generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(0,2*xi(1,1),0,0,0,0),(xi(3,1),xi(3,2),3*xi(1,1),0,0,0), (xi(4,1),xi(4,2),xi(3,2),4*xi(1,1),0,0),(xi(5,1),xi(5,2),xi(4,2) - xi(3,1),xi(3, 2),5*xi(1,1),0),(xi(6,1),xi(6,2),xi(5,2) - xi(4,1),xi(4,2) - xi(3,1),xi(3,2),6* xi(1,1)))$ $ [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] adx1 := [ ] [0 0 1 0 0 0] [ ] [0 0 0 1 0 0] [ ] [0 0 0 0 1 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] adx2 := [ ] [0 0 0 0 0 0] [ ] [0 0 1 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] adx3 := [ ] [-1 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 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 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] adx5 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 by subtracting adjoints one then may suppose: xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0,xi(6,1):=0 delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 xi(4,2) 0 0 0 0] [ ] [0 xi(5,2) xi(4,2) 0 0 0] [ ] [0 xi(6,2) xi(5,2) xi(4,2) 0 0] We denote this delta by the shortform shortformdelta:={xi(4,2), ss, xi(5,2), ss, xi(6,2)} paramindexeslist:={{4,2},{5,2},{6,2}} a:=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),(0,1,0,0,0,0),(0,1,1 ,0,0,0))$ $ shortformdelta:={0,ss,1,ss,1}$ Comm. relations of g_{6,19} :$ x(1)*x(2):=x(3)$ x(1)*x(3):=x(4)$ x(1)*x(4):=x(5)$ x(1)*x(5):=x(6)$ x(2)*x(3):=x(5)$ x(2)*x(4):=x(6)$ on resout l'equation {{0,1},3} 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},4} qui est maintenant AA:= - d(3,0)$ Unknown: d(3,0) Unknown: d(3,0) bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(4,0) + d(2,1)$ Unknowns: {d(4,0),d(2,1)} Unknowns: {d(4,0),d(2,1)} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(5,0) + d(3,1) + d(2, 1)$ Unknowns: {d(5,0),d(3,1),d(2,1)} Unknowns: {d(5,0),d(3,1),d(2,1)} bonne inconnue W:=d(5,0)$ sa valeur doit etre WW:=d(3,1) + d(2,1)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,6) + d(0,5))$ Unknowns: {d(0,6),d(0,5)} Unknowns: {d(0,6),d(0,5)} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:= - d(0,5)$ on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,6) + d(1,5))$ Unknowns: {d(1,6),d(1,5)} Unknowns: {d(1,6),d(1,5)} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:= - d(1,5)$ on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,6) + d(2,5))$ Unknowns: {d(2,6),d(2,5)} Unknowns: {d(2,6),d(2,5)} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:= - d(2,5)$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,6) - d(3,5) + d(1, 0)$ Unknowns: {d(3,6),d(3,5),d(1,0)} Unknowns: {d(3,6),d(3,5),d(1,0)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(3,5) + d(1,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - (d(4,6) + d(4,5))$ Unknowns: {d(4,6),d(4,5)} Unknowns: {d(4,6),d(4,5)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(4,5)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,6) - d(5,5) + d(2, 2) + d(0,0)$ Unknowns: {d(5,6),d(5,5),d(2,2),d(0,0)} Unknowns: {d(5,6),d(5,5),d(2,2),d(0,0)} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - d(5,5) + d(2,2) + d(0,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,6) - d(6,5) + d(3, 