generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),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) + xi(2,1),5*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),0,0,0,xi(6,6)))$ $ [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 0 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 0 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 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 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) xi(2,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) 0 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(4,2), ss, xi(5,2), xi(5,6), ss, xi(6,1), xi(6,2)} paramindexeslist:={{2,1},{4,2},{5,2},{5,6},{6,1},{6,2}} a:=1$ b:=1$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,1,0,0,1),(0,1,0 ,0,0,0))$ $ shortformdelta:={0,ss,1,ss,0,1,ss,0,1}$ 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) + d(2,1)$ Unknowns: {d(3,0),d(2,1)} Unknowns: {d(3,0),d(2,1)} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},5} qui est maintenant AA:=d(6,1) - d(4,0) + d(3,1)$ Unknowns: {d(6,1),d(4,0),d(3,1)} Unknowns: {d(6,1),d(4,0),d(3,1)} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(4,0) - d(3,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:=d(2,1)$ Unknown: d(2,1) Unknown: d(2,1) bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,6) + d(0,4))$ Unknowns: {d(0,6),d(0,4)} Unknowns: {d(0,6),d(0,4)} bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:= - d(0,4)$ on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,6) + d(1,4))$ Unknowns: {d(1,6),d(1,4)} Unknowns: {d(1,6),d(1,4)} bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,6) + d(2,4))$ Unknowns: {d(2,6),d(2,4)} Unknowns: {d(2,6),d(2,4)} bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:= - d(2,4)$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,6) - d(3,4) + d(1, 0)$ Unknowns: {d(3,6),d(3,4),d(1,0)} Unknowns: {d(3,6),d(3,4),d(1,0)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(3,4) + d(1,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,6) - d(4,4) + d(2, 2) + d(0,0)$ Unknowns: {d(4,6),d(4,4),d(2,2),d(0,0)} Unknowns: {d(4,6),d(4,4),d(2,2),d(0,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(4,4) + d(2,2) + d(0,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=d(6,2) - d(5,6) - d(5,4) + d(3,2)$ Unknowns: {d(6,2),d(5,6),d(5,4),d(3,2)} Unknowns: {d(6,2),d(5,6),d(5,4),d(3,2)} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=d(5,6) + d(5,4) - d(3,2)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,6) - d(6,4) + d(2, 2) + d(0,0)$ Unknowns: {d(6,6),d(6,4),d(2,2),d(0,0)} Unknowns: {d(6,6),d(6,4),d(2,2),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(6,4) + d(2,2) + 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)$ Unknown: d(3,5) Unknown: d(3,5) bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,5) + d(2,3) + d(1, 0)$ Unknowns: {d(4,5),d(2,3),d(1,0)} Unknowns: {d(4,5),d(2,3),d(1,0)} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=d(2,3) + d(1,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:=d(6,3) - d(5,5) + d(3,3) + d(0,0)$ Unknowns: {d(6,3),d(5,5),d(3,3),d(0,0)} Unknowns: {d(6,3),d(5,5),d(3,3),d(0,0)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,5) - d(3,3) - d(0,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,5) + d(2,3)$ Unknowns: {d(6,5),d(2,3)} Unknowns: {d(6,5),d(2,3)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(2,3)$ on resout l'equation {{0,4},4} 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 {{0,4},5} qui est maintenant AA:=d(6,4) + d(3,4) + d(1,0)$ Unknowns: {d(6,4),d(3,4),d(1,0)} Unknowns: {d(6,4),d(3,4),d(1,0)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - (d(3,4) + d(1,0))$ on resout l'equation {{0,5},5} 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 {{0,6},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 {{0,6},5} qui est maintenant AA:= - d(5,5) + d(2,2) + 2*d( 0,0)$ Unknowns: {d(5,5),d(2,2),d(0,0)} Unknowns: {d(5,5),d(2,2),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,2) + 2*d(0,0)$ 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},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) + d(0, 1)$ Unknowns: {d(4,3),d(3,2),d(0,1)} Unknowns: {d(4,3),d(3,2),d(0,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(3,2) + d(0,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,3) + d(4,2) - d(3, 1)$ Unknowns: {d(5,3),d(4,2),d(3,1)} Unknowns: {d(5,3),d(4,2),d(3,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:=d(1,1) + d(0,1) - d(0,0)$ Unknowns: {d(1,1),d(0,1),d(0,0)} Unknowns: {d(1,1),d(0,1),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:= - d(0,1) + d(0,0)$ 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},3} 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 {{1,3},4} qui est maintenant AA:= - d(4,4) + d(2,2) - 2*d( 0,1) + 2*d(0,0)$ Unknowns: {d(4,4),d(2,2),d(0,1),d(0,0)} Unknowns: {d(4,4),d(2,2),d(0,1),d(0,0)} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=( - d(4,4) + d(2,2) + 2*d(0,0))/2$ on resout l'equation {{1,3},5} qui est maintenant AA:= - d(5,4) - d(4,4) + d(3, 2) + d(2,2) + 2*d(0,0)$ Unknowns: {d(5,4),d(4,4),d(3,2),d(2,2),d(0,0)} Unknowns: {d(5,4),d(4,4),d(3,2),d(2,2),d(0,0)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(4,4) + d(3,2) + d(2,2) + 2*d(0,0)$ on resout l'equation {{1,4},5} qui est maintenant AA:=(3*d(4,4) - 3*d(2,2) - 4* 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:=(3*d(2,2) + 4*d(0,0))/3$ 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) + 3*d(0,2) - 4* d(0,0))/3$ 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:=( - 3*d(0,2) + 4*d(0,0))/3$ on resout l'equation {{2,6},5} 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$ 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},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},4},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},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},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},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},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,4},6},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},4},0}, {{{3,4},5},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,6},4},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(0,0)/3,0,0,0,0,0),(0,(2*d(0,0))/3,0,0,0,0,0),(0,0,(4*d(0,0))/3,0,0 ,0,0),(0,d(3,1),d(3,2),2*d(0,0),0,0,0),(d(4,0),d(4,1),d(4,2),(3*d(3,2) + d(0,0)) /3,(8*d(0,0))/3,0,( - d(0,0))/3),(d(5,0),d(5,1),d(5,2),d(4,2) - d(3,1),(3*d(3,2) + 2*d(0,0))/3,(10*d(0,0))/3,d(5,6)),(d(6,0),d(4,0) - d(3,1),(3*d(5,6) + 