%This is "program3_3.red"   (formerly "structcomplItex.tex") :
%computes the complex structures on the Lie algebra G_{6,3}
%in the case  xi(6,1) =  0 , xi(2,5)=0.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  OUTPUT   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off echo$     off nat$
 OUT  "rprogram3_3.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%CAS ou xi(6,1) =  0 , xi(2,5)=0 et xi(3,4) neq 0.
xi(1,6):=0$
xi(2,5):=0$

%DIM:=(dimension de l'algebre)$
DIM:= 6$
  in "6nilp/6nilp.3"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ECRITURE DES RELATIONS DE COMMUTATION EN TEX
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
WRITE "This output from the file \texttt{structcomplItex.tex}"$
WRITE "Computation of all complex    structures on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
WRITE "\par Case USD xi(1,6) = 0, xi(2,5) = 0.USD"$
write "USD \\ xi(1,6):=",xi(1,6),"USD"$
write "USD \\ xi(2,5):=",xi(2,5),"USD"$
WRITE "\smallskip  \par "$
WRITE "Commutation relations for"$
write "USD {\mathcal{G}}_{", dim,",", PART(REFALGTEX,1), "}:USD\\"$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=X(i)*x(j)$
%FOR j:=1:DIM DO X(j):=MKID(x_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j) NEQ 0 THEN WRITE
 %       "USD[x_",i,",x_",j,"]=", X_(i,j),"USD;"$
        "USD[x(",i,"),x(",j,")]=", X_(i,j),"USD;"$
WRITE "\P"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 operator x,f,A$
 noncom x,f$
 for all j let
  F(X(j))=FOR j1:=1:DIM SUM xi(j1,j)*X(j1)$
  let f(0)=0$
 for all j let f(-x(j))=-f(x(j))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(U*X(j))=U*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(U*V*X(j))=U*V*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(-U*V*X(j))=-U*V*f(X(j))$
 FOR ALL s,i,j,k,l
 LET f(X(s) *xi(i,j)*xi(k,l))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(xi(i,j)*xi(k,l)*X(s))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-X(s) *xi(i,j)*xi(k,l))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j))=(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j))=-(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j)**2)=(-xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j)**2)=(xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-xi(i,j)*xi(k,l)*X(s))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(-U*X(j))=-U*f(X(j))$
 FOR ALL A,B LET F(A+B)=F(A)+F(B)$
 %LIGNE A MODIFIER POUR LA LINEARITE SUR LE PARAMETRE CONTINU L
 for all j let f(L*x(j))=L*f(x(j))$
 for all j let f(-L*x(j))=-L*f(x(j))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%  collecting the torsion equations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 for all j,k let
  temp110(j,k) = x(j)*F(x(k))$
 for all j,k let
  temp120(j,k) = F(temp110(j,k))$
 for all j,k let
  temp101(j,k) = F(x(j))*x(k)$
 for all j,k let
  temp102(j,k) = F(temp101(j,k))$
 for all j,k let
  temp2(j,k) =    F(x(j))* F(x(k)) $
 for all j,k let  temp0(j,k) =  x(j)*x(k)$
 for all j,k let
  A(j,k)=  -temp2(j,k) + temp0(j,k) + temp120(j,k) + temp102(j,k);
COLLECT_TORSION:=FOR j1:=1:DIM-1 JOIN
   FOR j2:=j1+1:DIM JOIN
   IF A(j1,j2) NEQ 0 THEN
   {{{j1,j2},A(j1,j2)}}
   ELSE {}$
%IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion
 %  dans les cas suivants",COLLECT_TORSION

IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion"
   ELSE WRITE "Zero torsion"$
write "\par"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%la liste des équations non nulles
%write "list of the nonzero torsion equations"$
COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_TORSION) JOIN
   FOR j2:=1:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_TORSION,j1),2) NEQ 0 THEN
   {{{PART(PART(COLLECT_TORSION,j1),1),j2}, V(j2)*PART(PART(COLLECT_TORSION,j1),2)} }
    ELSE {}$

