generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),2*xi(1,1),0,0 ,0),(xi(4,1),xi(4,2), - xi(2,1),xi(2,2) + xi(1,1),0,0),(xi(5,1),xi(5,2),xi(5,3), xi(4,2) - xi(3,1),xi(2,2) + 2*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),0,0,xi(6 ,6)))$ [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [0 1 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 0 0 0] adx2 := [ ] [-1 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 0 0 0 0] adx3 := [ ] [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 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=0 delta:= [ 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 - xi(2,1) 0 0 0 ] [ ] [ 0 0 xi(5,3) - xi(3,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,3), xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,3},{5,6},{6,1},{6,2},{6,3}} a:=1$ delta:= mat((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,-1,0,-2),(1,1 ,0,0,0,0))$ shortformdelta:={0,ss,1,0,ss,0,-2,ss,1,1,0}$ on resout l'equation {{0,1},0} qui est maintenant AA:= - (d(0,6) + d(0,3))$ Unknowns: {d(0,6),d(0,3)} Unknowns: {d(0,6),d(0,3)} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:= - d(0,3)$ on resout l'equation {{0,1},1} qui est maintenant AA:= - (d(1,6) + d(1,3))$ Unknowns: {d(1,6),d(1,3)} Unknowns: {d(1,6),d(1,3)} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:= - d(1,3)$ on resout l'equation {{0,1},2} qui est maintenant AA:= - (d(2,6) + d(2,3))$ Unknowns: {d(2,6),d(2,3)} Unknowns: {d(2,6),d(2,3)} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:= - d(2,3)$ on resout l'equation {{0,1},3} qui est maintenant AA:= - d(3,6) - d(3,3) + d(1, 1) + d(0,0)$ Unknowns: {d(3,6),d(3,3),d(1,1),d(0,0)} Unknowns: {d(3,6),d(3,3),d(1,1),d(0,0)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(3,3) + d(1,1) + d(0,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,6) + d(4,3) + d(2 ,0))$ Unknowns: {d(4,6),d(4,3),d(2,0)} Unknowns: {d(4,6),d(4,3),d(2,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - (d(4,3) + d(2,0))$ on resout l'equation {{0,1},5} qui est maintenant AA:= - 2*d(6,1) - d(5,6) - d( 5,3) - d(4,1) - d(4,0)$ Unknowns: {d(6,1),d(5,6),d(5,3),d(4,1),d(4,0)} Unknowns: {d(6,1),d(5,6),d(5,3),d(4,1),d(4,0)} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=( - (d(5,6) + d(5,3) + d(4,1) + d(4,0)))/2$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,6) - d(6,3) + d(2, 1) + d(1,1) + d(0,0)$ Unknowns: {d(6,6),d(6,3),d(2,1),d(1,1),d(0,0)} Unknowns: {d(6,6),d(6,3),d(2,1),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(6,3) + d(2,1) + d(1,1) + d(0,0)$ on resout l'equation {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},3} qui est maintenant AA:=d(3,3) + d(1,2) - d(1,1) - d(0,0)$ Unknowns: {d(3,3),d(1,2),d(1,1),d(0,0)} Unknowns: {d(3,3),d(1,2),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:= - d(1,2) + d(1,1) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:=d(4,3) + d(2,0) + d(1,0)$ Unknowns: {d(4,3),d(2,0),d(1,0)} Unknowns: {d(4,3),d(2,0),d(1,0)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - (d(2,0) + d(1,0))$ on resout l'equation {{0,2},5} qui est maintenant AA:= - 2*d(6,2) - d(5,6) - d( 4,2) - d(3,0)$ Unknowns: {d(6,2),d(5,6),d(4,2),d(3,0)} Unknowns: {d(6,2),d(5,6),d(4,2),d(3,0)} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=( - (d(5,6) + d(4,2) + d(3,0)))/2$ on resout l'equation {{0,2},6} qui est maintenant AA:=d(6,3) + d(2,2) - d(2,1) + d(1,2) - d(1,1)$ Unknowns: {d(6,3),d(2,2),d(2,1),d(1,2),d(1,1)} Unknowns: {d(6,3),d(2,2),d(2,1),d(1,2),d(1,1)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - d(2,2) + d(2,1) - d(1,2) + d(1,1)$ on resout l'equation {{0,3},5} qui est maintenant AA:=2*d(2,2) - 2*d(2,1) + 2*d (2,0) + 2*d(1,2) - 2*d(1,1) + d(1,0)$ Unknowns: {d(2,2),d(2,1),d(2,0),d(1,2),d(1,1),d(1,0)} Unknowns: {d(2,2),d(2,1),d(2,0),d(1,2),d(1,1),d(1,0)} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=2*( - d(2,2) + d(2,1) - d(2,0) - d(1,2) + d(1,1))$ on resout l'equation {{0,4},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 {{0,4},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 {{0,4},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 {{0,4},3} qui est maintenant AA:=d(3,5) + d(1,4)$ Unknowns: {d(3,5),d(1,4)} Unknowns: {d(3,5),d(1,4)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,4},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 {{0,4},5} qui est maintenant AA:= - 2*d(6,4) + d(5,5) - d( 4,4) - 2*d(2,2) + 2*d(2,1) - 2*d(2,0) - 2*d(1,2) + 2*d(1,1) - d(0,0)$ Unknowns: {d(6,4),d(5,5),d(4,4),d(2,2),d(2,1),d(2,0),d(1,2),d(1,1),d(0,0)} Unknowns: {d(6,4),d(5,5),d(4,4),d(2,2),d(2,1),d(2,0),d(1,2),d(1,1),d(0,0)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=(d(5,5) - d(4,4) - 2*d(2,2) + 2*d(2,1) - 2*d(2,0) - 2*d( 1,2) + 2*d(1,1) - d(0,0))/2$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(6,5) + d(2,4) + d(1,4)$ Unknowns: {d(6,5),d(2,4),d(1,4)} Unknowns: {d(6,5),d(2,4),d(1,4)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - (d(2,4) + d(1,4))$ on resout l'equation {{0,5},5} qui est maintenant AA:=2*(d(2,4) + d(1,4))$ Unknowns: {d(2,4),d(1,4)} Unknowns: {d(2,4),d(1,4)} bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,6},3} qui est maintenant AA:= - 2*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 {{0,6},5} qui est maintenant AA:=2*(d(5,5) - d(2,1) + d(2, 0) - d(1,1) - 2*d(0,0))$ Unknowns: {d(5,5),d(2,1),d(2,0),d(1,1),d(0,0)} Unknowns: {d(5,5),d(2,1),d(2,0),d(1,1),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,1) - d(2,0) + d(1,1) + 2*d(0,0)$ 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},3} qui est maintenant AA:= - (d(3,4) + d(0,2))$ Unknowns: {d(3,4),d(0,2)} Unknowns: {d(3,4),d(0,2)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:= - d(0,2)$ 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) - d(3, 1)$ Unknowns: {d(5,4),d(4,2),d(3,1)} Unknowns: {d(5,4),d(4,2),d(3,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:=(3*d(2,2) - 3*d(2,1) + 3* d(2,0) + 2*d(1,2) - 2*d(1,1) - 2*d(0,2) + 2*d(0,1) - d(0,0))/2$ Unknowns: {d(2,2),d(2,1),d(2,0),d(1,2),d(1,1),d(0,2),d(0,1),d(0,0)} Unknowns: {d(2,2),d(2,1),d(2,0),d(1,2),d(1,1),d(0,2),d(0,1),d(0,0)} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=( - 3*d(2,2) + 3*d(2,1) - 3*d(2,0) + 2*d(1,1) + 2*d(0,2) - 2*d(0,1) + d(0,0))/2$ on resout l'equation {{1,3},5} qui est maintenant AA:= - d(2,2) + 2*d(2,1) - 2* d(2,0) + 2*d(0,2) - 2*d(0,1) + d(0,0)$ Unknowns: {d(2,2),d(2,1),d(2,0),d(0,2),d(0,1),d(0,0)} Unknowns: {d(2,2),d(2,1),d(2,0),d(0,2),d(0,1),d(0,0)} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:=2*d(2,1) - 2*d(2,0) + 2*d(0,2) - 2*d(0,1) + d(0,0)$ on resout l'equation {{1,4},5} qui est maintenant AA:=d(2,1) - d(2,0) + d(1,1) + 2*d(0,2) - 3*d(0,1) - d(0,0)$ Unknowns: {d(2,1),d(2,0),d(1,1),d(0,2),d(0,1),d(0,0)} Unknowns: {d(2,1),d(2,0),d(1,1),d(0,2),d(0,1),d(0,0)} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=d(2,0) - d(1,1) - 2*d(0,2) + 3*d(0,1) + d(0,0)$ on resout l'equation {{1,6},5} qui est maintenant AA:= - d(1,1) - 2*d(0,2) + d( 0,1) + d(0,0)$ Unknowns: {d(1,1),d(0,2),d(0,1),d(0,0)} Unknowns: {d(1,1),d(0,2),d(0,1),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:= - 2*d(0,2) + d(0,1) + d(0,0)$ on resout l'equation {{2,3},5} qui est maintenant AA:=6*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},3},0}, {{{0,3},5},0}, {{{0,3},6},0}, {{{0,4},0},0}, {{{0,4},1},0}, {{{0,4},2},0}, {{{0,4},3},0}, {{{0,4},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},0},0}, {{{0,6},1},0}, {{{0,6},2},0}, {{{0,6},3},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},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},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},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}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}, {{{5,6},5},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),(2*d(0,1),d(0,1) + d(0,0),0,0,0,0,0),(d(2,0),d(2,0 ) + 2*d(0,1),2*d(0,1) + d(0,0),0,0,0,0),(d(3,0),d(3,1),d(3,2),d(0,1) + 2*d(0,0), 0,0,0),(d(4,0),d(4,1),d(4,2), - d(2,0) - 2*d(0,1),3*d(0,1) + 