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),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 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 := [ ] [-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] 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(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 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),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,1,1,0,0,1),(0,0,0 ,0,0,0))$ $ shortformdelta:={1,ss,1,ss,1,1,ss,0,0}$ 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) + d(2,0))$ Unknowns: {d(3,2),d(2,0)} Unknowns: {d(3,2),d(2,0)} bonne inconnue W:=d(3,2)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - d(4,2) - d(3,0) + d(2, 1)$ Unknowns: {d(4,2),d(3,0),d(2,1)} Unknowns: {d(4,2),d(3,0),d(2,1)} bonne inconnue W:=d(4,2)$ sa valeur doit etre WW:= - d(3,0) + d(2,1)$ on resout l'equation {{0,1},5} qui est maintenant AA:=d(6,1) - d(5,2) - d(4,0) + d(3,1) + d(2,1)$ Unknowns: {d(6,1),d(5,2),d(4,0),d(3,1),d(2,1)} Unknowns: {d(6,1),d(5,2),d(4,0),d(3,1),d(2,1)} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(5,2) + d(4,0) - d(3,1) - 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) + 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(1, 1) + 2*d(0,0)$ Unknowns: {d(4,5),d(4,4),d(1,1),d(0,0)} Unknowns: {d(4,5),d(4,4),d(1,1),d(0,0)} bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:= - d(4,4) + d(1,1) + 2*d(0,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:= - d(5,5) - d(5,4) - d(2, 0) + d(1,1) + 2*d(0,0)$ Unknowns: {d(5,5),d(5,4),d(2,0),d(1,1),d(0,0)} Unknowns: {d(5,5),d(5,4),d(2,0),d(1,1),d(0,0)} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:= - d(5,4) - d(2,0) + d(1,1) + 2*d(0,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - (d(6,5) + d(6,4))$ Unknowns: {d(6,5),d(6,4)} Unknowns: {d(6,5),d(6,4)} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(6,4)$ 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) - 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(1,1) + d(1,0) - 2*d(0,0)$ Unknowns: {d(4,4),d(2,3),d(1,1),d(1,0),d(0,0)} Unknowns: {d(4,4),d(2,3),d(1,1),d(1,0),d(0,0)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:= - d(2,3) + d(1,1) - d(1,0) + 2*d(0,0)$ on resout l'equation {{0,3},5} qui est maintenant AA:=d(6,3) + d(5,4) + d(3,3) + d(2,3) + d(2,0) - d(1,1) - d(0,0)$ Unknowns: {d(6,3),d(5,4),d(3,3),d(2,3),d(2,0),d(1,1),d(0,0)} Unknowns: {d(6,3),d(5,4),d(3,3),d(2,3),d(2,0),d(1,1),d(0,0)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:= - d(5,4) - d(3,3) - d(2,3) - d(2,0) + d(1,1) + d(0,0)$ 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},4} 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 {{0,4},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},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(2,6) - d(2,3)$ Unknowns: {d(2,6),d(2,3)} Unknowns: {d(2,6),d(2,3)} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=d(2,6)$ on resout l'equation {{0,6},5} qui est maintenant AA:=d(6,6) + d(5,4) + d(3,6) + d(2,6) + d(2,0) - d(1,1) - d(0,0)$ Unknowns: {d(6,6),d(5,4),d(3,6),d(2,6),d(2,0),d(1,1),d(0,0)} Unknowns: {d(6,6),d(5,4),d(3,6),d(2,6),d(2,0),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:= - d(5,4) - d(3,6) - d(2,6) - d(2,0) + d(1,1) + 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},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 {{1,2},3} qui est maintenant AA:= - d(3,3) + 2*d(1,1) + d( 0,0)$ Unknowns: {d(3,3),d(1,1),d(0,0)} Unknowns: {d(3,3),d(1,1),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=2*d(1,1) + d(0,0)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,3) - d(2,0) + d(0, 1)$ Unknowns: {d(4,3),d(2,0),d(0,1)} Unknowns: {d(4,3),d(2,0),d(0,1)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:= - d(2,0) + d(0,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,3) - d(3,0) + d(2, 