COMMENT
 WRITE "Torsion 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"$
%%%%%%%%%%%%%%%%%%%% computing the complex structures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I USD in the case USD xi(1,6) = 0. USD"$
write "\\ One first gets"$
write "\\ from equation USD24|1USD :"$
%24|1
xi(1,4):=0$
write "\\ USD xi(1,4):=", xi(1,4)$

write "USD\\ and from equation USD26|3USD :"$
xi(3,6):=0$
write"\\USD xi(3,6):=", xi(3,6)$

write "USD\\ and from equation USD35|1USD :"$
xi(1,5):=0$
write"\\USD xi(1,5):=",xi(1,5)$

write "USD\\ and from equation USD36|2USD :"$
%36|2
xi(2,6):=0$
write"\\USD xi(2,6):=",xi(2,6),"USD"$

write "\\ and from equation USD14|2USD :"$
%14|2
xi(2,4):=0$
write"\\USD xi(2,4):=",xi(2,4),"USD"$

write "\\ and from equation USD15|3USD :"$
%15|3
xi(3,5):=0$
write"\\USD xi(3,5):=",xi(3,5),"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then, one gets from  equation USD13|3USD, as USD xi(3,4) \neq 0 USD :"$
% 13|3
% est : xi(3,4)*xi(2,3)) = 0
% premier cas : xi(3,4) NEQ  0
xi(2,3):=0$
write"\\USD xi(2,3):=",xi(2,3),"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ and from equation USD12|3USD :"$
% 12|3
xi(2,2):=-xi(1,1)$
write"\\USD xi(2,2):=",xi(2,2),"USD"$
%write "xi(2,2):=",xi(2,2)$

write "\\ and from equation USD14|4USD :"$
% 14|4
xi(4,5):=0$
%write "xi(4,5):=",xi(4,5)$
write"\\USD xi(4,5):=",xi(4,5),"USD"$

write "\\ and from equation USD14|5USD :"$
% 14|5
xi(5,5):=xi(1,1)$
%write "xi(5,5):=",xi(5,5)$
write"\\USD xi(5,5):=",xi(5,5),"USD"$

write "\\ and from equation USD14|6USD :"$
% 14|6
xi(6,5):=xi(2,1)$
%write "xi(6,5):=",xi(6,5)$
write"\\USD xi(6,5):=",xi(6,5),"USD"$


write "\\ and from equation USD24|4USD :"$
% 24|4
xi(4,6):=0$
%write "xi(4,6):=",xi(4,6)$
write"\\USD xi(4,6):=",xi(4,6),"USD"$

write "\\ and from equation USD24|5USD :"$
% 24|5
xi(5,6):=xi(1,2)$
%write "xi(5,6):=",xi(5,6)$
write"\\USD xi(5,6):=",xi(5,6),"USD"$

write "\\ and from equation USD24|6USD :"$
% 24|6
xi(6,6):=xi(2,2)$
%write "xi(6,6):=",xi(6,6)$
write"\\USD xi(6,6):=",xi(6,6),"USD"$

write "\\ and from equation USD34|5USD :"$
% 34|5
xi(1,3):=0$
%write "xi(1,3):=",xi(1,3)$
write"\\USD xi(1,3):=",xi(1,3),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par With these values, \textit{Reduce} computes again all equations."$
write "Then the only nonzero integrability equation left "$
write "USD xi(1,2)*xi(2,1) + xi(1,1)**2 +1 = 0 USD"$
write "makes sense only if USD xi(1,2) \neq 0 ,USD"$
write "and then gives USD xi(2,1) :USD"$
xi(2,1) := -(xi(1,1)**2 +1)/xi(1,2)$
write"\\USD xi(2,1):=",xi(2,1),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "The USD 3 \times 3 USD term in  USD J^2 USD is: "$
%terme 3x3 de J**2
%xi(3,4)*xi(4,3) + xi(3,3)**2
write "USD xi(3,4)*xi(4,3) + xi(3,3)**2. USD"$
%Donc xi(3,4) = 0 est impossible
write "This makes sense only if USD xi(3,4) \neq 0 ,USD"$
write "and then gives USD xi(4,3) :USD"$
xi(4,3) := -(xi(3,3)**2 +1)/xi(3,4)$
%write "xi(4,3):=",xi(4,3)$
write"\\USD xi(4,3):=",xi(4,3),"USD"$

