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),xi(2,2) + xi( 1,1),0,0,0),(xi(4,1),xi(4,2),xi(3,2),xi(2,2) + 2*xi(1,1),0,0),(xi(5,1),xi(5,2), xi(4,2),xi(3,2),xi(2,2) + 3*xi(1,1),0),(xi(6,1),xi(6,2), - xi(3,1),xi(2,1),0,2* xi(2,2) + 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 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 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] adx3 := [ ] [-1 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 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 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) 0 0 0] [ ] [xi(6,1) xi(6,2) 0 xi(2,1) 0 0] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(4,2), ss, xi(5,2), ss, xi(6,1), xi(6,2)} paramindexeslist:={{2,1},{4,2},{5,2},{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),(0,1,1,0,0,0),(1,0,0 ,0,0,0))$ $ shortformdelta:={0,ss,1,ss,1,ss,1,0}$ on resout l'equation {{0,1},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 {{0,1},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 {{0,1},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 {{0,1},3} qui est maintenant AA:= - (d(3,6) + d(2,0))$ Unknowns: {d(3,6),d(2,0)} Unknowns: {d(3,6),d(2,0)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - d(4,6) - d(3,0) + d(2, 1)$ Unknowns: {d(4,6),d(3,0),d(2,1)} Unknowns: {d(4,6),d(3,0),d(2,1)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(3,0) + d(2,1)$ on resout l'equation {{0,1},5} qui est maintenant AA:= - d(5,6) - d(4,0) + d(3, 1) + d(2,1)$ Unknowns: {d(5,6),d(4,0),d(3,1),d(2,1)} Unknowns: {d(5,6),d(4,0),d(3,1),d(2,1)} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:= - d(4,0) + d(3,1) + d(2,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,6) + d(1,1) + 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) + d(0,0)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - (d(0,5) + d(0,4))$ Unknowns: {d(0,5),d(0,4)} Unknowns: {d(0,5),d(0,4)} bonne inconnue W:=d(0,5)$ sa valeur doit etre WW:= - d(0,4)$ on resout l'equation {{0,2},1} qui est maintenant AA:= - (d(1,5) + d(1,4))$ Unknowns: {d(1,5),d(1,4)} Unknowns: {d(1,5),d(1,4)} bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:= - d(1,4)$ on resout l'equation {{0,2},2} qui est maintenant AA:= - (d(2,5) + d(2,4))$ Unknowns: {d(2,5),d(2,4)} Unknowns: {d(2,5),d(2,4)} bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:= - d(2,4)$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,5) - d(3,4) + d(1, 0)$ Unknowns: {d(3,5),d(3,4),d(1,0)} Unknowns: {d(3,5),d(3,4),d(1,0)} bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:= - d(3,4) + d(1,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,5) - d(4,4) + d(2, 2) + d(0,0)$ Unknowns: {d(4,5),d(4,4),d(2,2),d(0,0)} Unknowns: {d(4,5),d(4,4),d(2,2),d(0,0)} bonne inconnue W:=d(4,5)$ 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(5,5) - d(5,4) + d(3, 2) + d(2,2) + d(0,0)$ Unknowns: {d(5,5),d(5,4),d(3,2),d(2,2),d(0,0)} Unknowns: {d(5,5),d(5,4),d(3,2),d(2,2),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(5,4) + d(3,2) + d(2,2) + d(0,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,5) - d(6,4) - d(3, 0) + d(1,2)$ Unknowns: {d(6,5),d(6,4),d(3,0),d(1,2)} Unknowns: {d(6,5),d(6,4),d(3,0),d(1,2)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(6,4) - d(3,0) + d(1,2)$ 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)$ Unknown: d(2,4) Unknown: d(2,4) bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,3},3} qui est maintenant AA:=d(3,4) - d(1,0)$ Unknowns: {d(3,4),d(1,0)} Unknowns: {d(3,4),d(1,0)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,3},4} qui est maintenant AA:=d(4,4) + d(2,3) - d(2,2) + d(1,0) - d(0,0)$ Unknowns: {d(4,4),d(2,3),d(2,2),d(1,0),d(0,0)} Unknowns: {d(4,4),d(2,3),d(2,2),d(1,0),d(0,0)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:= - d(2,3) + d(2,2) - d(1,0) + d(0,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:=d(5,4) + d(3,3) - d(3,2) + d(2,3) - d(2,2)$ Unknowns: {d(5,4),d(3,3),d(3,2),d(2,3),d(2,2)} Unknowns: {d(5,4),d(3,3),d(3,2),d(2,3),d(2,2)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - d(3,3) + d(3,2) - d(2,3) + d(2,2)$ on resout l'equation {{0,3},6} qui est