2*d(0,0),0,2*d(0,1) ),(d(5,0),d(5,1),d(5,2),d(5,3),d(4,2) - d(3,1),3*(d(0,1) + d(0,0)),d(5,6)),(d(6, 0),( - (d(5,6) + d(5,3) + d(4,1) + d(4,0)))/2,( - (d(5,6) + d(4,2) + d(3,0)))/2, d(2,0) + d(0,1),d(0,1),0,2*(d(0,1) + d(0,0))))$ pour delta:= [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 -1 0 -2] [ ] [1 1 0 0 0 0 ] pour shortformdelta:={0,ss,1,0,ss,0,-2,ss,1,1,0} Unknowns: {d(6,0), d(5,6), d(5,3), 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(0,1), d(0,0)} Unknowns: {d(6,0), d(5,6), d(5,3), 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(0,1), d(0,0)} listeparametresMATD{d(6,0), d(5,6), d(5,3), 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(0,1), d(0,0)}$ dim Der(gtildedelta):=15$ t1:=D(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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] Unknowns: {d(2,0),d(0,1),d(0,0)} Unknowns: {d(2,0),d(0,1),d(0,0)} commutant de t1 dans der(gtildedelta): mat((d(0,0),d(0,1),0,0,0,0,0), (2*d(0,1),d(0,1) + d(0,0),0,0,0,0,0), (d(2,0),d(2,0) + 2*d(0,1),2*d(0,1) + d(0,0),0,0,0,0), (0,0,0,d(0,1) + 2*d(0,0),0,0,0), (0,0,0, - d(2,0) - 2*d(0,1),3*d(0,1) + 2*d(0,0),0,2*d(0,1)), (0,0,0,0,0,3*(d(0,1) + d(0,0)),0), (0,0,0,d(2,0) + d(0,1),d(0,1),0,2*(d(0,1) + d(0,0)))) Unknowns: {d(2,0),d(0,1),d(0,0)} Unknowns: {d(2,0),d(0,1),d(0,0)} t2:=D(0,1):= [0 1 0 0 0 0 0] [ ] [2 1 0 0 0 0 0] [ ] [0 2 2 0 0 0 0] [ ] [0 0 0 1 0 0 0] [ ] [0 0 0 -2 3 0 2] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 1 1 0 2] {{x - 3,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(149)] [ ] [ 0 ] }, {x - 2,2, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(150)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 4, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [2*arbcomplex(151)] [ ] [ 0 ] [ ] [ arbcomplex(151) ] }, {x - 1, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(152)] [ ] [ 0 ] [ ] [ arbcomplex(152) ] }, {x + 1, 1, [ 3*arbcomplex(153) ] [ ------------------- ] [ 2 ] [ ] [ - 3*arbcomplex(153) ] [----------------------] [ 2 ] [ ] [ arbcomplex(153) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }} Unknowns: {d(2,0),d(0,1),d(0,0)} Unknowns: {d(2,0),d(0,1),d(0,0)} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),d(0,1),0,0,0,0,0),(2*d(0,1),d(0,1) + d(0,0),0,0,0,0,0),(d(2,0),d(2,0 ) + 2*d(0,1),2*d(0,1) + d(0,0),0,0,0,0),(0,0,0,d(0,1) + 2*d(0,0),0,0,0),(0,0,0, - (d(2,0) + 2*d(0,1)),3*d(0,1) + 2*d(0,0),0,2*d(0,1)),(0,0,0,0,0,3*(d(0,1) + d(0 ,0)),0),(0,0,0,d(2,0) + d(0,1),d(0,1),0,2*(d(0,1) + d(0,0))))$ This t2 is not semisimple. Unknowns: {d(2,0),d(0,1),d(0,0)} Unknowns: {d(2,0),d(0,1),d(0,0)} t3:=D(2,0):= [0 0 0 0 0 0 0] [ ] [0 0 0 0 0 0 0] [ ] [1 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 