!The! generic! nilpotent! derivation! as! in! (!Cohomology! tables! page! 50)! : ! the! eigen\ values are! 0$ by! subtracting! adjoints! one! then! may! suppose! xi(4,1)=xi(4,2)=xi(5,1)=xi(6 ,1)=x\ i(6,2)=0$ delta:= mat((0,0,0,0,0,0),(xi(2,1),0,0,0,0,0),(xi(3,1),xi(3,2),0,0,0,0),(0,0,0,0,0,0),(0 ,xi(5,2), - xi(2,1),0,0,0),(0,0,xi(6,3),xi(5,2) - xi(3,1),xi(2,1),0))$ We denote this delta by the shortform$ shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,2), ss, xi(6,3)}$ paramindexeslist:={{2,1},{3,1},{3,2},{5,2},{6,3}}$ 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,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},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 {{0,1},6} qui est maintenant AA:= - d(5,0) + d(3,1)$ Unknowns: {d(5,0),d(3,1)} Unknowns: {d(5,0),d(3,1)} bonne inconnue W:=d(5,0)$ sa valeur doit etre WW:=d(3,1)$ 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(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(4,2)$ Unknowns: {d(5,4),d(4,2)} Unknowns: {d(5,4),d(4,2)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,4) + d(5,2) - d(4, 1) - d(3,1)$ Unknowns: {d(6,4),d(5,2),d(4,1),d(3,1)} Unknowns: {d(6,4),d(5,2),d(4,1),d(3,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(5,2) - d(4,1) - d(3,1)$ 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},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 {{1,3},6} qui est maintenant AA:=d(5,3) + d(2,1) + d(0,1)$ Unknowns: {d(5,3),d(2,1),d(0,1)} Unknowns: {d(5,3),d(2,1),d(0,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - (d(2,1) + d(0,1))$ on resout l'equation {{1,4},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 {{1,4},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 {{1,4},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 {{1,4},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 {{1,4},5} qui est maintenant AA:= - d(5,5) + d(2,2) + 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) + 2*d(1,1)$ on resout l'equation {{1,4},6} qui est maintenant AA:= - d(6,5) + d(4,2) + d(2, 1)$ Unknowns: {d(6,5),d(4,2),d(2,1)} Unknowns: {d(6,5),d(4,2),d(2,1)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(4,2) + d(2,1)$ on resout l'equation {{1,5},6} qui est maintenant AA:= - d(3,3) + d(2,2) + 3*d( 1,1) - d(0,0)$ Unknowns: {d(3,3),d(2,2),d(1,1),d(0,0)} Unknowns: {d(3,3),d(2,2),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,2) + 3*d(1,1) - d(0,0)$ 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},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},5} 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 {{2,4},6} qui est maintenant AA:= - 2*d(1,1) - d(0,2) + d( 0,0)$ Unknowns: {d(1,1),d(0,2),d(0,0)} Unknowns: {d(1,1),d(0,2),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:=( - d(0,2) + d(0,0))/2$ Derivation equations to cancel (Reduce output) : \\{{{{0,1},4},0}, {{{0,1},5},0}, {{{0,1},6},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},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},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},0},0}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},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},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},6},0}, {{{2,6},4},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}, {{{4,5},6},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(0,1),d(0,2),d(0,3),0,0,0),(0,( - (d(0,2) - d(0,0)))/2,0,0,0,0,0),( 