!The! generic! nilpotent! derivation! as! in! (!Cohomology! tables! page! 50)! : ! the! eigen\ values are! 0$ by! subtracting! adjoints! one! then! may! suppose! xi(4,1)=xi(4,2)=xi(5,1)=xi(6 ,1)=x\ i(6,2)=0$ delta:= mat((0,0,0,0,0,0),(xi(2,1),0,0,0,0,0),(xi(3,1),xi(3,2),0,0,0,0),(0,0,0,0,0,0),(0 ,xi(5,2), - xi(2,1),0,0,0),(0,0,xi(6,3),xi(5,2) - xi(3,1),xi(2,1),0))$ We denote this delta by the shortform$ shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,2), ss, xi(6,3)}$ paramindexeslist:={{2,1},{3,1},{3,2},{5,2},{6,3}}$ a neq {}$ a:=a$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,a ,1,0,0))$ shortformdelta:={0,ss,0,0,ss,1,ss,a}$ phase 1 de la resolution des equations$ on resout l'equation {{0,1},4} 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},5} qui est maintenant AA:= - d(4,0) + d(2,1)$ Unknowns: {d(4,0),d(2,1)} Unknowns: {d(4,0),d(2,1)} bonne inconnue W:=d(4,0)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(5,0) + d(4,1) + d(3, 1)*a$ Unknowns: {d(5,0),d(4,1),d(3,1),a} Unknowns: {d(5,0),d(4,1),d(3,1),a} bonne inconnue W:=d(5,0)$ sa valeur doit etre WW:=d(4,1) + d(3,1)*a$ 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(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) + d(0,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,5) + d(4,2) + d(3, 2)*a - d(3,0) - d(2,1)$ Unknowns: {d(6,5),d(4,2),d(3,2),d(3,0),d(2,1),a} Unknowns: {d(6,5),d(4,2),d(3,2),d(3,0),d(2,1),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:=d(4,2) + d(3,2)*a - d(3,0) - d(2,1)$ on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,6)*a$ Unknowns: {d(0,6),a} Unknowns: {d(0,6),a} pas de selection possible de variable a coefficient numerique dans - d(0,6)*a on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,6)*a$ Unknowns: {d(1,6),a} Unknowns: {d(1,6),a} pas de selection possible de variable a coefficient numerique dans - d(1,6)*a on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,6)*a$ Unknowns: {d(2,6),a} Unknowns: {d(2,6),a} pas de selection possible de variable a coefficient numerique dans - d(2,6)*a on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,6)*a$ Unknowns: {d(3,6),a} Unknowns: {d(3,6),a} pas de selection possible de variable a coefficient numerique dans - d(3,6)*a on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,6)*a$ Unknowns: {d(4,6),a} Unknowns: {d(4,6),a} pas de selection possible de variable a coefficient numerique dans - d(4,6)*a on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,6)*a + d(2,3)$ Unknowns: {d(5,6),d(2,3),a} Unknowns: {d(5,6),d(2,3),a} bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=d(5,6)*a$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,6)*a + d(4,3) + d( 3,3)*a + d(0,0)*a$ Unknowns: {d(6,6),d(4,3),d(3,3),d(0,0),a} Unknowns: {d(6,6),d(4,3),d(3,3),d(0,0),a} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=a*(d(6,6) - d(3,3) - d(0,0))$ on resout l'equation {{0,4},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 {{0,4},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 {{0,4},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 {{0,4},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 {{0,4},4} qui est maintenant AA:= - d(4,6)$ Unknown: d(4,6) Unknown: d(4,6) bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},5} qui est maintenant AA:= - d(5,6) + d(2,4) + d(1, 0)$ Unknowns: {d(5,6),d(2,4),d(1,0)} Unknowns: {d(5,6),d(2,4),d(1,0)} bonne inconnue W:=d(5,6)$ sa valeur doit etre WW:=d(2,4) + d(1,0)$ on resout l'equation {{0,4},6} qui est maintenant AA:= - d(6,6) + d(4,4) + d(3, 4)*a + d(0,0)$ Unknowns: {d(6,6),d(4,4),d(3,4),d(0,0),a} Unknowns: {d(6,6),d(4,4),d(3,4),d(0,0),a} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(4,4) + d(3,4)*a + d(0,0)$ on resout l'equation {{0,5},6} 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 {{1,2},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 {{1,2},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 {{1,2},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 {{1,2},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 {{1,2},4} qui est maintenant AA:= - d(4,4) + d(2,2) + d(1, 1)$ Unknowns: {d(4,4),d(2,2),d(1,1)} Unknowns: {d(4,4),d(2,2),d(1,1)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) + d(4,2) + d(0, 1)$ Unknowns: {d(5,4),d(4,2),d(0,1)} Unknowns: {d(5,4),d(4,2),d(0,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) + d(0,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:= - d(6,4) + d(5,2) - d(4, 1) - d(3,1)$ Unknowns: {d(6,4),d(5,2),d(4,1),d(3,1)} Unknowns: {d(6,4),d(5,2),d(4,1),d(3,1)} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:=d(5,2) - d(4,1) - d(3,1)$ on resout l'equation {{1,3},5} qui est maintenant AA:=a*( - d(3,3) + d(2,2) + d (1,1))$ Unknowns: {d(3,3),d(2,2),d(1,1),a} Unknowns: {d(3,3),d(2,2),d(1,1),a} pas de selection possible de variable a coefficient numerique dans a*( - d(3,3) + d(2,2) + d(1,1)) on resout l'equation {{1,3},6} qui est maintenant AA:=d(5,3) + d(2,1) + d(0,1)* a$ Unknowns: {d(5,3),d(2,1),d(0,1),a} Unknowns: {d(5,3),d(2,1),d(0,1),a} bonne inconnue W:=d(5,3)$ sa valeur doit etre WW:= - (d(2,1) + d(0,1)*a)$ on resout l'equation {{1,4},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)/2$ on resout l'equation {{1,4},6} qui est maintenant AA:= - d(3,2)*a + d(3,0) + 2* d(2,1) + 2*d(0,1)$ Unknowns: {d(3,2),d(3,0),d(2,1),d(0,1),a} Unknowns: {d(3,2),d(3,0),d(2,1),d(0,1),a} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:=(d(3,2)*a - d(3,0) - 2*d(0,1))/2$ on resout l'equation {{2,3},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 {{2,3},5} 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 {{2,3},6} qui est maintenant AA:=( - 2*d(3,3)*a + 2*d(3,3) + 2*d(2,2)*a + 2*d(0,2)*a + d(0,0)*a - 3*d(0,0))/2$ Unknowns: {d(3,3),d(2,2),d(0,2),d(0,0),a} Unknowns: {d(3,3),d(2,2),d(0,2),d(0,0),a} pas de selection possible de variable a coefficient numerique dans ( - 2*d(3,3)* a + 2*d(3,3) + 2*d(2,2)*a + 2*d(0,2)*a + d(0,0)*a - 3*d(0,0))/2 on resout l'equation {{2,4},5} 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 {{2,4},6} qui est maintenant AA:=d(2,2) + d(0,2) - d(0,0)$ 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:= - d(0,2) + d(0,0)$ Derivation