1 0 0 0] {{x, 7, [ - arbcomplex(154)] [ ] [ arbcomplex(154) ] [ ] [ arbcomplex(155) ] [ ] [ 0 ] [ ] [ arbcomplex(156) ] [ ] [ arbcomplex(157) ] [ ] [ arbcomplex(158) ] }} This t3 is nilpotent. Unknowns: {d(2,0),d(0,1),d(0,0)} Unknowns: {d(2,0),d(0,1),d(0,0)} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),d(0,1),0,0,0,0,0), (2*d(0,1),d(0,1) + d(0,0),0,0,0,0,0), (d(2,0),d(2,0) + 2*d(0,1),2*d(0,1) + d(0,0),0,0,0,0), (0,0,0,d(0,1) + 2*d(0,0),0,0,0), (0,0,0, - (d(2,0) + 2*d(0,1)),3*d(0,1) + 2*d(0,0),0,2*d(0,1)), (0,0,0,0,0,3*(d(0,1) + d(0,0)),0), (0,0,0,d(2,0) + d(0,1),d(0,1),0,2*(d(0,1) + d(0,0)))) alpha*t2 +beta*t3:= mat((0,alpha,0,0,0,0,0), (2*alpha,alpha,0,0,0,0,0), (beta,2*alpha + beta,2*alpha,0,0,0,0), (0,0,0,alpha,0,0,0), (0,0,0, - (2*alpha + beta),3*alpha,0,2*alpha), (0,0,0,0,0,3*alpha,0), (0,0,0,alpha + beta,alpha,0,2*alpha)) 2 2 - (4*alpha - x)*(3*alpha - x)*(2*alpha - x) *(alpha + x)*(alpha - x) As 2*alpha is for s a multiplicity 2 eigenvalue, a hand computation shows that s is semisimple if and only if 3*beta + 4*alpha = 0. We take : alpha:= 3 beta:= -4 Then s:= [0 3 0 0 0 0 0] [ ] [6 3 0 0 0 0 0] [ ] [-4 2 6 0 0 0 0] [ ] [0 0 0 3 0 0 0] [ ] [0 0 0 -2 9 0 6] [ ] [0 0 0 0 0 9 0] [ ] [0 0 0 -1 3 0 6] mateigen(s,x) := {{x - 9,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(159)] [ ] [ 0 ] }, {x - 6, 2, [ arbcomplex(160) ] [-----------------] [ 2 ] [ ] [ arbcomplex(160) ] [ ] [ arbcomplex(161) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 12, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [2*arbcomplex(162)] [ ] [ 0 ] [ ] [ arbcomplex(162) ] }, {x - 3, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [3*(arbcomplex(164) + arbcomplex(163))] [ ] [ arbcomplex(163) ] [ ] [ 0 ] [ ] [ arbcomplex(164) ] }, {x + 3, 1, [ 3*arbcomplex(165) ] [ ------------------- ] [ 2 ] [ ] [ - 3*arbcomplex(165) ] [----------------------] [ 2 ] [ ] [ arbcomplex(165) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }} We shift the notations to denote s as t2prime or simply t2 if no confusion ari\ ses t2:= [0 3 0 0 0 0 0] [ ] [6 3 0 0 0 0 0] [ ] [-4 2 6 0 0 0 0] [ ] [0 0 0 3 0 0 0] [ ] [0 0 0 -2 9 0 6] [ ] [0 0 0 0 0 9 0] [ ] [0 0 0 -1 3 0 6] The rank is 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 -1 0 0 0 0 0] [ ] [2 1 0 0 0 0 0] [ ] [ - 2 ] [0 ------ 1 0 0 0 0] [ 3 ] [ ] [0 0 0 1 0 0 3] [ ] [ 1 ] [0 0 0 --- 1 0 0] [ 3 ] [ ] [0 0 0 0 0 1 0] [ ] [ 1 ] [0 0 0 0 --- 0 1] [ 2 ] P**(-1)*t1*P:= [1 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 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t2*P:= [6 0 0 0 0 0 0] [ ] [0 -3 0 0 0 0 0] [ ] [0 0 6 0 0 0 0] [ ] [0 0 0 3 0 0 0] [ ] [0 0 0 0 12 0 0] [ ] [0 