off echo,nat$
out "rcalculderivgtildedelta6_7IX4.r"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  introduction of the generic derivation delta of the 6-dimensional algebra g
%as computed by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for all i,j let ksi(i,j)=xi(i,j)$


matrix delta(6,6)$

delta:=
mat((ksi(1,1),0,0,0,0,0),(ksi(2,1),ksi(2,2),0,0,0,0),(ksi(3,1),ksi(3,2),2*ksi(1,
1),0,0,0),(ksi(4,1),ksi(4,2),ksi(4,3),ksi(2,2) + ksi(1,1),0,0),(ksi(5,1),ksi(5,2
),ksi(5,3),ksi(3,2),3*ksi(1,1),0),(ksi(6,1),ksi(6,2),ksi(6,3),ksi(4,2) + ksi(3,1
),ksi(4,3) - ksi(2,1),ksi(2,2) + 2*ksi(1,1)))$
shortformdelta:={xi(2,1),SS,xi(3,1),xi(3,2),SS,xi(4,3),SS,xi(5,2),xi(5,3),SS,xi(6,2),xi(6,3)}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% restriction on the parameters for nilpotent derivation and mod(ad g).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(1,1):=0$
xi(2,2):=0$
%impose par nilpotence de delta
xi(4,2):=0$
xi(4,1):=0$
xi(5,1):=0$
xi(6,1):=0$
%reduction par adjoints
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%initial conditions for specific delta
xi(2,1):=1$

xi(3,1):=0$
xi(3,2):=0$

xi(4,3):=a$
a:=1$

xi(5,2):=0$
xi(5,3):=0$

xi(6,2):=1$
xi(6,3):=0$
write "delta:=",delta$
write "shortformdelta:=",shortformdelta$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% in "the commut relations file of g"
% extended to the commut. relations of $\tilde{g}_delta
%by the commutation relations involving X(0)=delta
% and the projector V(0) on X(0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
in "6nilp/6nilp.7"$
for j:=0:6 do x(0)*x(j):= if j neq 0 then  for k:=1:6 sum delta(k,j)*x(k) else 0
for j:=1:6 do x(j)*x(0):= - x(0)*x(j)$

FOR j:=0:DIM DO
       V(0)*x(j):= IF j=0 THEN 1 ELSE 0;
FOR i:=1:DIM DO  V(i)*x(0):=0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%MATRIXD will denote the generic derivation of the 7-dimensional {\tilde{g}}_{delta}
% Let MATRIXD=matrix(D(i,j)). Collecting the equations {i,j}   1\leqslant i < i \leqslant 6
%              -D [X(i),X(j)] + [DX(i), X(j)] + [X(i), DX(j)].
MATRIX MATRIXD(7,7)$
for i:=1:7 DO for j:=1:7  DO MATRIXD(i,j):=D(i-1,j-1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
COLLECT_DERIVGTILDEDELTA:=
for i:=0:DIM-1 JOIN for j:=i+1:DIM JOIN
     { {{i,j},
            -(WS:=for k:=0:DIM sum
            (WS:=V(k)*(WS:=X(i)*X(j))) * (for m:=0:DIM sum D(m,k)*X(m))
            )
            + (WS:=for k:=0:DIM sum D(k,i)*(WS:=X(k)*X(j)))
            + (WS:= for k:=0:DIM sum D(k,j)*(WS:=X(i)*X(k)))
            }}$
%COLLECT_DERIVGTILDEDELTA;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Collecting the nonzero equations {i,j}|k obtained by projecting
%on X(k) the derivation equation -D [X(i),X(j)] + [DX(i), X(j)] + [X(i), DX(j)].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$

%la liste des équations non nulles
%write "list of the nonzero derivation equations"$

COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_DERIVGTILDEDELTA) JOIN
   FOR j2:=0:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_DERIVGTILDEDELTA,j1),2) NEQ 0 THEN
         {{{PART(PART(COLLECT_DERIVGTILDEDELTA,j1),1),j2},
                  V(j2)*PART(PART(COLLECT_DERIVGTILDEDELTA,j1),2)} }
   ELSE  {}$

COMMENT
 WRITE "Derivation equations to cancel (Reduce output) : \\", COLLECT_EQ$