write "\\ From the USD 3 \times 4 USD term in  USD J^2 USD one gets: "$
%terme 3x4 de J**2 donne
xi(4,4) := -xi(3,3)$
%write "xi(4,4):=",xi(4,4)$
write"\\USD xi(4,4):=",xi(4,4),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 3 \times 2 USD term in  USD J^2 USD one gets : "$
%terme 3x2 de J**2
%xi(4,2)*xi(3,4)
xi(4,2):=
-(+ xi(3,3)*xi(3,2) - xi(3,2)*xi(1,1) + xi(3,1)*xi(1,2))/xi(3,4)$
%write "xi(4,2):=",xi(4,2)$
write"\\USD xi(4,2):=",xi(4,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 3 \times 1 USD term in  USD J^2 USD one gets : "$
%terme 3x1
xi(3,2):=
(xi(4,1)*xi(3,4)*xi(1,2) + xi(3,3)*xi(3,
1)*xi(1,2)
%- xi(3,2)*xi(1,1)**2 - xi(3,2)
+ xi(3,1)*xi(1,2)*xi(1,1)
)/(1+xi(1,1)**2)$
%)/xi(1,2)
%write "xi(3,2):=",xi(3,2)$
write"\\USD xi(3,2):=",xi(3,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 5 \times 4 USD term in  USD J^2 USD one gets : "$

%terme 5x4 de J**2 donne xi(5,3)
%xi(6,4)*xi(1,2) - xi(5,4)*xi(3,3) + xi(5,4)*xi(1,1) + xi(5,3)*xi(3,4),
xi(5,3):=-(xi(6,4)*xi(1,2) - xi(5,4)*xi(3,3) + xi(5,4)*xi(1,1))/xi(3,4)$
%write "xi(5,3):=",xi(5,3)$
write"\\USD xi(5,3):=",xi(5,3),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 5 \times 3 USD term in  USD J^2 USD one gets : "$
%terme 5x3 de J**2 donne xi(5,4)
xi(5,4):=
( - xi(6,4)*xi(3,3)*xi(1,2) - xi(6,4)*
xi(1,2)*xi(1,1) + xi(6,3)*xi(3,4)*xi(1,2)
                             %- xi(5,4)*xi(1,1)**2
                           %- xi(5,4)
)/(1+xi(1,1)**2)$
%)/xi(3,4)
%write "xi(5,4):=",xi(5,4)$
write"\\USD xi(5,4):=",xi(5,4),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 5 \times 2 USD term in  USD J^2 USD one gets : "$
%terme 5x2 donne xi(5,1)
  xi(5,1):=-
( - xi(6,4)*xi(4,1)*xi(1,2) -
 xi(6,3)*xi(3,1)*xi(1,2) + xi(6,2)*xi(1,1)**2 + xi(6,2)
            %+ xi(5,1)*xi(1,1)**2 + xi(5,1))
 )/(xi(1,1)**2 + 1)$
 %)/(xi(1,1)**2 + 1),0,0,-1,0),
%write "xi(5,1):=",xi(5,1)$
write"\\USD xi(5,1):=",xi(5,1),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 5 \times 1 USD term in  USD J^2 USD one gets : "$
%terme 5x1 de J**2
xi(5,2):=
( - xi(6,4)*
xi(4,1)*xi(3,4)*xi(3,3)*xi(1,2)**2 + xi(6,4)*xi(4,1)*xi(3,4)*xi(1,2)**2*
xi(1,1) - xi(6,4)*xi(3,3)**2*xi(3,1)*xi(1,2)**2 - xi(6,4)*xi(3,1)*xi(1,2
)**2 + xi(6,3)*xi(4,1)*xi(3,4)**2*xi(1,2)**2 + xi(6,3)*xi(3,4)*xi(3,3)*
xi(3,1)*xi(1,2)**2 + xi(6,3)*xi(3,4)*xi(3,1)*xi(1,2)**2*xi(1,1) - 2*xi(6
,2)*xi(3,4)*xi(1,2)*xi(1,1)**3 - 2*xi(6,2)*xi(3,4)*xi(1,2)*xi(1,1) + xi(
6,1)*xi(3,4)*xi(1,2)**2*xi(1,1)**2 + xi(6,1)*xi(3,4)*xi(1,2)**2
%- xi(5,2)*xi(3,4)*xi(1,1)**4 - 2*xi(5,2)*xi(3,4)*xi(1,1)**2 - xi(5,2)*xi(3,4)
)/(xi(3,4)*((xi(1,1)**2 + 1)**2))$
%write "xi(5,2):=",xi(5,2)$
write"\\USD xi(5,2):=",xi(5,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