0 0 0 0 9 0] [ ] [0 0 0 0 0 0 3] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((2*d(0,1) + d(0,0),0,0,0,0,0,0),(0, - (d(0,1) - d(0,0)),0,0,0,0,0),(3*d(2,0) + 4*d(0,1),0,2*d(0,1) + d(0,0),0,0,0,0),((2*(2*d(3,1) + d(3,0)) + 9*d(4,0) + 12 *d(4,1) + 6*d(5,3) + 6*d(5,6) - 6*d(6,0))/3,(9*d(5,3) - 12*d(4,2) + 18*d(4,1) - 4*d(3,2) + 6*d(3,1) - 12*d(3,0) + 3*d(5,6) + 18*d(6,0))/9,(2*d(3,2) + 3*d(3,0) + 6*d(4,2) + 3*d(5,6))/3, - (3*d(0,1) - 2*d(0,0) + 3*d(2,0)),0,0, - 3*(3*d(2,0) + 4*d(0,1))),(( - 2*(2*d(3,1) + d(3,0) - 3*d(4,1) + 3*d(5,3) + 3*d(5,6) - 3*d(6,0 )))/9,( - (27*d(4,0) - 4*d(3,2) + 6*d(3,1) - 12*d(3,0) - 9*d(4,1) + 6*d(4,2) + 9 *d(5,3) + 3*d(5,6) + 18*d(6,0)))/27,( - (2*d(3,2) + 3*d(3,0) - 3*d(4,2) + 3*d(5, 6)))/9,0,2*(2*d(0,1) + d(0,0)),0,0),(2*d(5,1) + d(5,0),(3*(d(5,1) - d(5,0)) - 2* d(5,2))/3,d(5,2),(d(4,2) - d(3,1) + 3*d(5,3))/3,(2*(d(4,2) - d(3,1)) + d(5,6))/2 ,3*(d(0,1) + d(0,0)),d(5,6) + 3*d(5,3)),((2*d(3,1) + d(3,0) - 9*d(4,0) - 12*d(4, 1) - 6*d(5,3) - 6*d(5,6) + 6*d(6,0))/9,(3*(d(3,1) + d(3,0)) - 2*d(3,2) - 18*d(4, 1) + 12*d(4,2) - 9*d(5,3) - 3*d(5,6) - 18*d(6,0))/27,(d(3,2) - 3*d(3,0) - 6*d(4, 2) - 3*d(5,6))/9,(3*d(2,0) + 4*d(0,1))/3,0,0,5*d(0,1) + 2*d(0,0) + 3*d(2,0)))$ PP:= [1 -1 0 0 0 0 0] [ ] [2 1 0 0 0 0 0] [ ] [ - 2 ] [0 ------ 1 0 0 0 0] [ 3 ] [ ] [0 0 0 1 0 0 3] [ ] [ 1 ] [0 0 0 --- 1 0 0] [ 3 ] [ ] [0 0 0 0 0 1 0] [ ] [ 1 ] [0 0 0 0 --- 0 1] [ 2 ] avec PP:=P*Q:= [1 -1 0 0 0 0 0] [ ] [2 1 0 0 0 0 0] [ ] [ - 2 ] [0 ------ 1 0 0 0 0] [ 3 ] [ ] [0 0 0 1 0 0 3] [ ] [ 1 ] [0 0 0 --- 1 0 0] [ 3 ] [ ] [0 0 0 0 0 1 0] [ ] [ 1 ] [0 0 0 0 --- 0 1] [ 2 ] MATDDIAGONALISE:= mat((2*d(0,1) + d(0,0),0,0,0,0,0,0), (0, - (d(0,1) - d(0,0)),0,0,0,0,0), (3*d(2,0) + 4*d(0,1),0,2*d(0,1) + d(0,0),0,0,0,0), ((2*(2*d(3,1) + d(3,0)) + 9*d(4,0) + 12*d(4,1) + 6*d(5,3) + 6*d(5,6) - 6*d(6,0))/3,(9*d(5,3) - 12*d(4,2) + 18*d(4,1) - 4*d(3,2) + 6*d(3,1) - 12*d(3,0) + 3*d(5,6) + 18*d(6,0))/9, 2*d(3,2) + 3*d(3,0) + 6*d(4,2) + 3*d(5,6) -------------------------------------------, 3 - (3*d(0,1) - 2*d(0,0) + 3*d(2,0)),0,0, - 3*(3*d(2,0) + 4*d(0,1))), - 2*(2*d(3,1) + d(3,0) - 3*d(4,1) + 3*d(5,3) + 3*d(5,6) - 3*d(6,0)) (----------------------------------------------------------------------,( - 9 (27*d(4,0) - 4*d(3,2) + 6*d(3,1) - 12*d(3,0) - 9*d(4,1) + 6*d(4,2) + 9*d(5,3) + 3*d(5,6) + 18*d(6,0)))/27, - (2*d(3,2) + 3*d(3,0) - 3*d(4,2) + 3*d(5,6)) ------------------------------------------------,0,2*(2*d(0,1) + d(0,0)),0, 9 0), 3*(d(5,1) - d(5,0)) - 2*d(5,2) (2*d(5,1) + d(5,0),--------------------------------,d(5,2), 3 d(4,2) - d(3,1) + 3*d(5,3) 2*(d(4,2) - d(3,1)) + d(5,6) ----------------------------,------------------------------, 3 2 