% WRITE "Torsion equations to cancel (Latex output) : \\USD"$
% for each A in COLLECT_EQ do
%        if PART(A,2) neq 0 then
%        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
%        %<<write "\\", A,"\\">>   $
% write "USD"$


 off factor$
 on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE OTEZERO(U)$  %enleve les 0 dans une liste
BEGIN$
LIST U$
RETURN FOR EACH A IN U JOIN IF A NEQ 0 THEN {A} ELSE {}$
END$


PROCEDURE distinct(U)$
BEGIN;INTEGER j;LIST UU$ j:=1;UU:={}$
S: IF U NEQ {} THEN    ZZZ:=PART(u,j)$
IF ZZZ MEMBER UU THEN <<GO TO P>> ELSE UU:= ZZZ. UU $
P: IF j<LENGTH(U) THEN <<j:=j+1,GO TO S>> ELSE <<RETURN REVERSE(UU)>>$
CLEAR ZZZ$
END$

PROCEDURE UNKNOWNSINLIST(U)$
BEGIN$ LIST U$
RETURN  IF U={} THEN U ELSE
        IF PART(PART(SOLVE(U),1),0) = LIST
          THEN FOR EACH A IN PART(SOLVE(U),1) COLLECT LHS A
        ELSE {LHS PART(SOLVE(U),1)} $
END$


PROCEDURE UNKNOWNSINEXPRESSION(U)$
BEGIN$
RETURN  IF NUMBERP U THEN {}  ELSE
        IF PART(PART(SOLVE(U),1),0) = LIST
          THEN FOR EACH A IN PART(SOLVE(U),1) COLLECT LHS A
        ELSE {LHS PART(SOLVE(U),1)} $
END$


PROCEDURE SSOLVE(n,p)$
BEGIN INTEGER k,j$
 k:=1$ j:=1$
S1: IF z(k,j) NEQ 0 THEN <<GO TO S2>> ELSE <<GO TO S3>>$
S2: CALLLET(z(k,j),0) $
S3: IF j<p THEN j:=j+1 ELSE IF k<n  THEN <<k:=k+1$ j:=1>> ELSE GO TO
 S4$
GO TO S1$
S4: END$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SELECTVAR(U)$
BEGIN$
integer i,j$
i:=6$ j:=6$
LI:=UNKNOWNSINEXPRESSION(U)$
S: IF D(i,j) MEMBER LI AND NUMBERP DF(U,D(i,j))
   THEN <<RETURN D(i,j), GO TO F>>
   ELSE
        IF j>0 THEN <<j:=j-1, GO TO S>>
        ELSE IF i>0 THEN <<i:=i-1,j:=6, GO TO S>>
             ELSE <<write "pas de selection possible de variable a coefficient independant des xi dans ", U, return 0>>$
F :
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
procedure calllet(U,V)$
let U=V$


PROCEDURE  resol(L)$
begin  integer k$
k:=1$
S:
  AA:=part(part(L,k),2)$
  If AA=0 then <<go to F>>$
  write "on resout l'equation " , PART(PART(L,k),1)," qui est  maintenant AA:=",AA$
  W:= SELECTVAR(AA)$
  if W=0 then <<go to F>>$
  write  "bonne inconnue W:=",W$
  WW:= RHS(part(solve(AA=0,W),1))$
  write "sa valeur doit etre WW:=",WW$
  calllet(W,WW)$
  F:
if k < length(L) then <<k:=k+1, go to S>>$
     clear AA,W,WW$
END$


write "phase 1 de la resolution des equations"$


resol(COLLECT_EQ)$



on factor$
 WRITE "Derivation equations to cancel (Reduce output) : \\", COLLECT_EQ$
 %bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "Il y a une phase 2"$
write "il n'y a pas de phase 2"$
on factor$
%write "collect_eq:=",collect_eq$
% bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


matrix MATD(7,7)$
for i:=1:7 do for j:=1:7 do MATD(i,j):=D(i-1,j-1)$

off  nat$
write "derivation generique de gtildedelta:"$
write "MATD:=",MATD$
write "pour delta:=",delta$
write "pour shortformdelta:=",shortformdelta$
off nat$