2*d(0,0 ))/3,d(0,0)/3,0,0,(7*d(0,0))/3))$ $ pour delta:= [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 0 1 0 0 1] [ ] [0 1 0 0 0 0] pour shortformdelta:={0,ss,1,ss,0,1,ss,0,1} Unknowns: {d(6,0), d(5,6), 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(0,0)} Unknowns: {d(6,0), d(5,6), 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(0,0)} listeparametresMATD{d(6,0), d(5,6), 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(0,0)}$ dim Der(gtildedelta):=11$ t1:=D(0,0):= [ 1 ] [1 --- 0 0 0 0 0 ] [ 3 ] [ ] [ 2 ] [0 --- 0 0 0 0 0 ] [ 3 ] [ ] [ 4 ] [0 0 --- 0 0 0 0 ] [ 3 ] [ ] [0 0 0 2 0 0 0 ] [ ] [ 1 8 - 1 ] [0 0 0 --- --- 0 ------] [ 3 3 3 ] [ ] [ 2 10 ] [0 0 0 0 --- ---- 0 ] [ 3 3 ] [ ] [ 2 1 7 ] [0 0 --- --- 0 0 --- ] [ 3 3 3 ] {{3*x - 2, 1, [ - arbcomplex(93)] [ ] [ arbcomplex(93) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {3*x - 4, 1, [ 0 ] [ ] [ 0 ] [ ] [ - 3*arbcomplex(94) ] [---------------------] [ 2 ] [ ] [ 0 ] [ ] [ arbcomplex(94) ] [ ---------------- ] [ 4 ] [ ] [ - arbcomplex(94) ] [ ------------------- ] [ 12 ] [ ] [ arbcomplex(94) ] }, {x - 2, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(95) ] [ ] [ arbcomplex(95) ] [ ] [ - arbcomplex(95) ] [-------------------] [ 2 ] [ ] [ arbcomplex(95) ] }, {x - 1,1, [arbcomplex(96)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {3*x - 10,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(97)] [ ] [ 0 ] }, {3*x - 8, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(98)] [ ] [ arbcomplex(98) ] [ ] [ 0 ] }, {3*x - 7, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(99) ] [ ] [ - 2*arbcomplex(99) ] [---------------------] [ 3 ] [ ] [ arbcomplex(99) ] }} Unknown: d(0,0) Unknown: d(0,0) commutant de t1 dans der(gtildedelta): [ d(0,0) ] [d(0,0) -------- 0 0 0 0 0 ] [ 3 ] [ ] [ 2*d(0,0) ] [ 0 ---------- 0 0 0 0 0 ] [ 3 ] [ ] [ 4*d(0,0) ] [ 0 0 ---------- 0 0 0 0 ] [ 3 ] [ ] [ 0 0 0 2*d(0,0) 0 0 0 ] [ ] [ d(0,0) 8*d(0,0) - d(0,0) ] [ 0 0 0 -------- ---------- 0 -----------] [ 3 3 3 ] [ ] [ 2*d(0,0) 10*d(0,0) ] [ 0 0 0 0 ---------- ----------- 0 ] [ 3 3 ] [ ] [ 2*d(0,0) d(0,0) 7*d(0,0) ] [ 0 0 ---------- -------- 0 0 ---------- ] [ 3 3 3 ] 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 -1 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 ] [ ] [ - 1 ] [0 0 ------ -1 1 0 1 ] [ 6 ] [ ] [ 1 1 - 2 ] [0 0 ---- --- -1 1 ------] [ 18 2 3 ] [ ] [ - 2 ] [0 0 ------ -1 0 0 1 ] [ 3 ] P**(-1)*t1*P:= [1 0 0 0 0 0 0 ] [ ] [ 2 ] [0 --- 0 0 0 0 0 ] [ 3 ] [ ] [ 4 ] [0 0 --- 0 0 0 0 ] [ 3 ] [ ] [0 0 0 2 0 0 0 ] [ ] [ 8 ] [0 0 0 0 --- 0 0 ] [ 3 ] [ ] [ 10 ] [0 0 0 0 0 ---- 0 ] [ 3 ] [ ] [ 7 ] [0 0 0 0 0 0 ---] [ 3 ] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((d(0,0),0,0,0,0,0,0),(0,(2*d(0,0))/3,0,0,0,0,0),(0,0,(4*d(0,0))/3,0,0,0,0),( 0,d(3,1),d(3,2),2*d(0,0),0,0,0),( - d(6,0) + d(4,0),d(6,0) + d(4,1) - 2*d(4,0) + d(3,1), - d(5,6) + d(4,2),d(3,2),(8*d(0,0))/3,0,0),(( - d(6,0) + 3*d(5,0) + 3*d (4,0))/3,(2*d(6,0) + 6*d(5,1) - 6*d(5,0) + 6*d(4,1) - 8*d(4,0) + 3*d(3,1))/6, - d(5,6) + d(5,2) + d(4,2), - d(5,6) + d(4,2) - d(3,1),d(3,2),(10*d(0,0))/3,d(5,6) + d(3,2)),(d(6,0), - d(6,0) + d(4,0),d(5,6) + d(3,2),0,0,0,(7*d(0,0))/3))$ $ PP:= [1 -1 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 ] [ ] [ - 1 ] [0 