Case a:=1/2$ a:=1/2$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1/2,0,0,0),(0,1,0,0,0,0),(0,0 ,0,0,( - 1)/2,0))$ shortformdelta:={1, ss, 0, 0, ss, 1/2, ss, 1, 0, ss, 0, 0}$ phase 1 de la resolution des equations$ on resout l'equation {{0,1},0} 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 {{0,1},1} 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 {{0,1},2} qui est maintenant AA:= - d(2,2) + d(1,1) + d(0, 0)$ Unknowns: {d(2,2),d(1,1),d(0,0)} Unknowns: {d(2,2),d(1,1),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=d(1,1) + d(0,0)$ on resout l'equation {{0,1},3} qui est maintenant AA:= - d(3,2)$ Unknown: d(3,2) Unknown: d(3,2) bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},4} qui est maintenant AA:=( - 2*d(4,2) + d(3,1) - 2 *d(2,0))/2$ Unknowns: {d(4,2),d(3,1),d(2,0)} Unknowns: {d(4,2),d(3,1),d(2,0)} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:=(d(3,1) - 2*d(2,0))/2$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,2) - d(3,0) + d(2, 1)$ Unknowns: {d(5,2),d(3,0),d(2,1)} Unknowns: {d(5,2),d(3,0),d(2,1)} bonne inconnue W:=d(5,2)$ sa valeur doit etre WW:= - d(3,0) + d(2,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:=( - 2*d(6,2) - d(5,1) - 2 *d(4,0))/2$ Unknowns: {d(6,2),d(5,1),d(4,0)} Unknowns: {d(6,2),d(5,1),d(4,0)} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=( - d(5,1) - 2*d(4,0))/2$ on resout l'equation {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},4} qui est maintenant AA:= - d(4,5) + d(1,0)$ Unknowns: {d(4,5),d(1,0)} Unknowns: {d(4,5),d(1,0)} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,5) + d(1,1) + 2*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:=d(1,1) + 2*d(0,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:=( - 2*d(6,5) + 3*d(3,0) - d(2,1))/2$ Unknowns: {d(6,5),d(3,0),d(2,1)} Unknowns: {d(6,5),d(3,0),d(2,1)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=(3*d(3,0) - d(2,1))/2$ on resout l'equation {{0,3},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 {{0,3},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 {{0,3},2} qui est maintenant AA:=( - d(2,4) + 2*d(1,3))/2$ Unknowns: {d(2,4),d(1,3)} Unknowns: {d(2,4),d(1,3)} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:=2*d(1,3)$ on resout l'equation {{0,3},3} qui est maintenant AA:=( - d(3,4))/2$ Unknown: d(3,4) Unknown: d(3,4) bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,3},4} qui est maintenant AA:=( - d(4,4) + d(3,3) + d(0 ,0))/2$ Unknowns: {d(4,4),d(3,3),d(0,0)} Unknowns: {d(4,4),d(3,3),d(0,0)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(3,3) + d(0,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:=( - d(5,4) + 2*d(2,3) + 2 *d(1,0))/2$ Unknowns: {d(5,4),d(2,3),d(1,0)} Unknowns: {d(5,4),d(2,3),d(1,0)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=2*(d(2,3) + d(1,0))$ on resout l'equation {{0,3},6} qui est maintenant AA:=( - d(6,4) - d(5,3) - 2*d (2,0))/2$ Unknowns: {d(6,4),d(5,3),d(2,0)} Unknowns: {d(6,4),d(5,3),d(2,0)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(5,3) - 2*d(2,0)$ on resout l'equation {{0,4},5} qui est maintenant AA:=2*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 {{0,4},6} 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 {{0,5},0} qui est maintenant AA:=d(0,6)/2$ 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)/2$ 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)/2$ 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)/2$ Unknown: d(3,6) Unknown: d(3,6) bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},4} qui est maintenant AA:=d(4,6)/2$ Unknown: d(4,6) Unknown: d(4,6) bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},5} qui est maintenant AA:=d(5,6)/2$ 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(1,1) - 3*d(0, 0))/2$ Unknowns: {d(6,6),d(1,1),d(0,0)} Unknowns: {d(6,6),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(1,1) + 3*d(0,0)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(3,3) + 2*d(1,1)$ Unknowns: {d(3,3),d(1,1)} Unknowns: {d(3,3),d(1,1)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=2*d(1,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - 2*d(1,0) + d(0,1)$ Unknowns: {d(1,0),d(0,1)} Unknowns: {d(1,0),d(0,1)} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(0,1)/2$ on resout l'equation {{1,2},6} qui est maintenant AA:=(2*d(5,3) + 3*d(3,1) + 2* d(2,0))/2$ Unknowns: {d(5,3),d(3,1),d(2,0)} Unknowns: {d(5,3),d(3,1),d(2,0)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=( - 3*d(3,1) - 2*d(2,0))/2$ on resout l'equation {{1,3},2} 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 {{1,3},5} qui est maintenant AA:=2*(d(1,1) - d(0,0))$ Unknowns: {d(1,1),d(0,0)} Unknowns: {d(1,1),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:=d(0,0)$ on resout l'equation {{1,3},6} qui est maintenant AA:=(2*d(4,3) - 3*d(3,0) - d( 2,1))/2$ Unknowns: {d(4,3),d(3,0),d(2,1)} Unknowns: {d(4,3),d(3,0),d(2,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=(3*d(3,0) + d(2,1))/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},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},2},0}, {{{0,4},4},0}, {{{0,4},5},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},2},0}, {{{0,6},4},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},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},2},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},2},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},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},4},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},4},0}, {{{3,6},5},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(0,1)/2,d(0,0),0,0,0,0,0),(d(2,0),d(2,1),2*d(0,0 ),0,0,0,0),(d(3,0),d(3,1),0,2*d(0,0),0,0,0),(d(4,0),d(4,1),(d(3,1) - 2*d(2,0))/2 ,(3*d(3,0) + d(2,1))/2,3*d(0,0),d(0,1)/2,0),(d(5,0),d(5,1), - (d(3,0) - d(2,1)), ( - (3*d(3,1) + 2*d(2,0)))/2,d(0,1),3*d(0,0),0),(d(6,0),d(6,1),( - (d(5,1) + 2*d (4,0)))/2,d(6,3),(3*d(3,1) - 2*d(2,0))/2,(3*d(3,0) - d(2,1))/2,4*d(0,0)))$ pour delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1/2,0,0,0),(0,1,0,0,0,0),(0,0 ,0,0,( - 1)/2,0))$ pour shortformdelta:={1, ss, 0, 0, ss, 1/2, ss, 1, 0, ss, 0, 0}$ Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)} dim Der(gtildedelta):=13$ Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)} listeparametresMATD{d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,1), d(0,0)}$ 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] [ ] [0 0 2 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 3 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 4] commutant de t1 := mat((d(0,0),d(0,1),0,0,0,0,0), d(0,1) (--------,d(0,0),0,0,0,0,0), 2 (d(2,0),d(2,1),2*d(0,0),0,0,0,0), (d(3,0),d(3,1),0,2*d(0,0),0,0,0), d(3,1) - 2*d(2,0) 3*d(3,0) + d(2,1) d(0,1) (d(4,0),d(4,1),-------------------,-------------------,3*d(0,0),--------,0), 2 2 2 - (3*d(3,1) + 2*d(2,0)) (d(5,0),d(5,1), - (d(3,0) - d(2,1)),--------------------------,d(0,1), 2 3*d(0,0),0), - (d(5,1) + 2*d(4,0)) 3*d(3,1) - 2*d(2,0) (d(6,0),d(6,1),------------------------,d(6,3),---------------------, 2 2 3*d(3,0) - d(2,1) -------------------,4*d(0,0))) 2 t2:=D(0,1) [ 0 1 0 0 0 0 0] [ ] [ 1 ] [--- 0 0 0 0 0 0] [ 2 ] [ ] [ 0 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0 0] [ ] [ 1 ] [ 0 0 0 0 0 --- 0] [ 2 ] [ ] [ 0 0 0 0 1 0 0] [ ] [ 0 0 0 0 0 0 0] le calcul du commutant de t1 et t2 dans der(gtildedelta): commutant de t1 et t2 dans der(gtildedelta): commutant de t1 et t2:= [ d(0,0) d(0,1) 0 0 0 0 0 ] [ ] [ d(0,1) ] [-------- d(0,0) 0 0 0 0 0 ] [ 2 ] [ ] [ 0 0 2*d(0,0) 0 0 0 0 ] [ ] [ 0 0 0 2*d(0,0) 0 0 0 ] [ ] [ d(0,1) ] [ d(4,0) d(4,1) 0 0 3*d(0,0) -------- 0 ] [ 2 ] [ ] [ d(4,1) 2*d(4,0) 0 0 d(0,1) 3*d(0,0) 0 ] [ ] [ 0 0 - 2*d(4,0) d(6,3) 0 0 4*d(0,0)] t1,t2 est un tore maximal. t1 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:= [sqrt(2) sqrt(2) 0 0 0 0 0] [ ] [ 1 -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(2) sqrt(2) 0] [ ] [ 0 0 0 0 2 -2 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 2 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 