WRITE " \par Now the nonzero torsion equations left are :"$

COMMENT
write COLLECT_EQ$

%Latex output
 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 "USD"$
write "\\ \P \\"$
%%%%%%%
%La matrice MATJ de J :
write "\par The matrix USD J USD is :\\"$

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

 %WRITE "Matrice de J:=",MATJ$

 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$

 write "USDUSD J^2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\end{pmatrix}USDUSD" >>$

 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$

%%%%%%%%%%%%%% writing the specific file containing the reduce data for J %%%%%%%%%%%%%
out "matJ3_3.red"$
WRITE "% This file contains the REDUCE data for the general complex structures"$
write "% on the Lie algebra g_{", dim, ",", PART(REFALGTEX,1), "} in the case xi(6,1) =  0 , xi(2,5)=0."$
WRITE "matrix J(6,6)"$
%WRITE "J:=",MATJ$
WRITE "% Writing the entries of J:"$
WRITE "% J(i,k) denotes the entry in the ith row , kth column"$
WRITE "% i.e. stands for  the  LaTex expression J^i_k"$
WRITE "% Similarly, xi(i,k) stands for the  LaTex expression \xi^i_k"$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "J(",i,",",j,"):=",MATJ(i,j)>>$


 %FOR i:=1:DIM DO FOR j:=1:DIM DO
 %<<WRITE MATJ(i,j):=MATJ(i,j)>>$

 WRITE "%where the parameters are subject to the following condition"$
 write "%USDxi(1,2)*xi(3,4)\neq 0. USD"$

 write "END"$
 shut "matJ3_3.red"$
 out "rprogram3_3.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 write "\par Now we'll use equivalence by automorphisms"$
 write "of the form"$
matrix PHI(6,6)$
for i:=1:6 do <<PHI(i,i):=1>>$
for i:=4:6 do for j:=1:3 do <<PHI(i,j):=b(i,j)>>$
FOR i:=1:DIM join for j:=1:dim collect b(i,j)$
%FOR each A in ws do <<FACTOR A>>$
write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write PHI(i,j), "&">>
                         else    <<write PHI(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write PHI(i,j), "&">>
                         else    <<write PHI(i,j), "\end{pmatrix}USDUSD" >>$

% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  \Phi^",i,"_",j,"=",PHI(i,j),";USD\\">>$

%write "PHI:=",PHI$

write "USD det \Phi:=",det(PHI),"USD"$
write "\\ USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
J2:=PHI**(-1)*MATJ*PHI$

for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$

write "From these formulas, one has that"$
write "USD xi(1,2),xi(3,4) USD are invariant under the action of USD \Phi USD"$
write "USD  J2^1_2  = ", J2(1,2), "J2^3_4 =",J2(3,4),"USD\\"$

write "Take the following values"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\USD J2(3,1):=b(4,1)*xi(3,4) + xi(3,1)USD\\$
               b(4,1):=-(xi(3,4))**(-1)*  xi(3,1)$
write "\\ USD b(4,1):=",b(4,1),"USD"$
write "in order to get USD J2^3_1=0 USD and "$

%\\USD J2(3,3):=b(4,3)*xi(3,4) + xi(3,3)USD\\$
b(4,3):=-xi(3,3)/xi(3,4)$
write "\\ USD b(4,3):=",b(4,3),"USD"$
write "in order to get USD J2^3_3=0 USD and "$

