generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),xi(2,3),0,0,0),(xi(3,1),xi(3,2),xi(3,3) ,0,0,0),(xi(4,1),xi(4,2),xi(4,3),xi(2,2) + xi(1,1),xi(2,3),xi(4,6)),(xi(5,1),xi( 5,2),xi(5,3),xi(3,2),xi(3,3) + 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 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] adx2 := [ ] [-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 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [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 And the matrix A:=(xi(2,2),xi(2,3)),(xi(3,2),xi(3,3)) is nilpotent We hence get 2 cases according to whether A neq 0 or A=0. We consider here the case 1 where A = 0. In that case, one may suppose A:=((0,0),(0,0)). xi(2,2):=0 xi(2,3):=0 xi(3,2):=0 xi(3,3):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 0 0 0 0 0 ] [ ] [xi(3,1) 0 0 0 0 0 ] [ ] [ 0 0 xi(4,3) 0 0 xi(4,6)] [ ] [ 0 xi(5,2) xi(5,3) 0 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), xi(3,1), xi(6,1), ss, xi(5,2), xi(6,2), ss, xi(4,3), xi(5,3), xi(6,3), ss, xi(4,6), xi(5,6)} paramindexeslist:={{2,1}, {3,1}, {6,1}, {5,2}, {6,2}, {4,3}, {5,3}, {6,3}, {4,6}, {5,6}} a:=1$ b:=1$ 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),(0,1,0,0,0,1),(0,0,0 ,0,0,0))$ shortformdelta:={1,0,0,ss,1,0,ss,1,0,0,ss,0,1}$ 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(6,1) - d(5,2) - d(3,0) + d(2,1)$ Unknowns: {d(6,1),d(5,2),d(3,0),d(2,1)} Unknowns: {d(6,1),d(5,2),d(3,0),d(2,1)} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(5,2) + d(3,0) - d(2,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,2)$ Unknown: d(6,2) Unknown: d(6,2) bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},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 {{0,2},5} qui est maintenant AA:= - d(5,5) + d(1,1) + 2*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:=d(1,1) + 2*d(0,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,5)$ Unknown: d(6,5) Unknown: d(6,5) bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=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(6,3) - d(5,4) + d(2,3) + d(1,0)$ Unknowns: {d(6,3),d(5,4),d(2,3),d(1,0)} Unknowns: {d(6,3),d(5,4),d(2,3),d(1,0)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(5,4) - d(2,3) - d(1,0)$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,4)$ Unknown: d(6,4) Unknown: d(6,4) bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},5} 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 {{0,6},2} 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 {{0,6},4} qui est maintenant AA:=d(3,6) - d(1,0)$ Unknowns: {d(3,6),d(1,0)} Unknowns: {d(3,6),d(1,0)} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(3,6)$ on resout l'equation {{0,6},5} qui est maintenant AA:=d(6,6) + d(2,6) - d(1,1) - d(0,0)$ Unknowns: {d(6,6),d(2,6),d(1,1),d(0,0)} Unknowns: {d(6,6),d(2,6),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(2,6) + d(1,1) + d(0,0)$ 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:= - d(5,4) + d(0,1)$ Unknowns: {d(5,4),d(0,1)} Unknowns: {d(5,4),d(0,1)} bonne inconnue W:=d(5,4)$ 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:= - d(3,6) + d(2,3) + d(0, 1)$ Unknowns: {d(3,6),d(2,3),d(0,1)} Unknowns: {d(3,6),d(2,3),d(0,1)} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=d(3,6) - d(2,3)$ on resout l'equation {{1,3},5} 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)$ on resout l'equation {{1,6},2} 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 {{1,6},4} 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 {{1,6},5} qui est maintenant AA:=2*d(3,6) - d(2,3)$ Unknowns: {d(3,6),d(2,3)} Unknowns: {d(3,6),d(2,3)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(2,3)/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},2},0}, {{{0,4},4},0}, {{{0,4},5},0}, {{{0,5},2},0}, {{{0,5},4},0}, {{{0,5},5},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},2},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,5},2},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,6},2},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,6},4},0}, {{{2,6},5},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}, {{{4,6},5},0}, {{{5,6},5},0}}$ Il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),( - d(2,3))/2,0,0,0,0,0),(d(2,3)/2,d(0,0),0,0,0,0,0),(d(2,0),d(2,1), 2*d(0,0),d(2,3),0,0,0),(d(3,0),d(3,1),0,2*d(0,0),0,0,d(2,3)/2),(d(4,0),d(4,1),d( 3,1) - d(2,0),d(4,3),3*d(0,0),d(2,3)/2,d(4,6)),(d(5,0),d(5,1),d(5,2),d(5,3),( - d(2,3))/2,3*d(0,0),d(5,6)),(d(6,0),d(5,2) + d(3,0) - d(2,1),0, - 2*d(2,3),0,0,2* d(0,0)))$ 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 1 0 0 0 1] [ ] [0 0 0 0 0 0] pour shortformdelta:={1,0,0,ss,1,0,ss,1,0,0,ss,0,1} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,6), d(4,3), d(4,1), d(4,0), d(3,1), d(3,0), d(2,3), d(2,1), d(2,0), d(0,0)} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,6), d(4,3), d(4,1), d(4,0), d(3,1), d(3,0), d(2,3), d(2,1), d(2,0), d(0,0)} listeparametresMATD{d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,6), d(4,3), d(4,1), d(4,0), d(3,1), d(3,0), d(2,3), d(2,1), d(2,0), d(0,0)}$ dim Der(gtildedelta):=16$ t1:=D(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 3 0] [ ] [0 0 0 0 0 0 2] Unknowns: {d(2,3),d(0,0)} Unknowns: {d(2,3),d(0,0)} commutant de t1 dans der(gtildedelta): [ - d(2,3) ] [ d(0,0) ----------- 0 0 0 0 0 ] [ 2 ] [ ] [ d(2,3) ] [-------- d(0,0) 0 0 0 0 0 ] [ 2 ] [ ] [ 0 0 2*d(0,0) d(2,3) 0 0 0 ] [ ] [ d(2,3) ] [ 0 0 0 2*d(0,0) 0 0 --------] [ 2 ] [ ] [ d(2,3) ] [ 0 0 0 0 3*d(0,0) -------- 0 ] [ 2 ] [ ] [ - d(2,3) ] [ 0 0 0 0 ----------- 3*d(0,0) 0 ] [ 2 ] [ ] [ 0 0 0 - 2*d(2,3) 0 0 2*d(0,0)] Unknowns: {d(2,3),d(0,0)} Unknowns: {d(2,3),d(0,0)} t2:=D(2,3):= [ - 1 ] [ 0 ------ 0 0 0 0 0 ] [ 2 ] [ ] [ 1 ] [--- 0 0 0 0 0 0 ] [ 2 ] [ ] [ 0 0 0 1 0 0 0 ] [ ] [ 1 ] [ 0 0 0 0 0 0 ---] [ 2 ] [ ] [ 1 ] [ 0 0 0 0 0 --- 0 ] [ 2 ] [ ] [ - 1 ] [ 0 0 0 0 ------ 0 0 ] [ 2 ] [ ] [ 0 0 0 -2 0 0 0 ] (x*( - 16*x**6 - 24*x**4 - 9*x**2 - 1))/16$ 2 {{4*x + 1, 2, [ - arbcomplex(53) ] [-------------------] [ 2*x ] [ ] [ arbcomplex(53) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(54) ] [ ---------------- ] [ 2*x ] [ ] [ arbcomplex(54) ] [ ] [ 0 ] }, {x,1, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(55)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, 2 {x + 1, 1, [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(56) ] [ ------------------- ] [ 2 ] [ ] [ - arbcomplex(56)*x ] [---------------------] [ 2 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(56) ] }} Unknowns: {d(2,3),d(0,0)} Unknowns: {d(2,3),d(0,0)} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),( - d(2,3))/2,0,0,0,0,0),(d(2,3)/2,d(0,0),0,0,0,0,0),(0,0,2*d(0,0),d (2,3),0,0,0),(0,0,0,2*d(0,0),0,0,d(2,3)/2),(0,0,0,0,3*d(0,0),d(2,3)/2,0),(0,0,0, 0,( - d(2,3))/2,3*d(0,0),0),(0,0,0, - 2*d(2,3),0,0,2*d(0,0)))$ 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 i 0 0 0 0 0 ] [ ] [i 1 0 0 0 0 0 ] [ ] [ - 1 ] [0 0 1 - i 0 0 ------] [ 2 ] [ ] [ i ] [0 0 0 1 0 0 --- ] [ 2 ] [ ] [0 0 0 0 1 i 0 ] [ ] [0 0 0 0 i 1 0 ] [ ] [0 0 0 2*i 0 0 1 ] P**(-1)*t1*P:= [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 3 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t2*P:= [ - i ] [------ 0 0 0 0 0 0 ] [ 2 ] [ ] [ i ] [ 0 --- 0 0 0 0 0 ] [ 2 ] [ ] [ 0 0 0 0 0 0 0 ] [ ] [ 0 0 0 i 0 0 0 ] [ ] [ i ] [ 0 0 0 0 --- 0 0 ] [ 2 ] [ ] [ - i ] [ 0 0 0 0 0 ------ 0 ] [ 2 ] [ ] [ 0 0 0 0 0 0 - i] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): P**(-1)*MATD*P:= mat((( - i*d(2,3) + 2*d(0,0))/2,0,0,0,0,0,0),(0,(i*d(2,3) + 2*d(0,0))/2,0,0,0,0, 0),((d(6,0) + i*d(5,2) + i*d(3,0) + i*d(2,1) + 2*d(2,0))/2,(i*d(6,0) + d(5,2) + d(3,0) + d(2,1) + 2*i*d(2,0))/2,2*d(0,0),0,0,0,0),(( - i*d(6,0) + d(5,2) + 2*i*d (3,1) + 3*d(3,0) - d(2,1))/4,(d(6,0) - i*d(5,2) + 2*d(3,1) + i*d(3,0) + i*d(2,1) )/4,0,i*d(2,3) + 2*d(0,0),0,0,0),((d(5,1) - i*d(5,0) + i*d(4,1) + d(4,0))/2,( - i*d(5,1) + d(5,0) + d(4,1) + i*d(4,0))/2,( - i*d(5,2) + d(3,1) - d(2,0))/2,(2*d( 5,6) - i*d(5,3) - d(5,2) + 2*i*d(4,6) + d(4,3) - i*d(3,1) + i*d(2,0))/2,(i*d(2,3 ) + 6*d(0,0))/2,0,( - 2*i*d(5,6) + d(5,3) + i*d(5,2) + 2*d(4,6) + i*d(4,3) - d(3 ,1) + d(2,0))/4),((i*d(5,1) + d(5,0) + d(4,1) - i*d(4,0))/2,(d(5,1) + i*d(5,0) - i*d(4,1) + d(4,0))/2,(d(5,2) - i*d(3,1) + i*d(2,0))/2,(2*i*d(5,6) + d(5,3) - i* d(5,2) + 2*d(4,6) - i*d(4,3) - d(3,1) + d(2,0))/2,0,( - i*d(2,3) + 6*d(0,0))/2,( 2*d(5,6) + i*d(5,3) - d(5,2) - 2*i*d(4,6) + d(4,3) + i*d(3,1) - i*d(2,0))/4),((d (6,0) + i*d(5,2) + 2*d(3,1) - i*d(3,0) - i*d(2,1))/2,(i*d(6,0) + d(5,2) - 2*i*d( 3,1) + 3*d(3,0) - d(2,1))/2,0,0,0,0, - i*d(2,3) + 2*d(0,0)))$ PP:= [1 i 0 0 0 0 0 ] [ ] [i 1 0 0 0 0 0 ] [ ] [ - 1 ] [0 0 1 - i 0 0 ------] [ 2 ] [ ] [ i ] [0 0 0 1 0 0 --- ] [ 2 ] [ ] [0 0 0 0 1 i 0 ] [ ] [0 0 0 0 i 1 0 ] [ ] [0 0 0 2*i 0 0 1 ] avec PP:=P*Q:= [1 i 0 0 0 0 0 ] [ ] [i 1 0 0 0 0 0 ] [ ] [ - 1 ] [0 0 1 - i 0 0 ------] [ 2 ] [ ] [ i ] [0 0 0 1 0 0 --- ] [ 2 ] [ ] [0 0 0 0 1 i 0 ] [ ] [0 0 0 0 i 1 0 ] [ ] [0 0 0 2*i 0 0 1 ] MATDDIAGONALISE:= - i*d(2,3) + 2*d(0,0) mat((------------------------,0,0,0,0,0,0), 2 i*d(2,3) + 2*d(0,0) (0,---------------------,0,0,0,0,0), 2 d(6,0) + i*d(5,2) + i*d(3,0) + i*d(2,1) + 2*d(2,0) (----------------------------------------------------, 2 i*d(6,0) + d(5,2) + d(3,0) + d(2,1) + 2*i*d(2,0) --------------------------------------------------,2*d(0,0),0,0,0,0), 2 - i*d(6,0) + d(5,2) + 2*i*d(3,1) + 3*d(3,0) - d(2,1) (-------------------------------------------------------, 4 d(6,0) - i*d(5,2) + 2*d(3,1) + i*d(3,0) + i*d(2,1) ----------------------------------------------------,0,i*d(2,3) + 2*d(0,0), 4 0,0,0), d(5,1) - i*d(5,0) + i*d(4,1) + d(4,0) (---------------------------------------, 2 - i*d(5,1) + d(5,0) + d(4,1) + i*d(4,0) - i*d(5,2) + d(3,1) - d(2,0) ------------------------------------------,-------------------------------, 