1) + d(0,1)$ Unknowns: {d(5,3),d(3,0),d(2,1),d(0,1)} Unknowns: {d(5,3),d(3,0),d(2,1),d(0,1)} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - d(3,0) + d(2,1) + d(0,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:=d(5,4) + d(2,0) + d(1,1)$ Unknowns: {d(5,4),d(2,0),d(1,1)} Unknowns: {d(5,4),d(2,0),d(1,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:= - (d(2,0) + d(1,1))$ 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 {{1,6},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(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 {{1,6},5} qui est maintenant AA:=(4*d(4,6) - d(0,0))/4$ Unknowns: {d(4,6),d(0,0)} Unknowns: {d(4,6),d(0,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=d(0,0)/4$ 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},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},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,6},2},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,6},3},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}, {{{4,5},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))/4,0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(d(2,0),d(2,1),(3*d( 0,0))/2,0,0,0,0),(d(3,0),d(3,1), - d(2,0),2*d(0,0),0,0,0),(d(4,0),d(4,1), - d(3, 0) + d(2,1),( - 4*d(2,0) - d(0,0))/4,(5*d(0,0))/2,0,d(0,0)/4),(d(5,0),d(5,1),d(5 ,2),( - 4*d(3,0) + 4*d(2,1) - d(0,0))/4,( - 2*d(2,0) - d(0,0))/2,3*d(0,0),d(5,6) ),(d(6,0),d(5,2) + d(4,0) - d(3,1) - d(2,1),0,0,0,0,2*d(0,0)))$ $ pour delta:= [0 0 0 0 0 0] [ ] [1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] [ ] [0 1 1 0 0 1] [ ] [0 0 0 0 0 0] pour shortformdelta:={1,ss,1,ss,1,1,ss,0,0} Unknowns: {d(6,0), d(5,6), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,0)} Unknowns: {d(6,0), d(5,6), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), d(0,0)} listeparametresMATD{d(6,0), d(5,6), d(5,2), d(5,1), d(5,0), d(4,1), d(4,0), d(3,1), d(3,0), d(2,1), d(2,0), 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 ] [ ] [ 3 ] [0 0 --- 0 0 0 0 ] [ 2 ] [ ] [0 0 0 2 0 0 0 ] [ ] [ - 1 5 1 ] [0 0 0 ------ --- 0 ---] [ 4 2 4 ] [ ] [ - 1 - 1 ] [0 0 0 ------ ------ 3 0 ] [ 4 2 ] [ ] [0 0 0 0 0 0 2 ] {{2*x - 1, 1, [ arbcomplex(107) ] [-----------------] [ 2 ] [ ] [ arbcomplex(107) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {2*x - 3,1, [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(108)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {x - 2, 2, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ arbcomplex(110) + 4*arbcomplex(109) ] [ ------------------------------------- ] [ 2 ] [ ] [ - arbcomplex(110) + 4*arbcomplex(109) ] [----------------------------------------] [ 4 ] [ ] [ arbcomplex(109) ] [ ] [ arbcomplex(110) ] }, {x - 3,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(111)] [ ] [ 0 ] }, {x - 1,1, [arbcomplex(112)] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, {2*x - 5, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(113)] [ ] [arbcomplex(113)] [ ] [ 0 ] }} 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 ] [ ] [ 3*d(0,0) ] [ 0 0 ---------- 0 0 0 0 ] [ 2 ] [ ] [ 0 0 0 2*d(0,0) 0 0 0 ] [ ] [ - d(0,0) 5*d(0,0) d(0,0) ] [ 0 0 0 ----------- ---------- 0 --------] [ 4 2 4 ] [ ] [ - d(0,0) - d(0,0) ] [ 0 0 0 ----------- ----------- 3*d(0,0) 0 ] [ 4 2 ] [ ] [ 0 0 0 0 0 0 2*d(0,0)] 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 ] [ ] [ 1 ] [0 0 0 1 0 0 --- ] [ 2 ] [ ] [ 1 - 1 ] [0 0 0 --- 1 0 ------] [ 2 4 ] [ ] [ 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 ] [ ] [ 3 ] [0 0 --- 0 0 0 0] [ 2 ] [ ] [0 0 0 2 0 0 0] [ ] [ 5 ] [0 0 0 0 --- 0 0] [ 2 ] [ ] [0 0 0 0 0 3 0] [ ] [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),0,0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(d(2,0),(2*d(2,1) + d(2,0))/2,(3 *d(0,0))/2,0,0,0,0),(( - d(6,0) + 