ws:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$
% mise des elements matriciels de MATD en liste pour calcul des parametres
%WS:=LENGTH(UNKNOWNSINLIST(WS))$
WS:=IF a MEMBER UNKNOWNSINLIST(WS) THEN LENGTH(UNKNOWNSINLIST(WS))-1 ELSE
                                                 WS:=LENGTH(UNKNOWNSINLIST(WS))$
% lorsqu il y a  a  qui est considere comme unknown
write "dim Der(gtildedelta):=",WS$
% bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%derivation generique de gtildedelta:$
%partie devenue sans objet
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$
listparametresMATDincluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$
listparametresMATD:=
FOR EACH U IN listparametresMATDincluantparamalgebre_a join if U neq a then {U} ELSE {}$
write "listeparametresMATD",WS$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
LISTparametrestemoinMATD:=
for each h in listparametresMATD collect dd(part(h,1),part(h,2))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   ecriture du premier membre du tore
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ecrit le tore t1=D(i,j)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE CALCUL_D(i,j)$
BEGIN$
integer i,j$

WS:=for each U in listparametresMATD  JOIN IF U neq d(i,j) and U neq a then {U} else {}$
for each U in WS DO <<calllet(U,0)>>$

d(i,j):=1$
off nat$
WS:=MATD;

for each U in listparametrestemoinMATD do <<clear d(part(U,1),part(U,2))>>$
RETURN WS$

END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%introduction de la matrice ID identite
matrix ID(7,7)$
for i:=1:7 do for j:=1:7 do <<ID(i,j):=if i=j then 1 else 0>>$

write "un element t1 d'un tore "$
%write "seul candidat a etre un element t1 d'un tore "$

write "t1:=D(0,0)"$
matrix t1(7,7)$
t1:=calcul_D(0,0)$

on nat$
write "t1:=", t1$

%%
%off nat$
%on factor$
%det(t1-x*ID);
%off factor$
%mateigen(t1,x);
%bye$

%%
%write "commutant de t1 dans der(gtildedelta):"$
%Z:=MATD*t1-t1*MATD$
%SSOLVE(7,7)$
%write "commutant de t1 :=",MATD;
%bye$


%write "Mais  t1 n'est pas semisimple "$
%write "gtildedelta est caracteristiquement nilpotente"$
%for j:=1:7 do <<write "MATD**",j,":=",MATD**j>>$

%write "on peut prendre comme element semi simple du commutant de t1 dans der(gtildedelta):"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   ecriture du second membre du tore
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%WS:=for each U in listparametresMATD  JOIN IF U neq d(1,1) and U neq a then {U} else {}$
%for each U in WS DO <<calllet(U,0)>>$
%d(1,1):=1$
%off nat$
%t2:=matd;
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on nat$
%matrix t2(7,7)$



t2:=calcul_D(1,1)$
on nat$
write "t2:=D(1,1)",t2$

%load_package normform$
%Jordansymbolic(t2);
%bye$

%%off nat$
%on factor$
%det(t2-x*ID);
%off factor$
%mateigen(t2,x);
%bye$

%
%write "le calcul du commutant simultane de t1 et t2 dans der(gtildedelta):"$
%write "commutant de t1 et t2 dans der(gtildedelta):"$
%Z:=MATD*t1-t1*MATD$
%SSOLVE(7,7)$
%Z:=MATD*t2-t2*MATD$
%SSOLVE(7,7)$
%write "commutant de t1 et t2:=",MATD;
%bye$


%write "le calcul du commutant de t1 et t2 dans der(gtildedelta) permet de prendre comme element semi-simple dans le commutant:"$
%matrix t3(7,7)$

%write "t3:=",t3$

%%off nat$
%on factor$
%det(t3-x*ID);
%off factor$
%mateigen(t3,x);
%bye$

%
%write "commutant de t1 et t2 et t3 dans der(gtildedelta):"$
%Z:=MATD*t3-t3*MATD$
%SSOLVE(7,7)$
%write "commutant de t1,t2,t3:=",MATD$

%write "le calcul du commutant de t1 et t2 et t3 dans der(gtildedelta) montre  que t1,t2,t3 est un tore maximal."$
write " t1,t2 est un tore maximal."$
%write " t1 est un tore maximal."$
%write "seul tore maximal: {0}."$