equations to cancel (Reduce output) : \\{{{{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},0},0}, {{{0,4},1},0}, {{{0,4},2},0}, {{{0,4},3},0}, {{{0,4},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},5},0}, {{{0,5},6},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},4},0}, {{{1,3},5}, ( - (2*d(0,2) - 3*d(0,0) + 2*d(3,3))*a)/2}, {{{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},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},( - (2*d(3,3) - 3*d(0,0))*(a - 1))/2}, {{{2,4},0},0}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},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},5},0}, {{{3,4},6},0}, {{{3,5},6},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 y a une phase 2$ a neq {0,1}$ collect_eq:={{{{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},0},0}, {{{0,4},1},0}, {{{0,4},2},0}, {{{0,4},3},0}, {{{0,4},4},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},5},0}, {{{0,5},6},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},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},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},0},0}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},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},5},0}, {{{3,4},6},0}, {{{3,5},6},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),d(0,1),0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0,( - (d(3,0) + 2*d(0,1) - d(3,2)*a))/2,d(0,0),0,0,0,0),(d(3,0),d(3,1),d(3,2),(3*d(0,0))/2,0,0,0),(( - (d( 3,0) + 2*d(0,1) - d(3,2)*a))/2,d(4,1),d(4,2),0,(3*d(0,0))/2,0,0),(d(4,1) + d(3,1 )*a,d(5,1),d(5,2),( - (2*(a - 1)*d(0,1) - d(3,0) + d(3,2)*a))/2,d(4,2) + d(0,1), 2*d(0,0),0),(d(6,0),d(6,1),d(6,2),d(6,3), - (d(4,1) + d(3,1) - d(5,2)),( - (d(3, 0) - 2*d(0,1) - d(3,2)*a - 2*d(4,2)))/2,(5*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 0 0 0 0 0] [ ] [0 1 0 0 0 0] [ ] [0 0 a 1 0 0] pour shortformdelta:={0,ss,0,0,ss,1,ss,a} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(3,0), d(0,1), d(0,0), a} Unknowns: {d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(3,0), d(0,1), d(0,0), a} listeparametresMATD{d(6,3), d(6,2), d(6,1), d(6,0), d(5,2), d(5,1), d(4,2), d(4,1), d(3,2), d(3,1), d(3,0), d(0,1), d(0,0)}$ dim Der(gtildedelta):=13$ t1:=D(0,0):= [1 0 0 0 0 0 0 ] [ ] [ 1 ] [0 --- 0 0 0 0 0 ] [ 2 ] [ ] [0 0 1 0 0 0 0 ] [ ] [ 3 ] [0 0 0 --- 0 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 5 ] [0 0 0 0 0 0 ---] [ 2 ] MATD:= mat((d(0,0),d(0,1),0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 - (d(3,0) + 2*d(0,1) - d(3,2)*a) (0,-----------------------------------,d(0,0),0,0,0,0), 2 3*d(0,0) (d(3,0),d(3,1),d(3,2),----------,0,0,0), 2 - (d(3,0) + 2*d(0,1) - d(3,2)*a) 3*d(0,0) (-----------------------------------,d(4,1),d(4,2),0,----------,0,0), 2 2 - (2*(a - 1)*d(0,1) - d(3,0) + d(3,2)*a) (d(4,1) + d(3,1)*a,d(5,1),d(5,2),------------------------------------------- 2 ,d(4,2) + d(0,1),2*d(0,0),0), (d(6,0),d(6,1),d(6,2),d(6,3), - (d(4,1) + d(3,1) - d(5,2)), - (d(3,0) - 2*d(0,1) - d(3,2)*a - 2*d(4,2)) 5*d(0,0) ----------------------------------------------,----------)) 2 2 