2) + d(2,2) - d(2,1) + d(0,0)$ Unknowns: {d(6,6),d(6,5),d(3,2),d(2,2),d(2,1),d(0,0)} Unknowns: {d(6,6),d(6,5),d(3,2),d(2,2),d(2,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(6,5) + d(3,2) + d(2,2) - d(2,1) + d(0,0)$ on resout l'equation {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},3} qui est maintenant AA:=d(3,5) - d(1,0)$ Unknowns: {d(3,5),d(1,0)} Unknowns: {d(3,5),d(1,0)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,3},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 {{0,3},5} qui est maintenant AA:=d(5,5) + d(2,3) - d(2,2) - d(0,0)$ Unknowns: {d(5,5),d(2,3),d(2,2),d(0,0)} Unknowns: {d(5,5),d(2,3),d(2,2),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(2,3) + d(2,2) + d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:=d(6,5) + d(3,3) - d(3,2) + d(2,3) - d(2,2) + d(2,1)$ Unknowns: {d(6,5),d(3,3),d(3,2),d(2,3),d(2,2),d(2,1)} Unknowns: {d(6,5),d(3,3),d(3,2),d(2,3),d(2,2),d(2,1)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(3,3) + d(3,2) - d(2,3) + d(2,2) - d(2,1)$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(2,4) + d(1,0)$ Unknowns: {d(2,4),d(1,0)} Unknowns: {d(2,4),d(1,0)} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:= - d(2,4)$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(3,4) + d(2,4)$ Unknowns: {d(3,4),d(2,4)} Unknowns: {d(3,4),d(2,4)} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:= - d(3,4)$ on resout l'equation {{0,5},6} qui est maintenant AA:=2*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 {{1,2},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 {{1,2},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 {{1,2},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 {{1,2},3} qui est maintenant AA:= - d(3,3) + d(2,2) + d(1, 1)$ Unknowns: {d(3,3),d(2,2),d(1,1)} Unknowns: {d(3,3),d(2,2),d(1,1)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,3) + d(3,2)$ Unknowns: {d(4,3),d(3,2)} Unknowns: {d(4,3),d(3,2)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(3,2)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,3) + d(4,2) - d(3, 1) + d(0,1)$ Unknowns: {d(5,3),d(4,2),d(3,1),d(0,1)} Unknowns: {d(5,3),d(4,2),d(3,1),d(0,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(4,2) - d(3,1) + d(0,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,3) + d(5,2) - d(4, 1) + d(0,1)$ Unknowns: {d(6,3),d(5,2),d(4,1),d(0,1)} Unknowns: {d(6,3),d(5,2),d(4,1),d(0,1)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,2) - d(4,1) + d(0,1)$ on resout l'equation {{1,3},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 {{1,3},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 {{1,3},4} qui est maintenant AA:= - d(4,4) + d(2,2) + 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) + 2*d(1,1)$ on resout l'equation {{1,3},5} qui est maintenant AA:= - d(5,4) + d(3,2) + d(2, 1)$ Unknowns: {d(5,4),d(3,2),d(2,1)} Unknowns: {d(5,4),d(3,2),d(2,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(3,2) + d(2,1)$ on resout l'equation {{1,3},6} qui est maintenant AA:= - d(6,4) + d(4,2) - d(3, 1) + 2*d(0,1)$ Unknowns: {d(6,4),d(4,2),d(3,1),d(0,1)} Unknowns: {d(6,4),d(4,2),d(3,1),d(0,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(4,2) - d(3,1) + 2*d(0,1)$ on resout l'equation {{1,4},5} qui est maintenant AA:=3*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)/3$ on resout l'equation {{1,4},6} qui est maintenant AA:=(9*d(2,1) + d(0,0))/3$ Unknowns: {d(2,1),d(0,0)} Unknowns: {d(2,1),d(0,0)} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=( - d(0,0))/9$ on resout l'equation {{2,3},4} 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 {{2,3},5} qui est maintenant AA:=(3*d(2,2) - 2*d(0,0))/3$ Unknowns: {d(2,2),d(0,0)} Unknowns: {d(2,2),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=(2*d(0,0))/3$ on resout l'equation {{2,3},6} qui est maintenant AA:=(9*d(0,2) + 2*d(0,0))/9$ Unknowns: {d(0,2),d(0,0)} Unknowns: {d(0,2),d(0,0)} bonne inconnue W:=d(0,0)$ sa valeur doit etre WW:=( - 9*d(0,2))/2$ Derivation equations to cancel (Reduce output) : \\{{{{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},5},0}, {{{0,4},6},0}, {{{0,5},5},0}, {{{0,5},6},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},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},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},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},3},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},3},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},5},0}, {{{4,5},6},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},6},0}}$ Il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((( - 9*d(0,2))/2,d(0,1),d(0,2),0,0,0,0),(0,( - 3*d(0,2))/2,0,0,0,0,0),(0,d(0 ,2)/2, - 3*d(0,2),0,0,0,0),(0,d(3,1),d(3,2),( - 9*d(0,2))/2,0,0,0),(d(0,2)/2,d(4 ,1),d(4,2),d(3,2), - 6*d(0,2),0,0),((2*d(3,1) + d(0,2))/2,d(5,1),d(5,2),d(4,2) - d(3,1) + d(0,1),(2*d(3,2) + d(0,2))/2,( - 15*d(0,2))/2,0),(d(6,0),d(6,1),d(6,2) ,d(5,2) - d(4,1) + d(0,1),d(4,2) - d(3,1) + 2*d(0,1),d(3,2) + d(0,2), - 9*d(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 1 0 0 0 0] [ ] [0 1 1 0 0 0] pour shortformdelta:={0,ss,1,ss,1} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,1)} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,1)} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,1)}$ dim Der(gtildedelta):=11$ t1:=D(0,2):= [ - 9 ] [------ 0 1 0 0 0 0 ] [ 2 ] [ ] [ - 3 ] [ 0 ------ 0 0 0 0 0 ] [ 2 ] [ ] [ 1 ] [ 0 --- -3 0 0 0 0 ] [ 2 ] [ ] [ - 9 ] [ 0 0 0 ------ 0 0 0 ] [ 2 ] [ ] [ 1 ] [ --- 0 0 0 -6 0 0 ] [ 2 ] [ ] [ 1 1 - 15 ] [ --- 0 0 0 --- ------- 0 ] [ 2 2 2 ] [ ] [ 0 0 0 0 0 1 -9] {{2*x + 3, 1, [ 81*arbcomplex(99) ] [ ] [729*arbcomplex(99) ] [ ] [243*arbcomplex(99) ] [ ] [ 0 ] [ ] [ 9*arbcomplex(99) ] [ ] [ 15*arbcomplex(99) ] [-------------------] [ 2 ] [ ] [ arbcomplex(99) ] }, {x + 3, 1, [ 324*arbcomplex(100) ] [---------------------] [ 7 ] [ ] [ 0 ] [ ] [ 486*arbcomplex(100) ] [---------------------] [ 7 ] [ ] [ 0 ] [ ] [ 54*arbcomplex(100) ] [-------------------- ] [ 7 ] [ ] [ 6*arbcomplex(100) ] [ ] [ arbcomplex(100) ] }, {x + 6, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [9*arbcomplex(101)] [ ] [3*arbcomplex(101)] [ ] [ arbcomplex(101) ] }, {x + 