3 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 4] P**(-1)*t2*P:= [ 1 ] [--------- 0 0 0 0 0 0] [ sqrt(2) ] [ ] [ - 1 ] [ 0 --------- 0 0 0 0 0] [ sqrt(2) ] [ ] [ 0 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0 0] [ ] [ 1 ] [ 0 0 0 0 --------- 0 0] [ sqrt(2) ] [ ] [ - 1 ] [ 0 0 0 0 0 --------- 0] [ sqrt(2) ] [ ] [ 0 0 0 0 0 0 0] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= d(0,1) + sqrt(2)*d(0,0) mat((-------------------------,0,0,0,0,0,0), sqrt(2) - (d(0,1) - sqrt(2)*d(0,0)) (0,------------------------------,0,0,0,0,0), sqrt(2) (0,0,2*d(0,0),0,0,0,0), (0,0,0,2*d(0,0),0,0,0), d(4,1) + sqrt(2)*d(4,0) d(0,1) + 3*sqrt(2)*d(0,0) (-------------------------,0,0,0,---------------------------,0,0), sqrt(2) sqrt(2) - (d(4,1) - sqrt(2)*d(4,0)) - (d(0,1) - 3*sqrt(2)*d(0,0)) (0,------------------------------,0,0,0,--------------------------------,0), sqrt(2) sqrt(2) (0,0, - 2*d(4,0),d(6,3),0,0,4*d(0,0))) PP:= [sqrt(2) sqrt(2) 0 0 0 0 0] [ ] [ 1 -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(2) sqrt(2) 0] [ ] [ 0 0 0 0 2 -2 0] [ ] [ 0 0 0 0 0 0 1] avec PP:=P*Q:= [sqrt(2) sqrt(2) 0 0 0 0 0] [ ] [ 1 -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(2) sqrt(2) 0] [ ] [ 0 0 0 0 2 -2 0] [ ] [ 0 0 0 0 0 0 1] MATDDIAGONALISE:= d(0,1) + sqrt(2)*d(0,0) mat((-------------------------,0,0,0,0,0,0), sqrt(2) - (d(0,1) - sqrt(2)*d(0,0)) (0,------------------------------,0,0,0,0,0), sqrt(2) (0,0,2*d(0,0),0,0,0,0), (0,0,0,2*d(0,0),0,0,0), d(4,1) + sqrt(2)*d(4,0) d(0,1) + 3*sqrt(2)*d(0,0) (-------------------------,0,0,0,---------------------------,0,0), sqrt(2) sqrt(2) - (d(4,1) - sqrt(2)*d(4,0)) - (d(0,1) - 3*sqrt(2)*d(0,0)) (0,------------------------------,0,0,0,--------------------------------,0), sqrt(2) sqrt(2) (0,0, - 2*d(4,0),d(6,3),0,0,4*d(0,0))) on voit apparaitre les poids sur la diagonale d(0,1) + sqrt(2)*d(0,0) r(1) := ------------------------- sqrt(2) - (d(0,1) - sqrt(2)*d(0,0)) r(2) := ------------------------------ sqrt(2) r(3) := 2*d(0,0) r(4) := 2*d(0,0) d(0,1) + 3*sqrt(2)*d(0,0) r(5) := --------------------------- sqrt(2) - (d(0,1) - 3*sqrt(2)*d(0,0)) r(6) := -------------------------------- sqrt(2) r(7) := 4*d(0,0) r(1) := gamma1 r(2) := gamma2 r(3) := gamma1 + gamma2 r(4) := gamma1 + gamma2 r(5) := 2*gamma1 + gamma2 r(6) := gamma1 + 2*gamma2 r(7) := 2*(gamma1 + gamma2) Le systeme de poids est le systeme 2.37 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(2)}, {{0,2},x(5)}, x(4) {{0,3},------}, 2 {{0,4},0}, - x(6) {{0,5},---------}, 2 {{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(1) + sqrt(2)*x(0) diaY(2):= - x(1) + sqrt(2)*x(0) diaY(3):=x(2) diaY(4):=x(3) diaY(5):=2*x(5) + sqrt(2)*x(4) diaY(6):= - 2*x(5) + sqrt(2)*x(4) diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2}, - 2*sqrt(2)*diay(3)}, {{1,3},diay(5)/sqrt(2)}, {{1,4},diay(5)/2}, {{1,5},0}, {{1,6},2*sqrt(2)*diay(7)}, {{1,7},0}, {{2,3},( - diay(6))/sqrt(2)}, {{2,4},diay(6)/2}, {{2,5}, - 2*sqrt(2)*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_{7,2.37}$ (iL)$ and that for a neqconditionssura$ 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, - sqrt(2)*u1*u2, - 2*sqrt(2)*u1*u2,0 ,0,0),(0,0, - 2*u1*u2,0,0,0,0),(0,0,0,0, - 2*u1**2*u2,0,0),(0,0,0,0,0,2*u1*u2**2 ,0),(0,0,0,0,0,0,4*sqrt(2)*u1**2*u2**2))$ det(isom):= 128*u1**8*u2**8$ ZZ(1):=diay(1)*u1$ ZZ(2):=diay(2)*u2$ ZZ(3):= - (2*diay(4) + sqrt(2)*diay(3))*u1*u2$ ZZ(4):= - 2*sqrt(2)*diay(3)*u1*u2$ ZZ(5):= - 2*diay(5)*u1**2*u2$ ZZ(6):=2*diay(6)*u1*u2**2$ ZZ(7):=4*sqrt(2)*diay(7)*u1**2*u2**2$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(5)}$ {{1,4},zz(5)}$ {{1,5},0}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},0}$ {{2,4},zz(6)}$ {{2,5},zz(7)}$ {{2,6},0}$ {{2,7},0}$ {{3,4},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}$ We get the commutation relations of$ g_{7,2.37}$ Et cela pour a:=1/2$ shortformdelta:={1, ss, 0, 0, ss, 1/2, ss, 1, 0, ss, 0, 0}$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,1/2,0,0,0),(0,1,0,0,0,0),(0,0 ,0,0,( - 1)/2,0))$