0 ------ -1 1 0 1 ] [ 6 ] [ ] [ 1 1 - 2 ] [0 0 ---- --- -1 1 ------] [ 18 2 3 ] [ ] [ - 2 ] [0 0 ------ -1 0 0 1 ] [ 3 ] avec PP:=P*Q:= [1 -1 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 ] [ ] [ - 1 ] [0 0 ------ -1 1 0 1 ] [ 6 ] [ ] [ 1 1 - 2 ] [0 0 ---- --- -1 1 ------] [ 18 2 3 ] [ ] [ - 2 ] [0 0 ------ -1 0 0 1 ] [ 3 ] MATDDIAGONALISE:= mat((d(0,0),0,0,0,0,0,0), 2*d(0,0) (0,----------,0,0,0,0,0), 3 4*d(0,0) (0,0,----------,0,0,0,0), 3 (0,d(3,1),d(3,2),2*d(0,0),0,0,0), ( - d(6,0) + d(4,0),d(6,0) + d(4,1) - 2*d(4,0) + d(3,1), - d(5,6) + d(4,2), 8*d(0,0) d(3,2),----------,0,0), 3 - d(6,0) + 3*d(5,0) + 3*d(4,0) (---------------------------------, 3 2*d(6,0) + 6*d(5,1) - 6*d(5,0) + 6*d(4,1) - 8*d(4,0) + 3*d(3,1) -----------------------------------------------------------------, 6 10*d(0,0) - d(5,6) + d(5,2) + d(4,2), - d(5,6) + d(4,2) - d(3,1),d(3,2),-----------, 3 d(5,6) + d(3,2)), 7*d(0,0) (d(6,0), - d(6,0) + d(4,0),d(5,6) + d(3,2),0,0,0,----------)) 3 on voit apparaitre les poids sur la diagonale *** r declared operator r(1) := d(0,0) 2*d(0,0) r(2) := ---------- 3 4*d(0,0) r(3) := ---------- 3 r(4) := 2*d(0,0) 8*d(0,0) r(5) := ---------- 3 10*d(0,0) r(6) := ----------- 3 7*d(0,0) r(7) := ---------- 3 3*gamma1 r(1) := ---------- 2 r(2) := gamma1 r(3) := 2*gamma1 r(4) := 3*gamma1 r(5) := 4*gamma1 r(6) := 5*gamma1 7*gamma1 r(7) := ---------- 2 Le systeme de poids est le systeme 1.8 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},0}, {{0,2},x(6) + x(4)}, {{0,3},x(5)}, {{0,4},0}, {{0,5},0}, {{0,6},x(5)}, {{1,2},x(3)}, {{1,3},x(4)}, {{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):=x(0) diaY(2):=x(1) - x(0) - 12*x(6) + x(5) - 3*x(4) + 18*x(2) diaY(3):=-------------------------------------- 18 - 2*x(6) + x(5) - 2*x(4) + 2*x(3) diaY(4):=------------------------------------ 2 diaY(5):= - x(5) + x(4) diaY(6):=x(5) 3*x(6) - 2*x(5) + 3*x(4) diaY(7):=-------------------------- 3 liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},0}, {{1,3},diay(7)}, {{1,4},0}, {{1,5},0}, {{1,6},0}, {{1,7},diay(6)}, {{2,3},diay(4)}, {{2,4},diay(5)}, {{2,5},diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},diay(6)}, {{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_{7,1.8}$ 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),(0,0,0,0,0 ,1,0),(0,0,0,0,0,0,-1),(0,0,0,0,-1,0,0))$ $ det(isom):= 1$ ZZ(1):= - diay(2)$ ZZ(2):=diay(3)$ ZZ(3):=diay(1)$ ZZ(4):= - diay(4)$ ZZ(5):= - diay(7)$ ZZ(6):=diay(5)$ ZZ(7):= - diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},0}$ {{1,4},zz(6)}$ {{1,5},0}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(5)}$ {{2,4},zz(7)}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4},0}$ {{3,5},zz(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}$ We get the commutation relations of$ g_{7,1.8}$ Et cela pour a:=1, b:=1.$ shortformdelta:={0,ss,1,ss,0,1,ss,0,1}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,1,0,0,1),(0,1,0 ,0,0,0))$ $ The isomorphism from g_{7,1.8} to gtildedelta$ was constructed in 2 steps and is given by$ the product matrix P*isom:= mat((1,0,1,0,0,0,0),(-1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,0,-1,0,0,0),(0,( - 1)/ 6,0,1,-1,1,0),(0,1/18,0,( - 1)/2,2/3,-1,-1),(0,( - 2)/3,0,1,-1,0,0))$ $ which we record here under the name PSI$