0,d(2,1), - (d(0,2) - d(0,0)),0,0,0,0),(d(3,0),d(3,1),d(3,0),( - (5*d(0,2) - 3*d (0,0)))/2,0,0,0),(0,d(4,1),d(4,2),0,( - 3*(d(0,2) - d(0,0)))/2,0,0),(d(3,1),d(5, 1),d(5,2), - (d(2,1) + d(0,1)),d(4,2), - 2*(d(0,2) - d(0,0)),0),(d(6,0),d(6,1),d (6,2),d(6,3), - (d(4,1) + d(3,1) - d(5,2)),d(4,2) + d(2,1),( - 5*(d(0,2) - d(0,0 )))/2))$ 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,1} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,1), d(3,0), d(2,1), d(0,3), d(0,2), d(0,1), d(0,0)} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,1), d(3,0), d(2,1), d(0,3), d(0,2), d(0,1), d(0,0)} listeparametresMATD{d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,1), d(3,0), d(2,1), d(0,3), d(0,2), d(0,1), d(0,0)}$ dim Der(gtildedelta):=15$ t1:=D(0,0):= [1 0 0 0 0 0 0 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [0 0 1 0 0 0 0 ] [ ] [ 3 ] [0 0 0 --- 0 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 5 ] [0 0 0 0 0 0 ---] [ 2 ] MATD:= mat((d(0,0),d(0,1),d(0,2),d(0,3),0,0,0), - (d(0,2) - d(0,0)) (0,----------------------,0,0,0,0,0), 2 (0,d(2,1), - (d(0,2) - d(0,0)),0,0,0,0), - (5*d(0,2) - 3*d(0,0)) (d(3,0),d(3,1),d(3,0),--------------------------,0,0,0), 2 - 3*(d(0,2) - d(0,0)) (0,d(4,1),d(4,2),0,------------------------,0,0), 2 (d(3,1),d(5,1),d(5,2), - (d(2,1) + d(0,1)),d(4,2), - 2*(d(0,2) - d(0,0)),0), (d(6,0),d(6,1),d(6,2),d(6,3), - (d(4,1) + d(3,1) - d(5,2)),d(4,2) + d(2,1), - 5*(d(0,2) - d(0,0)) ------------------------)) 2 Unknowns: {d(0,2),d(0,0)} Unknowns: {d(0,2),d(0,0)} commutant de t1 dans der(gtildedelta): mat((d(0,0),0,d(0,2),0,0,0,0), - (d(0,2) - d(0,0)) (0,----------------------,0,0,0,0,0), 2 (0,0, - (d(0,2) - d(0,0)),0,0,0,0), - (5*d(0,2) - 3*d(0,0)) (0,0,0,--------------------------,0,0,0), 2 - 3*(d(0,2) - d(0,0)) (0,0,0,0,------------------------,0,0), 2 (0,0,0,0,0, - 2*(d(0,2) - d(0,0)),0), - 5*(d(0,2) - d(0,0)) (0,0,0,0,0,0,------------------------)) 2 Unknowns: {d(0,2),d(0,0)} Unknowns: {d(0,2),d(0,0)} t2:=D(0,2):= [0 0 1 0 0 0 0 ] [ ] [ - 1 ] [0 ------ 0 0 0 0 0 ] [ 2 ] [ ] [0 0 -1 0 0 0 0 ] [ ] [ - 5 ] [0 0 0 ------ 0 0 0 ] [ 2 ] [ ] [ - 3 ] [0 0 0 0 ------ 0 0 ] [ 2 ] [ ] [0 0 0 0 0 -2 0 ] [ ] [ - 5 ] [0 0 0 0 0 0 ------] [ 2 ] {{2*x + 1,1, [ 0 ] [ ] [arbcomplex(47)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x + 1, 1, [ - arbcomplex(48)] [ ] [ 0 ] [ ] [ arbcomplex(48) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {2*x + 3,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(49)] [ ] [ 0 ] [ ] [ 0 ] }, {x + 2,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(50)] [ ] [ 0 ] }, {2*x + 5, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(51)] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(52)] }, {x,1, [arbcomplex(53)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }} Unknowns: {d(0,2),d(0,0)} Unknowns: {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), - (d(0,2) - d(0,0)) (0,----------------------,0,0,0,0,0), 2 (0,0, - (d(0,2) - d(0,0)),0,0,0,0), - (5*d(0,2) - 