%\\USD J2(6,4):= - b(6,3)*xi(3,4) + xi(6,4)USD\\$
b(6,3):=xi(6,4)/xi(3,4)$
write "\\ USD b(6,3):=",b(6,3),"USD"$
write "in order to get USD J2^6_4=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%USD  J2^4_1=(b(4,2)*xi(3,4)*xi(1,1)**2 + b(4,2)*xi(3,4) + xi(4,1)*xi(3,4)*xi(1,2
%) + xi(3,3)*xi(3,1)*xi(1,2) + xi(3,1)*xi(1,2)*xi(1,1))/(xi(3,4)*xi(1,2));USD\\$

b(4,2):= -(xi(4,1)*xi(3,4)*xi(1,2
) + xi(3,3)*xi(3,1)*xi(1,2) + xi(3,1)*xi(1,2)*xi(1,1))
/(xi(3,4)*(xi(1,1)**2 +1))$


write "\\ USD b(4,2):=",b(4,2),"USD"$
write "in order to get USD J2^4_1=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%USD  J2^6_1=(b(6,2)*xi(3,4)*xi(1,1)**2 + b(6,2)*xi(3,4) - 2*b(6,1)*xi(3,4)*xi(1,
%2)*xi(1,1) - b(5,1)*xi(3,4)*xi(1,1)**2 - b(5,1)*xi(3,4) - xi(6,4)*xi(3,1)*xi(1,2
%) + xi(6,1)*xi(3,4)*xi(1,2))/(xi(3,4)*xi(1,2));USD\\$

b(6,2):=
-(- 2*b(6,1)*xi(3,4)*xi(1,
2)*xi(1,1) - b(5,1)*xi(3,4)*xi(1,1)**2 - b(5,1)*xi(3,4) - xi(6,4)*xi(3,1)*xi(1,2
) + xi(6,1)*xi(3,4)*xi(1,2))/(xi(3,4)*(xi(1,1)**2 +1))$
write "\\ USD b(6,2):=",b(6,2),"USD"$
write "in order to get USD J2^6_1=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%USD  J2^6_2=( - b(6,1)*xi(3,4)*xi(1,2)**2*xi(1,1)**2 - b(6,1)*xi(3,4)*xi(1,2)**2
% - b(5,2)*xi(3,4)*xi(1,1)**4 - 2*b(5,2)*xi(3,4)*xi(1,1)**2 - b(5,2)*xi(3,4) - xi
%(6,4)*xi(4,1)*xi(3,4)*xi(1,2)**2 - xi(6,4)*xi(3,3)*xi(3,1)*xi(1,2)**2 - xi(6,4)*
%xi(3,1)*xi(1,2)**2*xi(1,1) + xi(6,2)*xi(3,4)*xi(1,2)*xi(1,1)**2 + xi(6,2)*xi(3,4
%)*xi(1,2))/(xi(3,4)*xi(1,2)*(xi(1,1)**2 + 1));USD\\$

b(6,1):=
 (- b(5,2)*xi(3,4)*xi(1,1)**4 - 2*b(5,2)*xi(3,4)*xi(1,1)**2 - b(5,2)*xi(3,4) - xi
(6,4)*xi(4,1)*xi(3,4)*xi(1,2)**2 - xi(6,4)*xi(3,3)*xi(3,1)*xi(1,2)**2 - xi(6,4)*
xi(3,1)*xi(1,2)**2*xi(1,1) + xi(6,2)*xi(3,4)*xi(1,2)*xi(1,1)**2 + xi(6,2)*xi(3,4
)*xi(1,2))/(xi(3,4)*xi(1,2)**2*(xi(1,1)**2 + 1))$
write "\\ USD b(6,1):=",b(6,1),"USD"$
write "in order to get USD J2^6_2=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%USD  J2^6_3=( - b(5,3)*xi(3,4)*xi(1,1)**2 - b(5,3)*xi(3,4) - xi(6,4)*xi(3,3)*xi(
%1,2) - xi(6,4)*xi(1,2)*xi(1,1) + xi(6,3)*xi(3,4)*xi(1,2))/(xi(3,4)*xi(1,2))
%;USD\\$

b(5,3):=
(- xi(6,4)*xi(3,3)*xi(
1,2) - xi(6,4)*xi(1,2)*xi(1,1) + xi(6,3)*xi(3,4)*xi(1,2))
/(xi(3,4)*(xi(1,1)**2 +1))$
write "\\ USD b(5,3):=",b(5,3),"USD"$
write "in order to get USD J2^6_3=0 .USD  "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ The matrix USD \Phi USD is with these values"$
write "\\"$