3*(d(0,1) + d(0,0)),d(5,6) + 3*d(5,3)), 2*d(3,1) + d(3,0) - 9*d(4,0) - 12*d(4,1) - 6*d(5,3) - 6*d(5,6) + 6*d(6,0) (--------------------------------------------------------------------------- 9 ,(3*(d(3,1) + d(3,0)) - 2*d(3,2) - 18*d(4,1) + 12*d(4,2) - 9*d(5,3) d(3,2) - 3*d(3,0) - 6*d(4,2) - 3*d(5,6) - 3*d(5,6) - 18*d(6,0))/27,-----------------------------------------, 9 3*d(2,0) + 4*d(0,1) ---------------------,0,0,5*d(0,1) + 2*d(0,0) + 3*d(2,0))) 3 on voit apparaitre les poids sur la diagonale Note that d(0,1)/3 = - d(2,0)/4 on the torus - 4*d(0,1) d(2,0):= ------------- 3 r(1) := 2*d(0,1) + d(0,0) r(2) := - (d(0,1) - d(0,0)) r(3) := 2*d(0,1) + d(0,0) r(4) := d(0,1) + 2*d(0,0) r(5) := 2*(2*d(0,1) + d(0,0)) r(6) := 3*(d(0,1) + d(0,0)) r(7) := d(0,1) + 2*d(0,0) r(1) := gamma2 r(2) := gamma1 r(3) := gamma2 r(4) := gamma1 + gamma2 r(5) := 2*gamma2 r(6) := gamma1 + 2*gamma2 r(7) := gamma1 + gamma2 Le systeme de poids est le systeme 2.2 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(6) + x(3)}, {{0,2},x(6)}, {{0,3},0}, {{0,4}, - x(5)}, {{0,5},0}, {{0,6}, - 2*x(5)}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},0}, {{1,6},0}, {{2,3},x(5)}, {{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):=2*x(1) + x(0) - 2*x(2) + 3*x(1) - 3*x(0) diaY(2):=----------------------------- 3 diaY(3):=x(2) x(4) + 3*x(3) diaY(4):=--------------- 3 x(6) + 2*x(4) diaY(5):=--------------- 2 diaY(6):=x(5) diaY(7):=x(6) + 3*x(3) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},(7*diay(7) - 12*diay(4))/3}, {{1,3},2*diay(5)}, {{1,4},diay(6)/3}, {{1,5},0}, {{1,6},0}, {{1,7}, - 2*diay(6)}, {{2,3}, - diay(7) + 3*diay(4)}, {{2,4},0}, {{2,5},3*diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},diay(6)}, {{3,5},0}, {{3,6},0}, {{3,7},3*diay(6)}, {{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_{7,2.2}$ (iL)$ 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 :$ This isom computed by calculisom6_53xCXIII2.red$ mat((0,0,1,0,0,0,0),(1,0,0,0,0,0,0),(0,-2,0,0,0,0,0),(0,0,0,0,-6,4,0),(0,0,0,4,0 ,0,0),(0,0,0,0,0,0,6),(0,0,0,0,2,( - 7)/3,0))$ det(isom):= -288$ ZZ(1):=diay(2)$ ZZ(2):= - 2*diay(3)$ ZZ(3):=diay(1)$ ZZ(4):=4*diay(5)$ ZZ(5):=2*(diay(7) - 3*diay(4))$ ZZ(6):=( - (7*diay(7) - 12*diay(4)))/3$ ZZ(7):=6*diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(5)}$ {{1,3},zz(6)}$ {{1,4},2*zz(7)}$ {{1,5},0}$ {{1,6},0}$ {{1,7},0}$ {{2,3},zz(4)}$ {{2,4},0}$ {{2,5},0}$ {{2,6},zz(7)}$ {{2,7},0}$ {{3,4},0}$ {{3,5}, - zz(7)}$ {{3,6},zz(7)}$ {{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_{7,2.2}$ Et cela pour a:=1$ shortformdelta:={0,ss,1,0,ss,0,-2,ss,1,1,0}$ delta:= mat((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,-1,0,-2),(1,1 ,0,0,0,0))$