off echo,nat$
out "rcalculderivgtildedeltaXV.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),ksi(1,2),0,0,0,0),(0,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),0,ksi(2,2) + ksi(1,1),0,0),(ksi(5,1),ksi(5,2
),ksi(5,3),ksi(4,2) - ksi(3,1),ksi(2,2) + 2*ksi(1,1),ksi(1,2)),(ksi(6,1),ksi(6,2
),ksi(6,3), - ksi(4,1),0,2*ksi(2,2) + ksi(1,1)))$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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(5,2):=0$
%reduction par adjoints
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%initial conditions for specific delta
xi(1,2):=0$
xi(3,1):=0$
xi(3,2):=1$
xi(6,1):=a$
xi(6,2):=0$
xi(6,3):=0$
xi(5,3):=1$
a:=0$
write "delta:=",delta$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% in the commut relations file of g
% extended to the commut. relations of $\tilde{g}_delta
%by the commutation relations involving X(0)=delta
% and the projector V(0) on X(0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
in "6nilp/6nilp.8"$
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 resoud 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)$


 WRITE "Derivation equations to cancel (Reduce output) : \\", COLLECT_EQ$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "phase 2"$
write "pas de phase 2"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
write "collect_eq:=",collect_eq$

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

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

ws:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$
WS:=LENGTH(UNKNOWNSINLIST(WS))$
%WS:=LENGTH(UNKNOWNSINLIST(WS))-1$
% lorsqu il y a xi(5,1)=a  qui est considere comme unknown
write "dim Der(gtildedelta):=",WS$
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%derivation generique de gtildedelta:
%MATD:=
COMMENT
mat((d(0,0),0,d(0,2),0,0,0,0),

    (0,d(0,0),d(1,2),0,0,0,0),

    (0,0,d(0,0) - d(0,2),0,0,0,0),

    (d(3,0),d(3,1),d(3,2),2*d(0,0) - d(0,2),0,0,0),

    (d(3,1),d(4,1),d(4,2),0,2*d(0,0) - d(0,2),0,0),

    (d(5,0),d(5,1),d(5,2), - d(3,0) + d(3,2), - d(3,1) + d(4,2),

     3*d(0,0) - d(0,2),d(1,2)),

    (d(6,0),d(6,1),d(6,2), - d(3,1), - d(4,1),0,3*d(0,0) - 2*d(0,2))) $


%pour delta:=

%[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  1  0  0  0]
%[                ]
%[0  0  0  0  0  0]
%dim Der(gtildedelta):=14$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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 "t1:=D(0,0)"$
matrix t1(7,7)$
t1(1,1):=1$
t1(2,2):=1$
t1(3,3):=1$
t1(4,4):=2$
t1(5,5):=2$
t1(6,6):=3$
t1(7,7):=3$
t1(7,4):=-a$
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 "on peut prendre comme element semi simple du commutant de t1 dans der(gtildedelta):"$
%t2:= MATD;
%bye$
%%
%%off nat$


matrix t2(7,7)$
t2(1,3):=1$
for i:=3:6 do <<t2(i,i):=-1>>$
t2(7,7):=-2$

write "t2:=D(0,2)",t2$

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

%
%write "le calcul du commutant de t1 et t2 dans der(gtildedelta):"$
%write "commutant de t1 et t2 dans der(gtildedelta):"$
%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)$
%t3(1,3):=-1$
%t3(3,3):=1$
%t3(4,2):=-1$
%t3(4,4):=1$
%t3(5,1):=-1$
%t3(5,3):=-1$
%t3(5,5):=1$
%t3(6,1):=1/2$
%t3(6,3):=1/2$
%t3(6,6):=1$
%t3(7,7):=2$

%write "t3:=-D(0,2)+(D(5,0)+D(5,2))/2:=",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."$

%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$

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

write "P:=",P$
write "P**(-1)*t1*P:=", P**(-1)*t1*P;
write "P**(-1)*t2*P:=", P**(-1)*t2*P;
%rite "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 un peu seulement, il faut mettre Y(3) en 1, Y(1) en 3, Y(7) en 6 et Y(6) en 7 i.e. faire le changement de base de matrice de passage Q:"$
%matrix Q(7,7)$
%Q(3,1):=1$
%Q(2,2):=1$
%Q(1,3):=1$
%Q(4,4):=1$
%Q(5,5):=1$
%Q(7,6):=1$
%Q(6,7):=1$

 Q:=ID$  %si pas besoin de deuxieme diagonalisation
%%%%%%%%%%%%%%%% si besoin de la seconde base diagY trigonalisant les MATD passer a PP
%%%%%%%%%%%%%%%%% sinon rester a P
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(3)$
ZZ(2):=diaY(2)$
ZZ(3):=diaY(1)$
ZZ(4):=-diaY(5)$
ZZ(5):=-diaY(4)$
ZZ(6):=-diaY(7)$
ZZ(7):=-diaY(6)$
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(1):=ZZZ(3)$
diadiaY(2):=ZZZ(2)$
diadiaY(3):=ZZZ(1)$
diadiaY(4):=-ZZZ(5)$
diadiaY(5):=-ZZZ(4)$
diadiaY(6):=-ZZZ(7)$
diadiaY(7):=-ZZZ(6)$


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 "C'est les relations de commutations de g_{7,2.12} page 511."$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
WRITE "Le passage aux Z au lieu des diagY induit le changement sur les poids suivant:"$
beta1:=(gamma1+gamma2)/2$
beta2:=(gamma1-gamma2)/2$
write "beta1-beta2:=",beta1-beta2$
write "beta1+beta2:=",beta1+beta2$
write "beta1:=",beta1$
write "beta2:=",beta2$
write "les poids sont pour les Z:"$
ladiag:=for i:=1:7 collect {i,MATDDIAGONALISE(i,i)};
bye$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%