maintenant AA:=d(6,4) + d(3,0) + d(2,0) + d(1,3) - d(1,2)$ Unknowns: {d(6,4),d(3,0),d(2,0),d(1,3),d(1,2)} Unknowns: {d(6,4),d(3,0),d(2,0),d(1,3),d(1,2)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - d(3,0) - d(2,0) - d(1,3) + d(1,2)$ on resout l'equation {{0,4},5} qui est maintenant AA:=2*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(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 {{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) + 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(0, 1)$ Unknowns: {d(5,3),d(4,2),d(0,1)} Unknowns: {d(5,3),d(4,2),d(0,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:=d(4,2) + d(0,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - (d(6,3) + d(3,1) + d(0 ,2))$ Unknowns: {d(6,3),d(3,1),d(0,2)} Unknowns: {d(6,3),d(3,1),d(0,2)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - (d(3,1) + d(0,2))$ on resout l'equation {{1,3},4} qui est maintenant AA:=2*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)/2$ on resout l'equation {{1,3},5} qui est maintenant AA:=(4*d(0,1) + d(0,0))/2$ Unknowns: {d(0,1),d(0,0)} Unknowns: {d(0,1),d(0,0)} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=( - d(0,0))/4$ on resout l'equation {{1,3},6} qui est maintenant AA:=d(3,0) + d(2,1) - d(1,2)$ Unknowns: {d(3,0),d(2,1),d(1,2)} Unknowns: {d(3,0),d(2,1),d(1,2)} bonne inconnue W:=d(3,0)$ sa valeur doit etre WW:= - d(2,1) + d(1,2)$ on resout l'equation {{1,6},5} qui est maintenant AA:=2*d(2,1) - d(1,2)$ Unknowns: {d(2,1),d(1,2)} Unknowns: {d(2,1),d(1,2)} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=d(1,2)/2$ 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:=d(4,0) - d(3,1) + d(0,2)$ Unknowns: {d(4,0),d(3,1),d(0,2)} Unknowns: {d(4,0),d(3,1),d(0,2)} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(3,1) - d(0,2)$ on resout l'equation {{2,3},6} qui est maintenant AA:=2*d(2,2) - d(0,0)$ 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:=d(0,0)/2$ 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},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},4},0}, {{{0,5},5},0}, {{{0,5},6},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}, {{{1,5},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},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}, {{{3,4},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}, {{{4,6},5},0}}$ Il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),( - d(0,0))/4,d(0,2),0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,0,d(0,0)/2,0 ,0,0,0),(0,d(3,1),d(3,2),d(0,0),0,0,0),(d(3,1) - d(0,2),d(4,1),d(4,2),(4*d(3,2) - d(0,0))/4,(3*d(0,0))/2,0,0),(d(5,0),d(5,1),d(5,2),(4*d(4,2) - d(0,0))/4,(2*d(3 ,2) - d(0,0))/2,2*d(0,0),d(0,2)),(d(6,0),d(6,1),d(6,2), - (d(3,1) + d(0,2)),0,0, (3*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 1 0 0 0 0] [ ] [0 1 1 0 0 0] [ ] [1 0 0 0 0 0] pour shortformdelta:={0,ss,1,ss,1,ss,1,0} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,0)} Unknowns: {d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,0)} listeparametresMATD{d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(3,2), d(3,1), d(0,2), d(0,0)}$ dim Der(gtildedelta):=12$ t1:=D(0,0):= [ - 1 ] [1 ------ 0 0 0 0 0 ] [ 4 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [ 1 ] [0 0 --- 0 0 0 0 ] [ 2 ] [ ] [0 0 0 1 0 0 0 ] [ ] [ - 1 3 ] [0 0 0 ------ --- 0 0 ] [ 4 2 ] [ ] [ - 1 - 1 ] [0 0 0 ------ ------ 2 0 ] [ 4 2 ] [ ] [ 3 ] [0 0 0 0 0 0 ---] [ 2 ] {{2*x - 1, 2, [ arbcomplex(105) ] [-----------------] [ 2 ] [ ] [ arbcomplex(105) ] [ ] [ arbcomplex(106) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 2,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(107)] [ ] [ 0 ] }, {x - 1, 2, [ arbcomplex(108) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [2*arbcomplex(109)] [ ] [ arbcomplex(109) ] [ ] [ arbcomplex(109) ] [ ] [ 0 ] }, {2*x - 3, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(110)] [ ] [arbcomplex(110)] [ ] [arbcomplex(111)] }} 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 ] [ 4 ] [ ] [ d(0,0) ] [ 0 -------- 0 0 0 0 0 ] [ 2 ] [ ] [ d(0,0) ] [ 0 0 -------- 0 0 0 0 ] [ 2 ] [ ] [ 0 0 0 d(0,0) 0 0 0 ] [ ] [ - d(0,0) 3*d(0,0) ] [ 0 0 0 ----------- ---------- 0 0 ] [ 4 2 ] [ ] [ - d(0,0) - d(0,0) ] [ 0 0 0 ----------- ----------- 2*d(0,0) 0 ] [ 4 2 ] [ ] [ 3*d(0,0) ] [ 0 0 0 0 0 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 ] [1 --- 0 0 0 0 0] [ 2 ] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 0 0 0] [ ] [ 1 ] [0 0 0 --- 1 0 0] [ 2 ] [ ] [ 1 ] [0 0 0 --- 1 1 0] [ 2 ] [ ] [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 ] [ ] [ 1 ] [0 0 --- 0 0 0 0 ] [ 2 ] [ ] [0 0 0 1 0 0 0 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 3 ] [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),0,d(0,2),0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,0,d(0,0)/2,0,0,0,0),(0,d (3,1),d(3,2),d(0,0),0,0,0),(d(3,1) - d(0,2),(2*d(4,1) - d(0,2))/2,(2*d(4,2) - d( 3,2))/2,d(3,2),(3*d(0,0))/2,0,0),( - (d(3,1) - d(0,2) - d(5,0)),(d(5,0) - 2*d(4, 1) + 2*d(5,1) - (d(3,1) - d(0,2)))/2,d(5,2) - d(4,2),(2*d(4,2) - d(3,2))/2,d(3,2 ),2*d(0,0),d(0,2)),(d(6,0),(2*d(6,1) + d(6,0))/2,d(6,2), - (d(3,1) + d(0,2)),0,0 ,(3*d(0,0))/2))$ $ PP:= [ 1 ] [1 --- 0 0 0 0 0] [ 2 ] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 0 0 0] [ ] [ 1 ] [0 0 0 --- 1 0 0] [ 2 ] [ ] [ 1 ] [0 0 0 --- 1 1 0] [ 2 ] [ ] [0 0 0 0 0 0 1] avec PP:=P*Q:= [ 1 ] [1 --- 0 0 0 0 0] [ 2 ] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 1 0 0 0] [ ] [ 1 ] [0 0 0 --- 1 0 0] [ 2 ] [ ] [ 1 ] [0 0 0 --- 1 1 0] [ 2 ] [ ] [0 0 0 0 0 0 1] MATDDIAGONALISE:= mat((d(0,0),0,d(0,2),0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 d(0,0) (0,0,--------,0,0,0,0), 2 (0,d(3,1),d(3,2),d(0,0),0,0,0), 2*d(4,1) - d(0,2) 2*d(4,2) - d(3,2) 3*d(0,0) (d(3,1) - d(0,2),-------------------,-------------------,d(3,2),----------,0 2 2 2 ,0), ( - (d(3,1) - d(0,2) - d(5,0)), d(5,0) - 2*d(4,1) + 2*d(5,1) - (d(3,1) - d(0,2)) --------------------------------------------------,d(5,2) - d(4,2), 2 2*d(4,2) - d(3,2) -------------------,d(3,2),2*d(0,0),d(0,2)), 2 2*d(6,1) + d(6,0) 3*d(0,0) (d(6,0),-------------------,d(6,2), - (d(3,1) + d(0,2)),0,0,----------)) 2 2 on voit apparaitre les poids sur la diagonale r(1) := d(0,0) d(0,0) r(2) := -------- 2 d(0,0) r(3) := -------- 2 r(4) := d(0,0) 3*d(0,0) r(5) := ---------- 2 r(6) := 2*d(0,0) 3*d(0,0) r(7) := ---------- 2 r(1) := 2*gamma1 r(2) := gamma1 r(3) := gamma1 r(4) := 2*gamma1 r(5) := 3*gamma1 r(6) := 4*gamma1 r(7) := 3*gamma1 Le systeme de poids est le systeme 1.2 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(6)}, {{0,2},x(5) + x(4)}, {{0,3},x(5)}, {{0,4},0}, {{0,5},0}, {{0,6},0}, {{1,2},x(3)}, {{1,3},x(4)}, {{1,4},x(5)}, {{1,5},0}, {{1,6},0}, {{2,3},x(6)}, {{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) 2*x(1) + x(0) diaY(2):=--------------- 2 diaY(3):=x(2) x(5) + x(4) + 2*x(3) diaY(4):=---------------------- 2 diaY(5):=x(5) + x(4) diaY(6):=x(5) diaY(7):=x(6) liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},diay(7)}, {{1,3},diay(5)}, {{1,4},diay(6)}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},diay(4)}, {{2,4},diay(5)}, {{2,5},diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},diay(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}}$ Now we make explicit the isomorphism with an algebra of the book:$ Namely g_{7,1.2}$ (iii)$ 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),(0,0,0,0,0,0,1),(0,0,0,0,0,1,0))$ $ 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(7)$ ZZ(7):=diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},zz(6)}$ {{1,4},zz(5)}$ {{1,5},zz(7)}$ {{1,6},0}$ {{1,7},0}$ {{2,3},zz(5)}$ {{2,4},zz(6)}$ {{2,5},0}$ {{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_{7,1.2}$ (iii)$ Et cela pour a:=1, b:=1.$ shortformdelta:={0,ss,1,ss,1,ss,1,0}$ 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),(0,1,1,0,0,0),(1,0,0 ,0,0,0))$ $ The isomorphism from g_{7,rkgtildedelta.1} to gtildedelta$ was constructed in 2 steps and is given by$ the product matrix P*isom:= mat((1/2,0,-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),(0,0,0,1/ 2,1,0,0),(0,0,0,1/2,1,0,1),(0,0,0,0,0,1,0))$ $ which we record here under the name PSI$ PSI_III4:= mat((1/2,0,-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),(0,0,0,1/ 2,1,0,0),(0,0,0,1/2,1,0,1),(0,0,0,0,0,1,0))$ $