,2)) + 3*d(5,1)) --------------------------------------------------, 6 2*(d(3,2) - d(3,0)) - d(5,2),d(4,2) - d(3,1), 2*(d(1,1) + 2*d(0,0)) + 3*d(2,0) ----------------------------------)) 2 on voit apparaitre les poids sur la diagonale r(1) := d(0,0) r(2) := d(1,1) 3*d(2,0) + 2*d(0,0) r(3) := --------------------- 2 r(4) := d(1,1) + d(0,0) 3*d(2,0) + 4*d(0,0) r(5) := --------------------- 2 2*(d(1,1) + d(0,0)) + 3*d(2,0) r(6) := -------------------------------- 2 2*(d(1,1) + 2*d(0,0)) + 3*d(2,0) r(7) := ---------------------------------- 2 r(1) := gamma1 r(2) := gamma2 r(3) := gamma3 r(4) := gamma1 + gamma2 r(5) := gamma1 + gamma3 r(6) := gamma2 + gamma3 r(7) := gamma1 + gamma2 + gamma3 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},x(3)}, x(4) {{0,2},------}, 2 {{0,3},x(6)}, {{0,4},0}, - x(6) {{0,5},---------}, 2 {{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}} - 2*x(2) + 3*x(0) diaY(1):=-------------------- 3 diaY(2):=x(1) diaY(3):=x(2) 2*x(5) + 3*x(3) diaY(4):=----------------- 3 diaY(5):=x(4) diaY(6):=x(5) diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},diay(4)}, {{1,3},diay(5)/2}, {{1,4},0}, {{1,5},0}, {{1,6},( - diay(7))/2}, {{1,7},0}, {{2,3},diay(6)}, {{2,4},0}, {{2,5},diay(7)}, {{2,6},0}, {{2,7},0}, {{3,4},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)$ with L:=-1.$ 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((u1,0,0,0,0,0,0),(0,u2,0,0,0,0,0),(0,0,u3,0,0,0,0),(0,0,0,u1*u2,0,0,0),(0,0, 0,0,(u1*u3)/2,0,0),(0,0,0,0,0,u2*u3,0),(0,0,0,0,0,0,( - u1*u2*u3)/2))$ det(isom):= ( - u1**4*u2**4*u3**4)/4$ ZZ(1):=diay(1)*u1$ ZZ(2):=diay(2)*u2$ ZZ(3):=diay(3)*u3$ ZZ(4):=diay(4)*u1*u2$ ZZ(5):=(diay(5)*u1*u3)/2$ ZZ(6):=diay(6)*u2*u3$ ZZ(7):=( - diay(7)*u1*u2*u3)/2$ 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}, - zz(7)}$ {{2,6},0}$ {{2,7},0}$ {{3,4}, - 2*zz(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 obtient donc les relations de commutations de $ g_{7,3.1}$ (iL)$ with L:=-1.$ Et cela pour a:=1/2, b:=1.$ shortformdelta:={1,0,ss,0,1/2,ss,0,0,ss,1,0}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(1,0,0,0,0,0),(0,1/2,0,0,0,0),(0,0,0,0,0,0),(0,0 ,1,0,( - 1)/2,0))$