generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(0,0,2*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(4,1),xi(4,2), xi(2,2) + 2*xi(1,1),0),(xi(6,1),xi(6,2),xi(6,3),xi(5,2),xi(4,2),xi(2,2) + 3*xi(1 ,1)))$ $ [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 1 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 -1 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 1 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 xi(2,2):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=0,xi(6,1):=0 delta:= [ 0 0 0 0 0 0] [ ] [xi(2,1) 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 - xi(2,1) 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 xi(6,2) xi(6,3) 0 0 0] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(6,2), ss, xi(6,3)} paramindexeslist:={{2,1},{6,2},{6,3}} a:=0$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,-1,0,0,0),(0,0,0,0,0,0),(0,1, 0,0,0,0))$ $ shortformdelta:={1,ss,1,ss,0}$ 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:= - (d(4,2) + d(3,1) + d(2 ,0))$ 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) + d(2,0))$ on resout l'equation {{0,1},5} qui est maintenant AA:= - (d(5,2) + d(4,0))$ Unknowns: {d(5,2),d(4,0)} Unknowns: {d(5,2),d(4,0)} bonne inconnue W:=d(5,2)$ sa valeur doit etre WW:= - d(4,0)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,2) - d(5,0) + d(2, 1)$ Unknowns: {d(6,2),d(5,0),d(2,1)} Unknowns: {d(6,2),d(5,0),d(2,1)} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:= - d(5,0) + d(2,1)$ on resout l'equation {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},4} qui est maintenant AA:= - d(4,6) + d(1,0)$ Unknowns: {d(4,6),d(1,0)} Unknowns: {d(4,6),d(1,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - (d(5,6) + d(3,0))$ Unknowns: {d(5,6),d(3,0)} Unknowns: {d(5,6),d(3,0)} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - d(3,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,6) + d(1,1) + 2*d( 0,0)$ 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) + 2*d(0,0)$ on resout l'equation {{0,3},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 {{0,3},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 {{0,3},2} qui est maintenant AA:=d(2,4) + d(1,3)$ 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:= - d(1,3)$ on resout l'equation {{0,3},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 {{0,3},4} qui est maintenant AA:=d(4,4) - d(3,3) - d(0,0)$ 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) + d(2,0)$ Unknowns: {d(5,4),d(2,0)} Unknowns: {d(5,4),d(2,0)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:=d(6,4) + d(4,0) + d(2,3)$ Unknowns: {d(6,4),d(4,0),d(2,3)} Unknowns: {d(6,4),d(4,0),d(2,3)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - (d(4,0) + d(2,3))$ on resout l'equation {{0,4},5} 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 {{0,4},6} qui est maintenant AA:= - (d(3,0) + d(1,3))$ Unknowns: {d(3,0),d(1,3)} Unknowns: {d(3,0),d(1,3)} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:= - d(1,3)$ on resout l'equation {{0,5},2} 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 {{0,5},4} 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 {{0,5},6} 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 {{1,2},2} 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},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(3,1)$ Unknown: d(3,1) Unknown: d(3,1) bonne inconnue W:=d(3,1)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,2},6} qui est maintenant AA:=d(2,3) + d(0,1)$ Unknowns: {d(2,3),d(0,1)} Unknowns: {d(2,3),d(0,1)} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:= - d(0,1)$ 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},4} qui est maintenant AA:= - 2*d(0,1)$ Unknown: d(0,1) Unknown: d(0,1) bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,3},5} qui est maintenant AA:=d(4,3) + d(2,1)$ Unknowns: {d(4,3),d(2,1)} Unknowns: {d(4,3),d(2,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(2,1)$ on resout l'equation {{1,3},6} qui est maintenant AA:=d(5,3) + d(4,1)$ Unknowns: {d(5,3),d(4,1)} Unknowns: {d(5,3),d(4,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(4,1)$ 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},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) + 3*d(1,1) + 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:=3*d(1,1) + d(0,0)$ on resout l'equation {{1,4},6} qui est maintenant AA:= - (d(6,5) + d(2,0))$ Unknowns: {d(6,5),d(2,0)} Unknowns: {d(6,5),d(2,0)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{1,5},6} 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$ 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},2},0}, {{{0,5},4},0}, {{{0,5},6},0}, {{{0,6},2},0}, {{{0,6},4},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},2},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},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},0},0}, {{{3,4},1},0}, {{{3,4},2},0}, {{{3,4},3},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$ collect_eq:={{{{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},2},0}, {{{0,5},4},0}, {{{0,5},6},0}, {{{0,6},2},0}, {{{0,6},4},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},2},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},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},0},0}, {{{3,4},1},0}, {{{3,4},2},0}, {{{3,4},3},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}}$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),0,0,0,0,0,0),(0,d(0,0)/3,0,0,0,0,0),(d(2,0),d(2,1),(4*d(0,0))/3,0,0, 0,0),(0,0,0,(2*d(0,0))/3,0,0,0),(d(4,0),d(4,1), - d(2,0), - d(2,1),(5*d(0,0))/3, 0,0),(d(5,0),d(5,1), - d(4,0), - d(4,1), - d(2,0),2*d(0,0),0),(d(6,0),d(6,1), - (d(5,0) - d(2,1)),d(6,3), - d(4,0), - d(2,0),(7*d(0,0))/3))$ $ pour delta:= [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 0 0 0 0] [ ] [0 1 0 0 0 0] pour shortformdelta:={1,ss,1,ss,0} Unknowns: {d(6,3), d(6,1), d(6,0), d(5,1), d(5,0), d(4,1), d(4,0), d(2,1), d(2,0), 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(2,1), d(2,0), 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(2,1), d(2,0), d(0,0)}$ dim Der(gtildedelta):=10$ t1:=D(0,0):= [1 0 0 0 0 0 0 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 3 ] [ ] [ 4 ] [0 0 --- 0 0 0 0 ] [ 3 ] [ ] [ 2 ] [0 0 0 --- 0 0 0 ] [ 3 ] [ ] [ 5 ] [0 0 0 0 --- 0 0 ] [ 3 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 7 ] [0 0 0 0 0 0 ---] [ 3 ] Unknown: d(0,0) Unknown: d(0,0) commutant de t1 dans der(gtildedelta): [d(0,0) 0 0 0 0 0 0 ] [ ] [ d(0,0) ] [ 0 -------- 0 0 0 0 0 ] [ 3 ] [ ] [ 4*d(0,0) ] [ 0 0 ---------- 0 0 0 0 ] [ 3 ] [ ] [ 2*d(0,0) ] [ 0 0 0 ---------- 0 0 0 ] [ 3 ] [ ] [ 5*d(0,0) ] [ 0 0 0 0 ---------- 0 0 ] [ 3 ] [ ] [ 0 0 0 0 0 2*d(0,0) 0 ] [ ] [ 7*d(0,0) ] [ 0 0 0 0 0 0 ----------] [ 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 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 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 ] [ 3 ] [ ] [ 4 ] [0 0 --- 0 0 0 0 ] [ 3 ] [ ] [ 2 ] [0 0 0 --- 0 0 0 ] [ 3 ] [ ] [ 5 ] [0 0 0 0 --- 0 0 ] [ 3 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 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,d(0,0)/3,0,0,0,0,0),(d(2,0),d(2,1),(4*d(0,0))/3,0,0, 0,0),(0,0,0,(2*d(0,0))/3,0,0,0),(d(4,0),d(4,1), - d(2,0), - d(2,1),(5*d(0,0))/3, 0,0),(d(5,0),d(5,1), - d(4,0), - d(4,1), - d(2,0),2*d(0,0),0),(d(6,0),d(6,1), - (d(5,0) - d(2,1)),d(6,3), - d(4,0), - d(2,0),(7*d(0,0))/3))$ $ PP:= [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 0] [ ] [0 0 0 0 0 0 1] avec PP:=P*Q:= [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 0] [ ] [0 0 0 0 0 0 1] MATDDIAGONALISE:= mat((d(0,0),0,0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 3 4*d(0,0) (d(2,0),d(2,1),----------,0,0,0,0), 3 2*d(0,0) (0,0,0,----------,0,0,0), 3 5*d(0,0) (d(4,0),d(4,1), - d(2,0), - d(2,1),----------,0,0), 3 (d(5,0),d(5,1), - d(4,0), - d(4,1), - d(2,0),2*d(0,0),0), 7*d(0,0) (d(6,0),d(6,1), - (d(5,0) - d(2,1)),d(6,3), - d(4,0), - d(2,0),----------)) 3 on voit apparaitre les poids sur la diagonale r(1) := d(0,0) d(0,0) r(2) := -------- 3 4*d(0,0) r(3) := ---------- 3 2*d(0,0) r(4) := ---------- 3 5*d(0,0) r(5) := ---------- 3 r(6) := 2*d(0,0) 7*d(0,0) r(7) := ---------- 3 r(1) := 3*gamma1 r(2) := gamma1 r(3) := 4*gamma1 r(4) := 2*gamma1 r(5) := 5*gamma1 r(6) := 6*gamma1 r(7) := 7*gamma1 Le systeme de poids est le systeme 1.1 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(2)}, {{0,2},x(6)}, {{0,3}, - x(4)}, {{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(5)}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{3,4}, - x(6)}, {{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) 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},diay(3)}, {{1,3},diay(7)}, {{1,4}, - diay(5)}, {{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},diay(6)}, {{3,5},0}, {{3,6},0}, {{3,7},0}, {{4,5}, - diay(7)}, {{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_{7,1.1}$ (v)$ 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((0,0,-1,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,1,0,0,0),(0,-1,0,0,0,0,0),(0,0,0,0,1 ,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1))$ $ det(isom):= -1$ ZZ(1):=diay(2)$ ZZ(2):= - diay(4)$ ZZ(3):= - diay(1)$ ZZ(4):=diay(3)$ ZZ(5):=diay(5)$ ZZ(6):=diay(6)$ ZZ(7):=diay(7)$ listcommutateursdesZZ:=$ {{1,2},0}$ {{1,3},zz(4)}$ {{1,4},zz(5)}$ {{1,5},zz(6)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(5)}$ {{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_{7,1.1}$ (v)$ Et cela pour a:=0$ and that for a neq conditionssura$ shortformdelta:={1,ss,1,ss,0}$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,0,-1,0,0,0),(0,0,0,0,0,0),(0,1, 0,0,0,0))$ $ The isomorphism from g_{7,1.1} to gtildedelta$ was constructed in 2 steps and is given by$ the product matrix P*isom:= mat((0,0,-1,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,1,0,0,0),(0,-1,0,0,0,0,0),(0,0,0,0,1 ,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1))$ $ which we record here under the name PSI$ PSI_I:= mat((0,0,-1,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,1,0,0,0),(0,-1,0,0,0,0,0),(0,0,0,0,1 ,0,0),(0,0,0,0,0,1,0),(0,0,0,0,0,0,1))$ $