2*d(3,0))/2,( - d(6,0) - 2*d(5,2) - 2*d(4,0) + 6*d(3,1) + 2*d(3,0) + 2*d(2,1))/4, - d(2,0),2*d(0,0),0,0,0),((d(6,0) + 2*d(4,0) - d(3,0))/2,(d(6,0) + 2*d(5,2) + 4*d(4,1) + 4*d(4,0) - 4*d(3,1) - d(3,0) - 2*d( 2,1))/4,( - 2*d(3,0) + 2*d(2,1) + d(2,0))/2, - d(2,0),(5*d(0,0))/2,0,( - d(2,0)) /2),(( - d(6,0) + 4*d(5,0) - 4*d(4,0))/4,( - d(6,0) - 2*d(5,2) + 8*d(5,1) + 4*d( 5,0) - 8*d(4,1) - 6*d(4,0) + 2*d(3,1) + 2*d(2,1))/8,d(5,2) + d(3,0) - d(2,1),( - 2*d(3,0) + 2*d(2,1) + d(2,0))/2, - d(2,0),3*d(0,0),(4*d(5,6) - 2*d(3,0) + 2*d(2 ,1) + 3*d(2,0))/4),(d(6,0),(d(6,0) + 2*d(5,2) + 2*d(4,0) - 2*d(3,1) - 2*d(2,1))/ 2,0,0,0,0,2*d(0,0)))$ $ PP:= [ 1 ] [1 --- 0 0 0 0 0 ] [ 2 ] [ ] [0 1 0 0 0 0 0 ] [ ] [0 0 1 0 0 0 0 ] [ ] [ 1 ] [0 0 0 1 0 0 --- ] [ 2 ] [ ] [ 1 - 1 ] [0 0 0 --- 1 0 ------] [ 2 4 ] [ ] [ 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 ] [ ] [ 1 ] [0 0 0 1 0 0 --- ] [ 2 ] [ ] [ 1 - 1 ] [0 0 0 --- 1 0 ------] [ 2 4 ] [ ] [ 1 ] [0 0 0 --- 1 1 0 ] [ 2 ] [ ] [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), 2 2*d(2,1) + d(2,0) 3*d(0,0) (d(2,0),-------------------,----------,0,0,0,0), 2 2 - d(6,0) + 2*d(3,0) (----------------------, 2 - d(6,0) - 2*d(5,2) - 2*d(4,0) + 6*d(3,1) + 2*d(3,0) + 2*d(2,1) ------------------------------------------------------------------, 4 - d(2,0),2*d(0,0),0,0,0), d(6,0) + 2*d(4,0) - d(3,0) (----------------------------, 2 d(6,0) + 2*d(5,2) + 4*d(4,1) + 4*d(4,0) - 4*d(3,1) - d(3,0) - 2*d(2,1) ------------------------------------------------------------------------, 4 - 2*d(3,0) + 2*d(2,1) + d(2,0) 5*d(0,0) - d(2,0) ---------------------------------, - d(2,0),----------,0,-----------), 2 2 2 - d(6,0) + 4*d(5,0) - 4*d(4,0) (---------------------------------,( - d(6,0) - 2*d(5,2) + 8*d(5,1) 4 + 4*d(5,0) - 8*d(4,1) - 6*d(4,0) + 2*d(3,1) + 2*d(2,1))/8, - 2*d(3,0) + 2*d(2,1) + d(2,0) d(5,2) + d(3,0) - d(2,1),---------------------------------, - d(2,0), 2 4*d(5,6) - 2*d(3,0) + 2*d(2,1) + 3*d(2,0) 3*d(0,0),-------------------------------------------), 4 d(6,0) + 2*d(5,2) + 2*d(4,0) - 2*d(3,1) - 2*d(2,1) (d(6,0),----------------------------------------------------,0,0,0,0, 2 2*d(0,0))) on voit apparaitre les poids sur la diagonale r(1) := d(0,0) d(0,0) r(2) := -------- 2 3*d(0,0) r(3) := ---------- 2 r(4) := 2*d(0,0) 5*d(0,0) r(5) := ---------- 2 r(6) := 3*d(0,0) r(7) := 2*d(0,0) r(1) := 2*gamma1 r(2) := gamma1 r(3) := 3*gamma1 r(4) := 4*gamma1 r(5) := 5*gamma1 r(6) := 6*gamma1 r(7) := 4*gamma1 Le systeme de poids est le systeme 1.12 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(2)}, {{0,2},x(5) + 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},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):=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) 4*x(6) - x(4) + 2*x(3) diaY(7):=------------------------ 4 liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},diay(3)}, {{1,3},diay(5)}, {{1,4},diay(6)}, {{1,5},0}, {{1,6},0}, {{1,7},(3*diay(6))/2}, {{2,3},diay(4)}, {{2,4},diay(5)}, {{2,5},diay(6)}, {{2,6},0}, {{2,7},diay(5)/2}, {{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_{7,1.12}$ 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),(0,0,0,-1,0,0,0),(0,0,1/2,0,-1,0,0),(0,0,0,0 ,0,-1,0),(0,0,0,0,0,0,-1),(0,0,-1,0,0,0,0))$ $ det(isom):= 1$ ZZ(1):=diay(2)$ ZZ(2):=diay(1)$ ZZ(3):=( - (2*diay(7) - diay(4)))/2$ ZZ(4):= - diay(3)$ ZZ(5):= - diay(4)$ ZZ(6):= - diay(5)$ ZZ(7):= - diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(4)}$ {{1,3},0}$ {{1,4},zz(5)}$ {{1,5},zz(6)}$ {{1,6},zz(7)}$ {{1,7},0}$ {{2,3},zz(7)}$ {{2,4},zz(6)}$ {{2,5},zz(7)}$ {{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_{7,1.12}$ Et cela pour a:=1, b:=1.$ shortformdelta:={1,ss,1,ss,1,1,ss,0,0}$ delta:= mat((0,0,0,0,0,0),(1,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,1,1,0,0,1),(0,0,0 ,0,0,0))$ $