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 ] [ 2 ] [ ] [ 0 0 d(0,0) 0 0 0 0 ] [ ] [ 3*d(0,0) ] [ 0 0 0 ---------- 0 0 0 ] [ 2 ] [ ] [ 3*d(0,0) ] [ 0 0 0 0 ---------- 0 0 ] [ 2 ] [ ] [ 0 0 0 0 0 2*d(0,0) 0 ] [ ] [ 5*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 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 ] [ 2 ] [ ] [0 0 1 0 0 0 0 ] [ ] [ 3 ] [0 0 0 --- 0 0 0 ] [ 2 ] [ ] [ 3 ] [0 0 0 0 --- 0 0 ] [ 2 ] [ ] [0 0 0 0 0 2 0 ] [ ] [ 5 ] [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),d(0,1),0,0,0,0,0),(0,d(0,0)/2,0,0,0,0,0),(0, - d(0,1),d(0,0),0,0,0,0 ),(d(3,0),d(3,1),d(3,2),(3*d(0,0))/2,0,0,0),( - d(0,1),d(4,1),d(4,2),0,(3*d(0,0) )/2,0,0),(d(4,1) + d(3,1)*a,d(5,1),d(5,2), - (a - 1)*d(0,1),d(4,2) + d(0,1),2*d( 0,0),0),(d(6,0),d(6,1),d(6,2),d(6,3), - (d(4,1) + d(3,1) - d(5,2)),d(4,2) + d(0, 1),(5*d(0,0))/2))$ 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),d(0,1),0,0,0,0,0), d(0,0) (0,--------,0,0,0,0,0), 2 (0, - d(0,1),d(0,0),0,0,0,0), 3*d(0,0) (d(3,0),d(3,1),d(3,2),----------,0,0,0), 2 3*d(0,0) ( - d(0,1),d(4,1),d(4,2),0,----------,0,0), 2 (d(4,1) + d(3,1)*a,d(5,1),d(5,2), - (a - 1)*d(0,1),d(4,2) + d(0,1),2*d(0,0), 0), (d(6,0),d(6,1),d(6,2),d(6,3), - (d(4,1) + d(3,1) - d(5,2)),d(4,2) + d(0,1), 5*d(0,0) ----------)) 2 on voit apparaitre les poids sur la diagonale r(1) := d(0,0) d(0,0) r(2) := -------- 2 r(3) := d(0,0) 3*d(0,0) r(4) := ---------- 2 3*d(0,0) r(5) := ---------- 2 r(6) := 2*d(0,0) 5*d(0,0) r(7) := ---------- 2 r(1) := 2*gamma1 r(2) := gamma1 r(3) := 2*gamma1 r(4) := 3*gamma1 r(5) := 3*gamma1 r(6) := 4*gamma1 r(7) := 5*gamma1 Le systeme de poids est le systeme 1.3 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},0}, {{0,2},x(5)}, {{0,3},a*x(6)}, {{0,4},x(6)}, {{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(6)}, {{2,4},x(6)}, {{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) 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},0}, {{1,3},diay(6)}, {{1,4},diay(7)*a}, {{1,5},diay(7)}, {{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(7)}, {{3,5},diay(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}}$ Now we make explicit the isomorphism with an algebra of the book:$ Namely g_{7,1.3}$ (v)$ pour a neq{0,1}$ 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 :$ This isom computed by calculisom6_12VI.red$ mat((0,( - 1)/(a - 1),-1,0,0,0,0),(1,0,0,0,0,0,0),(0,a/(a - 1),0,0,0,0,0),(0,0,0 ,0,1/(a - 1),0,0),(0,0,0,a/(a - 1),0,0,0),(0,0,0,0,0,a/(a - 1),0),(0,0,0,0,b(7,5 ),0,a/(a - 1)))$ det(isom):= a**4/(a - 1)**5$ ZZ(1):=diay(2)$ ZZ(2):=(diay(3)*a - diay(1))/(a - 1)$ ZZ(3):= - diay(1)$ ZZ(4):=(diay(5)*a)/(a - 1)$ ZZ(5):=((a - 1)*b(7,5)*diay(7) + diay(4))/(a - 1)$ ZZ(6):=(diay(6)*a)/(a - 1)$ ZZ(7):=(diay(7)*a)/(a - 1)$ 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(6)}$ {{2,4},zz(7)}$ {{2,5},0}$ {{2,6},0}$ {{2,7},0}$ {{3,4}, - zz(7)}$ {{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_{7,1.3}$ (v)$ Et cela pour a:=a.$ Et cela pour a different de {0,1}.$ shortformdelta:={0,ss,0,0,ss,1,ss,a}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,a ,1,0,0))$