 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  \Phi^",i,"_",j,"=",PHI(i,j),";USD\\">>$

write "Then one gets for USDJ2USD \\"$

 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J2^",i,"_",j,"=",J2(i,j),";USD\\">>$


write "\\"$
 write "USDUSD J2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write J2(i,j), "&">>
                         else    <<write J2(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write J2(i,j), "&">>
                         else    <<write J2(i,j), "\end{pmatrix}USDUSD" >>$


write "Hence we are in the case where"$
write "USD xi(3,1)=xi(3,3)=xi(4,1)=xi(6,1)=xi(6,2)=xi(6,3)=xi(6,4)=  0 USD: \\"$
 xi(3,1):=0$
 xi(3,3):=0$
 xi(4,1):=0$
 xi(6,1):=0$
 xi(6,2):=0$
 xi(6,3):=0$
 xi(6,4):=0$

 %FOR i:=1:DIM DO FOR j:=1:DIM DO
 %<<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

write "\\"$
 write "USDUSD J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$


 clear b(5,1),b(5,3),b(4,2),b(5,2),b(6,3)$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 write "\par Now, to get USD J2(3,4) = 1, USDwe'll use equivalence by the automorphism"$
matrix psi1(6,6)$
psi1(1,1):=1$
psi1(2,2):=1$
psi1(3,3):=xi(3,4)$
psi1(4,4):=1$
psi1(5,5):=xi(3,4)$
psi1(6,6):=xi(3,4)$
write "\\"$
 write "USDUSD \Psi_1 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write psi1(i,j), "&">>
                         else    <<write psi1(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write psi1(i,j), "&">>
                         else    <<write psi1(i,j), "\end{pmatrix}USDUSD" >>$


J2:=psi1**(-1)*MATJ*psi1$

write "Then USD J2 = \Psi_1^{-1}J\Psi_1 USD is the matrix"$
write "\\"$
 write "USDUSD J2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write J2(i,j), "&">>
                         else    <<write J2(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write J2(i,j), "&">>
                         else    <<write J2(i,j), "\end{pmatrix}USDUSD" >>$

 write "\par Then, to get USD J2(1,2) = 1, USD we'll use equivalence by the automorphism"$
matrix psi2(6,6)$
psi2(1,1):=xi(1,2)$
psi2(2,2):=1$
psi2(3,3):=xi(1,2)$
psi2(4,4):=xi(1,2)$
psi2(5,5):=xi(1,2)**2$
psi2(6,6):=xi(1,2)$

write "\\"$
 write "USDUSD \Psi_2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write psi2(i,j), "&">>
                         else    <<write psi2(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write psi2(i,j), "&">>
                         else    <<write psi2(i,j), "\end{pmatrix}USDUSD" >>$

% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  \Psi^",i,"_",j,"=",psi(i,j),";USD\\">>$

%write "USD det \Psi:=",det(psi),"USD"$

NJ2:=psi2**(-1)*J2*psi2$

write "Then USD NJ2 = \Psi_2^{-1} J2 \Psi_2 USD is the matrix"$
write "\\"$
 write "USDUSD NJ2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write NJ2(i,j), "&">>
                         else    <<write NJ2(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write NJ2(i,j), "&">>
                         else    <<write NJ2(i,j), "\end{pmatrix}USDUSD" >>$


 write "\par Finally, to get USD J2(1,1) = 0, USD"$
 write "we'll use equivalence by the automorphism"$
matrix psi3(6,6)$
psi3(1,1):=1$
psi3(2,1):=-xi(1,1)$
psi3(2,2):=1$
psi3(3,3):=psi3(1,1)*psi3(2,2)-psi3(1,2)*psi3(2,1)$
psi3(4,4):=psi3(3,3)$
psi3(5,5):=psi3(3,3)*psi3(1,1)$
psi3(5,6):=psi3(3,3)*psi3(1,2)$
psi3(6,5):=psi3(3,3)*psi3(2,1)$
psi3(6,6):=psi3(3,3)*psi3(2,2)$

write "\\"$
 write "USDUSD \Psi_3 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write psi3(i,j), "&">>
                         else    <<write psi3(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write psi3(i,j), "&">>
                         else    <<write psi3(i,j), "\end{pmatrix}USDUSD" >>$

% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  \Psi^",i,"_",j,"=",psi(i,j),";USD\\">>$