%bye$

write "matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base  diagonalisant le tore maximal: on peut prendre"$


matrix P(7,7)$
%P:=ID$

for i:=1:7 do <<P(i,i):=1>>$
P(4,1):=-1$
P(6,3):=1$
on nat$

write "P:=",P$
write "P**(-1)*t1*P:=", P**(-1)*t1*P;
write "P**(-1)*t2*P:=", P**(-1)*t2*P;
%write "P**(-1)*t3*P:=", P**(-1)*t3*P;

write "matrice des derivations dans cette base diagonalisante Y(1),...,Y(7):"$
write "P**(-1)*MATD*P:=",P**(-1)*MATD*P$
%bye$

%write " A la lecture de cette matrice, on constate qu'elle n'est pas triangulaire; pour l'arranger, il faut faire le changement de base de matrice de passage Q:"$

matrix Q(7,7)$

Q:=ID$  %si pas besoin de deuxieme diagonalisation
%for i:=5:7 do <<Q(i,i):=1>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PP:=P*Q$
%write "matrice des derivations dans cette seconde base diagonalisante diaY(1),...,diaY(7):"$
write "PP**(-1)*MATD*PP:=",PP**(-1)*MATD*PP$

MATDDIAGONALISE:=PP**(-1)*MATD*PP$
%MATDDIAGONALISE:=P**(-1)*MATD*P$
write "avec PP:=P*Q:=",PP$
write "MATDDIAGONALISE:=",MATDDIAGONALISE$

write "on voit apparaitre les poids sur la diagonale"$
ladiag:=for i:=1:7 collect {i,MATDDIAGONALISE(i,i)};
%bye$

write "calcul de relations de commutation de la base diaY(j) diagonalisant le tore"$

listcommutateursdesX:=for i:=0:5 join for j:=i+1:6 collect {{i,j},X(i)*X(j)};
%bye

for j:=1:7 do diaY(j):=for k:=1:7 sum PP(k,j)*X(k-1)$
%for j:=1:7 do diaY(j):=for k:=1:7 sum P(k,j)*X(k-1)$
for j:=1:7 do write "diaY(",j,"):=",diaY(j)$

listcommutateursdesdiaY:=for i:=1:6 join for j:=i+1:7 collect {{i,j},diaY(i)*diaY(j)}$
%passage a Y rebaptise YY    Y(j):=X(j-1)
%(decalage des indices)
for j:=1:6 do X(j):=YY(j+1)$
%(X(0) n'intervient pas comme commutateur
%passage aux diaY
QQ:=PP**(-1)$
%Q:=P**(-1)$
For j:=1:7 do YY(j):=for k:=1:7 sum QQ(k,j)*diadiaY(k)$
%For j:=1:7 do YY(j):=for k:=1:7 sum Q(k,j)*diadiaY(k)$
write "liste des commutateurs des diaY(i) := (diadiaY=diaY"$
listcommutateursdesdiaY;
for j:=1:6 do <<clear X(j)>>$
%  bye$


for j:=1:7 do <<clear diaY(j)>>$

write "on pose :"$

ZZ(1):=diaY(2)$
ZZ(2):=diaY(1)$
ZZ(3):=-diaY(4)/a$
ZZ(4):=-diaY(3)$
ZZ(5):=-diaY(6)/a$
ZZ(6):=-diaY(5)$
ZZ(7):=-diaY(7)$
For i:=1:7 do <<write "ZZ(",i,"):=",ZZ(i)>>$
for j:=1:7 do diaY(j):=for k:=1:7 sum PP(k,j)*X(k-1)$
%for j:=1:7 do diaY(j):=for k:=1:7 sum P(k,j)*X(k-1)$


%formules d'inversion
diadiaY(3):=-ZZZ(4)$
diadiaY(4):=-a*ZZZ(3)$
diadiaY(5):=-ZZZ(6)$
diadiaY(6):=-a*ZZZ(5)$
diadiaY(7):=-ZZZ(7)$


listcommutateursdesZZ:=for i:=1:6 join for j:=i+1:7 collect {{i,j},ZZ(i)*ZZ(j)}$
for j:=0:6 do X(j):=YY(j+1)$
%For j:=1:7 do YY(j):=for k:=1:7 sum QQ(k,j)*diadiaY(k)$
For j:=1:7 do YY(j):=for k:=1:7 sum Q(k,j)*diadiaY(k)$
write "liste des commutateurs des ZZ(i) (ZZZ=ZZ:="$
listcommutateursdesZZ;


write "On obtient donc   les relations de commutation de g_{7,2.1(i_L)} pour L=0 page 419."$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%