9,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(102)] }, {2*x + 9, 2, [ 81*arbcomplex(104) ] [--------------------] [ 4 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(103) ] [ ] [ 27*arbcomplex(104) ] [--------------------] [ 4 ] [ ] [ 9*arbcomplex(104) ] [------------------- ] [ 2 ] [ ] [ arbcomplex(104) ] }, {2*x + 15, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 3*arbcomplex(105) ] [-------------------] [ 2 ] [ ] [ arbcomplex(105) ] }} Unknown: d(0,2) Unknown: d(0,2) commutant de t1 dans der(gtildedelta): - 9*d(0,2) mat((-------------,0,d(0,2),0,0,0,0), 2 - 3*d(0,2) (0,-------------,0,0,0,0,0), 2 d(0,2) (0,--------, - 3*d(0,2),0,0,0,0), 2 - 9*d(0,2) (0,0,0,-------------,0,0,0), 2 d(0,2) (--------,0,0,0, - 6*d(0,2),0,0), 2 d(0,2) d(0,2) - 15*d(0,2) (--------,0,0,0,--------,--------------,0), 2 2 2 (0,0,0,0,0,d(0,2), - 9*d(0,2))) rank 1 with maximal torus t1 1 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 2 ] [ 1 --- --- 0 0 0 0] [ 9 3 ] [ ] [ 0 1 0 0 0 0 0] [ ] [ 1 ] [ 0 --- 1 0 0 0 0] [ 3 ] [ ] [ 0 0 0 1 0 0 0] [ ] [ 1 1 1 ] [--- ---- --- 0 1 0 0] [ 3 81 9 ] [ ] [ 2 5 7 1 ] [--- ----- ---- 0 --- 1 0] [ 9 486 81 3 ] [ ] [ 4 1 7 1 2 ] [---- ----- ----- 0 --- --- 1] [ 81 729 486 9 3 ] P**(-1)*t1*P:= [ - 9 ] [------ 0 0 0 0 0 0 ] [ 2 ] [ ] [ - 3 ] [ 0 ------ 0 0 0 0 0 ] [ 2 ] [ ] [ 0 0 -3 0 0 0 0 ] [ ] [ - 9 ] [ 0 0 0 ------ 0 0 0 ] [ 2 ] [ ] [ 0 0 0 0 -6 0 0 ] [ ] [ - 15 ] [ 0 0 0 0 0 ------- 0 ] [ 2 ] [ ] [ 0 0 0 0 0 0 -9] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((( - 9*d(0,2))/2,d(0,1),0,0,0,0,0),(0,( - 3*d(0,2))/2,0,0,0,0,0),(0,0, - 3*d (0,2),0,0,0,0),(0,(d(3,2) + 3*d(3,1))/3,d(3,2),( - 9*d(0,2))/2,0,0,0),(0,(d(4,2) + 3*d(4,1) - d(0,1))/3,d(4,2),d(3,2), - 6*d(0,2),0,0),((d(3,2) + 3*d(3,1))/3,( 27*d(5,2) + 81*d(5,1) - 9*d(4,2) - 27*d(4,1) + d(3,2) + 9*d(3,1) - 9*d(0,1))/81, (9*d(5,2) - 3*d(4,2) + d(3,2) + 6*d(3,1))/9,(3*d(4,2) - d(3,2) - 3*d(3,1) + 3*d( 0,1))/3,d(3,2),( - 15*d(0,2))/2,0),((3*d(6,0) + d(4,2) - 3*d(3,1) + 2*d(0,1))/3, (162*d(6,2) + 486*d(6,1) + 54*d(6,0) - 108*d(5,2) - 324*d(5,1) + 24*d(4,2) + 54* d(4,1) + d(3,2) - 42*d(3,1) + 42*d(0,1))/486,(81*d(6,2) + 54*d(6,0) - 54*d(5,2) + 18*d(4,2) + d(3,2) - 45*d(3,1) + 18*d(0,1))/81,(9*d(5,2) - 6*d(4,2) - 9*d(4,1) + d(3,2) + 6*d(3,1) + 3*d(0,1))/9,(3*d(4,2) - d(3,2) - 3*d(3,1) + 6*d(0,1))/3,d (3,2), - 9*d(0,2)))$ $ PP:= [ 1 2 ] [ 1 --- --- 0 0 0 0] [ 9 3 ] [ ] [ 0 1 0 0 0 0 0] [ ] [ 1 ] [ 0 --- 1 0 0 0 0] [ 3 ] [ ] [ 0 0 0 1 0 0 0] [ ] [ 1 1 1 ] [--- ---- --- 0 1 0 0] [ 3 81 9 ] [ ] [ 2 5 7 1 ] [--- ----- ---- 0 --- 1 0] [ 9 486 81 3 ] [ ] [ 4 1 7 1 2 ] [---- ----- ----- 0 --- --- 1] [ 81 729 486 9 3 ] avec PP:=P*Q:= [ 1 2 ] [ 1 --- --- 0 0 0 0] [ 9 3 ] [ ] [ 0 1 0 0 0 0 0] [ ] [ 1 ] [ 0 --- 1 0 0 0 0] [ 3 ] [ ] [ 0 0 0 1 0 0 0] [ ] [ 1 1 1 ] [--- ---- --- 0 1 0 0] [ 3 81 9 ] [ ] [ 2 5 7 1 ] [--- ----- ---- 0 --- 1 0] [ 9 486 81 3 ] [ ] [ 4 1 7 1 2 ] [---- ----- ----- 0 --- --- 1] [ 81 729 486 9 3 ] MATDDIAGONALISE:= - 9*d(0,2) mat((-------------,d(0,1),0,0,0,0,0), 2 - 3*d(0,2) (0,-------------,0,0,0,0,0), 2 (0,0, - 3*d(0,2),0,0,0,0), d(3,2) + 3*d(3,1) - 9*d(0,2) (0,-------------------,d(3,2),-------------,0,0,0), 