2 2 2*d(5,6) - i*d(5,3) - d(5,2) + 2*i*d(4,6) + d(4,3) - i*d(3,1) + i*d(2,0) --------------------------------------------------------------------------, 2 i*d(2,3) + 6*d(0,0) ---------------------,0, 2 - 2*i*d(5,6) + d(5,3) + i*d(5,2) + 2*d(4,6) + i*d(4,3) - d(3,1) + d(2,0) --------------------------------------------------------------------------- 4 ), i*d(5,1) + d(5,0) + d(4,1) - i*d(4,0) (---------------------------------------, 2 d(5,1) + i*d(5,0) - i*d(4,1) + d(4,0) d(5,2) - i*d(3,1) + i*d(2,0) ---------------------------------------,------------------------------, 2 2 2*i*d(5,6) + d(5,3) - i*d(5,2) + 2*d(4,6) - i*d(4,3) - d(3,1) + d(2,0) ------------------------------------------------------------------------,0, 2 - i*d(2,3) + 6*d(0,0) ------------------------, 2 2*d(5,6) + i*d(5,3) - d(5,2) - 2*i*d(4,6) + d(4,3) + i*d(3,1) - i*d(2,0) --------------------------------------------------------------------------) 4 , d(6,0) + i*d(5,2) + 2*d(3,1) - i*d(3,0) - i*d(2,1) (----------------------------------------------------, 2 i*d(6,0) + d(5,2) - 2*i*d(3,1) + 3*d(3,0) - d(2,1) ----------------------------------------------------,0,0,0,0, 2 - i*d(2,3) + 2*d(0,0))) on voit apparaitre les poids sur la diagonale - i*d(2,3) + 2*d(0,0) r(1) := ------------------------ 2 i*d(2,3) + 2*d(0,0) r(2) := --------------------- 2 r(3) := 2*d(0,0) r(4) := i*d(2,3) + 2*d(0,0) i*d(2,3) + 6*d(0,0) r(5) := --------------------- 2 - i*d(2,3) + 6*d(0,0) r(6) := ------------------------ 2 r(7) := - i*d(2,3) + 2*d(0,0) r(1) := gamma2 r(2) := gamma1 r(3) := gamma1 + gamma2 r(4) := 2*gamma1 r(5) := 2*gamma1 + gamma2 r(6) := gamma1 + 2*gamma2 r(7) := 2*gamma2 Le systeme de poids est le systeme 2.38 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(2)}, {{0,2},x(5)}, {{0,3},x(4)}, {{0,4},0}, {{0,5},0}, {{0,6},x(5)}, {{1,2},x(4)}, {{1,3},x(5)}, {{1,4},0}, {{1,5},0}, {{1,6},0}, {{2,3},0}, {{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):=i*x(1) + x(0) diaY(2):=x(1) + i*x(0) diaY(3):=x(2) diaY(4):=2*i*x(6) + x(3) - i*x(2) diaY(5):=i*x(5) + x(4) diaY(6):=x(5) + i*x(4) 2*x(6) + i*x(3) - x(2) diaY(7):=------------------------ 2 liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},2*diay(3)}, {{1,3},diay(6)}, {{1,4},2*diay(5)}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},diay(5)}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{2,7},i*diay(6)}, {{3,4},0}, {{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_{7,2.38}$ 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,-1,0,0,0,0,0),(1,0,0,0,0,0,0),(0,0,2,0,0,0,0),(0,0,0,-1,0,0,0),(0,0,0,0,0 ,2,0),(0,0,0,0,0,0,-2),(0,0,0,0,2*i,0,0))$ det(isom):= 16*i$ ZZ(1):=diay(2)$ ZZ(2):= - diay(1)$ ZZ(3):=2*diay(3)$ ZZ(4):= - diay(4)$ ZZ(5):=2*i*diay(7)$ ZZ(6):=2*diay(5)$ ZZ(7):= - 2*diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(3)}$ {{1,3},zz(6)}$ {{1,4},0}$ {{1,5},zz(7)}$ {{1,6},0}$ {{1,7},0}$ {{2,3},zz(7)}$ {{2,4},zz(6)}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4},0}$ {{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_{7,2.38}$ Et cela pour a:=1, b:=1.$ shortformdelta:={1,0,0,ss,1,0,ss,1,0,0,ss,0,1}$ 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),(0,1,0,0,0,1),(0,0,0 ,0,0,0))$