%write "USD det \Psi:=",det(psi),"USD"$

NNJ2:=psi3**(-1)*NJ2*psi3$

write "Then USD NNJ2 = \Psi_3^{-1} NJ2 \Psi_3 USD is the matrix"$
write "\\"$
 write "USDUSD NNJ2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write NNJ2(i,j), "&">>
                         else    <<write NNJ2(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write NNJ2(i,j), "&">>
                         else    <<write NNJ2(i,j), "\end{pmatrix}USDUSD" >>$

COMMENT
write "\\ USD NJ2:=\psi^{-1}*J2*\psi USD has entries :"$


%for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$


 write "\par Introduce the automorphism USD \Psi = \Psi_1 \Psi_2 \Psi_3 :USD"$

psi:=psi1*psi2*psi3$

write "\\"$
 write "USDUSD \Psi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write psi(i,j), "&">>
                         else    <<write psi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write psi(i,j), "&">>
                         else    <<write psi(i,j), "\end{pmatrix}USDUSD" >>$

write "\\"$
write "USD det \Psi:=",det(psi),"USD"$

J2:=psi**(-1)*MATJ*psi$

write "Then USD J2 = \Psi^{-1}J\Psi USD is the matrix"$
write "\\"$
 write "USDUSD J2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write J2(i,j), "&">>
                         else    <<write J2(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write J2(i,j), "&">>
                         else    <<write J2(i,j), "\end{pmatrix}USDUSD" >>$



write "We've thus got into the case of a complex structure USD J USD where"$
write "USD xi(3,1)=xi(3,3)=xi(4,1)=xi(6,1)=xi(6,2)=xi(6,3)=xi(6,4)=  0 USD: \\"$
write "xi(1,1) = 0, USD and USD  xi(1,2) = xi(3,4) = 1 .USD "$

xi(1,1):=0$
xi(1,2):=1$
xi(3,4):=1$

write "\\"$
 write "USDUSD J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$



write "Hence, we have that"$
write "any integrable complex structure with USD xi(1,6) = 0 , xi(2,5) = 0 USD"$
write "is equivalent to the structure defined by "$

 write "USDUSD J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Recall the matrix USDJ_1 USD"$
matrix J_1(6,6)$
J_1(1,6):=1$
J_1(2,3):=1$
J_1(3,2):=-1$
J_1(4,5):=1$
J_1(5,4):=-1$
J_1(6,1):=-1$

 write "USDUSD J_1 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write J_1(i,j), "&">>
                         else    <<write J_1(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write J_1(i,j), "&">>
                         else    <<write J_1(i,j), "\end{pmatrix}USDUSD" >>$

write "Observe now that USD J USD is equivalent to USD J_1 USD by the automorphism"$
matrix M(6,6)$
M(1,2):=1$
M(2,3):=1$
M(3,1):=1$
M(4,6):=1$
M(5,4):=-1$
M(6,5):=-1$

 write "USDUSD M = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write M(i,j), "&">>
                         else    <<write M(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write M(i,j), "&">>
                         else    <<write M(i,j), "\end{pmatrix}USDUSD" >>$

write "\textit{i.e. } USD M^{-1}J M = J_1 USD, since"$
write "%USD M J_1 - J M =", M*J_1 - MATJ*M $
write "USD M J_1 - J M =0.USD"$
write "\par Hence, any integrable complex structure with "$
write "USD xi(1,6) =0, xi(2,5) \neq 0 USD"$
write "is equivalent to the structure defined by USD J_1 USD"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% write "\end{document}"$
% bye;
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ check of torsion"$

 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"$

%il faut enlever les quantites nulles de COLLECT_TORSION
%pour avoir la longueur veritable
COLLECT_TORSION:=FOR j1:=1:DIM-1 JOIN
   FOR j2:=j1+1:DIM JOIN
   IF A(j1,j2) NEQ 0 THEN
   {{{j1,j2},A(j1,j2)}}
   ELSE {}$

IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion"
   ELSE WRITE "Zero torsion"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "USD[\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),"USD;"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let B(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM JOIN
   FOR j2:=1:DIM JOIN
   IF B(j1,j2) NEQ 0 THEN
   {{{j1,j2},B(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 write "\end{document}"$
 bye;
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%