3 2 d(4,2) + 3*d(4,1) - d(0,1) (0,----------------------------,d(4,2),d(3,2), - 6*d(0,2),0,0), 3 d(3,2) + 3*d(3,1) (-------------------,(27*d(5,2) + 81*d(5,1) - 9*d(4,2) - 27*d(4,1) + d(3,2) 3 9*d(5,2) - 3*d(4,2) + d(3,2) + 6*d(3,1) + 9*d(3,1) - 9*d(0,1))/81,-----------------------------------------, 9 3*d(4,2) - d(3,2) - 3*d(3,1) + 3*d(0,1) - 15*d(0,2) -----------------------------------------,d(3,2),--------------,0), 3 2 3*d(6,0) + d(4,2) - 3*d(3,1) + 2*d(0,1) (-----------------------------------------,(162*d(6,2) + 486*d(6,1) 3 + 54*d(6,0) - 108*d(5,2) - 324*d(5,1) + 24*d(4,2) + 54*d(4,1) + d(3,2) - 42*d(3,1) + 42*d(0,1))/486,(81*d(6,2) + 54*d(6,0) - 54*d(5,2) + 18*d(4,2) + d(3,2) - 45*d(3,1) + 18*d(0,1))/81, 9*d(5,2) - 6*d(4,2) - 9*d(4,1) + d(3,2) + 6*d(3,1) + 3*d(0,1) ---------------------------------------------------------------, 9 3*d(4,2) - d(3,2) - 3*d(3,1) + 6*d(0,1) -----------------------------------------,d(3,2), - 9*d(0,2))) 3 on voit apparaitre les poids sur la diagonale *** r declared operator - 9*d(0,2) r(1) := ------------- 2 - 3*d(0,2) r(2) := ------------- 2 r(3) := - 3*d(0,2) - 9*d(0,2) r(4) := ------------- 2 r(5) := - 6*d(0,2) - 15*d(0,2) r(6) := -------------- 2 r(7) := - 9*d(0,2) r(1) := 3*gamma1 r(2) := gamma1 r(3) := 2*gamma1 r(4) := 3*gamma1 r(5) := 4*gamma1 r(6) := 5*gamma1 r(7) := 6*gamma1 Le systeme de poids est le systeme 1.11 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},0}, {{0,2},x(6) + x(5)}, {{0,3},x(6)}, {{0,4},0}, {{0,5},0}, {{0,6},0}, {{1,2},x(3)}, {{1,3},x(4)}, {{1,4},x(5)}, {{1,5},x(6)}, {{1,6},0}, {{2,3},x(5)}, {{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}} 4*x(6) + 18*x(5) + 27*x(4) + 81*x(0) diaY(1):=-------------------------------------- 81 2*x(6) + 15*x(5) + 18*x(4) + 486*x(2) + 1458*x(1) + 162*x(0) diaY(2):=-------------------------------------------------------------- 1458 7*x(6) + 42*x(5) + 54*x(4) + 486*x(2) + 324*x(0) diaY(3):=-------------------------------------------------- 486 diaY(4):=x(3) x(6) + 3*x(5) + 9*x(4) diaY(5):=------------------------ 9 2*x(6) + 3*x(5) diaY(6):=----------------- 3 diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},0}, {{1,3},diay(6)}, {{1,4},diay(7)}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},diay(4)}, {{2,4},diay(5)}, {{2,5},diay(6)}, {{2,6},diay(7)}, {{2,7},0}, {{3,4},diay(6)}, {{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_{7,1.11}$ 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(4)}$ {{1,3},0}$ {{1,4},zz(5)}$ {{1,5},zz(6)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(6)}$ {{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_{7,1.11}$ Et cela pour a:=1$ shortformdelta:={0,ss,1,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),(0,1,0,0,0,0),(0,1,1 ,0,0,0))$ $ The isomorphism from g_{7,1.11} to gtildedelta$ was constructed in 2 steps and is given by$ the product matrix P*isom:= mat((1/9,2/3,-1,0,0,0,0),(1,0,0,0,0,0,0),(1/3,1,0,0,0,0,0),(0,0,0,1,0,0,0),(1/81 ,1/9,( - 1)/3,0,1,0,0),(5/486,7/81,( - 2)/9,0,1/3,1,0),(1/729,7/486,( - 4)/81,0, 1/9,2/3,1))$ $ which we record here under the name PSI$