3*d(0,0)) (0,0,0,--------------------------,0,0,0), 2 - 3*(d(0,2) - d(0,0)) (0,0,0,0,------------------------,0,0), 2 (0,0,0,0,0, - 2*(d(0,2) - d(0,0)),0), - 5*(d(0,2) - d(0,0)) (0,0,0,0,0,0,------------------------)) 2 rank 2 with maximal torus t1,t2 2 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 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [0 0 1 0 0 0 0 ] [ ] [ 3 ] [0 0 0 --- 0 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 5 ] [0 0 0 0 0 0 ---] [ 2 ] P**(-1)*t2*P:= [0 0 0 0 0 0 0 ] [ ] [ - 1 ] [0 ------ 0 0 0 0 0 ] [ 2 ] [ ] [0 0 -1 0 0 0 0 ] [ ] [ - 5 ] [0 0 0 ------ 0 0 0 ] [ 2 ] [ ] [ - 3 ] [0 0 0 0 ------ 0 0 ] [ 2 ] [ ] [0 0 0 0 0 -2 0 ] [ ] [ - 5 ] [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(2,1) + d(0,1),0,d(0,3),0,0,0),(0,( - (d(0,2) - d(0,0)))/2,0,0,0,0, 0),(0,d(2,1), - (d(0,2) - d(0,0)),0,0,0,0),(d(3,0),d(3,1),0,( - (5*d(0,2) - 3*d( 0,0)))/2,0,0,0),(0,d(4,1),d(4,2),0,( - 3*(d(0,2) - d(0,0)))/2,0,0),(d(3,1),d(5,1 ),d(5,2) - d(3,1), - (d(2,1) + d(0,1)),d(4,2), - 2*(d(0,2) - d(0,0)),0),(d(6,0), d(6,1),d(6,2) - d(6,0),d(6,3), - (d(4,1) + d(3,1) - d(5,2)),d(4,2) + d(2,1),( - 5*(d(0,2) - d(0,0)))/2))$ PP:= [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:= [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),d(2,1) + d(0,1),0,d(0,3),0,0,0), - (d(0,2) - d(0,0)) (0,----------------------,0,0,0,0,0), 2 (0,d(2,1), - (d(0,2) - d(0,0)),0,0,0,0), - (5*d(0,2) - 3*d(0,0)) (d(3,0),d(3,1),0,--------------------------,0,0,0), 2 - 3*(d(0,2) - d(0,0)) (0,d(4,1),d(4,2),0,------------------------,0,0), 2 (d(3,1),d(5,1),d(5,2) - d(3,1), - (d(2,1) + d(0,1)),d(4,2), - 2*(d(0,2) - d(0,0)),0), (d(6,0),d(6,1),d(6,2) - d(6,0),d(6,3), - (d(4,1) + d(3,1) - d(5,2)), - 5*(d(0,2) - d(0,0)) d(4,2) + d(2,1),------------------------)) 2 on voit apparaitre les poids sur la diagonale r(1) := d(0,0) - (d(0,2) - d(0,0)) r(2) := ---------------------- 2 r(3) := - (d(0,2) - d(0,0)) - (5*d(0,2) - 3*d(0,0)) r(4) := -------------------------- 2 - 3*(d(0,2) - d(0,0)) r(5) := ------------------------ 2 r(6) := - 2*(d(0,2) - d(0,0)) - 5*(d(0,2) - d(0,0)) r(7) := ------------------------ 2 r(1) := gamma2 r(2) := gamma1 r(3) := 2*gamma1 r(4) := 5*gamma1 - gamma2 r(5) := 3*gamma1 r(6) := 4*gamma1 r(7) := 5*gamma1 Le systeme de poids est le systeme 2.30 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},x(5)}, {{1,5},x(6)}, {{1,6},0}, {{2,3},x(6)}, {{2,4},x(6)}, {{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},diay(6)}, {{2,6},diay(7)}, {{2,7},0}, {{3,4},0}, {{3,5},diay(7)}, {{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_{7,2.30}$ 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((0,0,1,0,0,0,0),(1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(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))$ det(isom):= 1$ ZZ(1):=diay(2)$ ZZ(2):=diay(3)$ ZZ(3):=diay(1)$ ZZ(4):=diay(4)$ ZZ(5):=diay(5)$ ZZ(6):=diay(6)$ ZZ(7):=diay(7)$ listcommutateursdesZZ:=$ {{1,2},zz(5)}$ {{1,3},0}$ {{1,4},0}$ {{1,5},zz(6)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},0}$ {{2,4},0}$ {{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_{7,2.30}$ shortformdelta:={0,ss,0,0,ss,0,ss,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))$