%This is "program9.red" (formerly "structcompl6_8.tex") : %computes the complex structures on the Lie algebra G_{6,8} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% OUTPUT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off echo$ off nat$ OUT "rprogram9.tex"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ON RAT$ OFF MSG$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% Loading the commutation relations file %%%%%%%%%%%%%% %DIM:=(dimension de l'algebre)$ DIM:= 6$ in "6nilp/6nilp.8"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %ECRITURE DES RELATIONS DE COMMUTATION EN TEX WRITE "\documentclass{article}"$ WRITE "\usepackage{amsmath,amssymb}"$ write "\setlength{\textwidth}{15cm}"$ write "\setlength{\textheight}{25cm}"$ write "\setlength{\voffset}{-2cm}" $ write " \setlength{\oddsidemargin}{0in}"$ write " \setlength{\evensidemargin}{0.36in}"$ WRITE "\sloppy"$ WRITE "\begin{document}"$ %WRITE "This output from the file \texttt{structcompl6\_8.tex}.\\"$ WRITE "This output from the file \texttt{program9.red}.\\"$ WRITE "Computation of all complex structures on the real Lie Algebra"$ write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.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"$ %FOR j:=1:DIM DO <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 % in the following cases :",COLLECT_TORSION IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion" ELSE WRITE "Zero torsion"$ write "\par"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off exp$ on factor$ %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 "USD"$ %write "We don't write down again the torsion equations (see case 1)."$ off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% computing the complex structures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %write "*****************************************************************"$ write "\\ \starline"$ write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$ write "equation USD J^2 = -I . USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "First, one gets from equation USD35|2USD :"$ xi(2,5):=0$ write "\\ USD xi(2,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ Then one gets from equation USD45|1USD :"$ xi(1,5):=0$ write "\\ USD xi(1,5):=", ws,"USD"$ write "\\ and from equation USD46|2USD :"$ xi(2,6):=0$ write "\\ USD xi(2,6):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ Then one gets from equation USD14|2USD :"$ xi(2,4):=0$ write "\\ USD xi(2,4):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ Then one gets from equation USD15|4USD :"$ xi(4,5):=0$ write "\\ USD xi(4,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ Then one gets from equation USD25|3USD :"$ xi(3,5):=0$ write "\\ USD xi(3,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 5x5 entry in USD J**2 USD is "$ write "USD {J**2}^5_5=xi(5,5)**2 + xi(6,5)*xi(5,6);USD\\"$ write "Hence USD xi(6,5)xi(5,6)\neq 0 USD"$ write "and USD xi(5,6) =(-1-xi(5,5)**2)/xi(6,5). USD"$ xi(5,6) :=(-1-xi(5,5)**2)/xi(6,5)$ write "\\ USD xi(5,6):=", ws,"USD"$ write "Moreover, the 6x5 entry in USD J**2 USD is "$ write "USD (xi(6,6) + xi(5,5))*xi(6,5);USD hence\\"$ xi(6,6):= -xi(5,5)$ write "\\ USD xi(6,6):=", ws,"USD"$ write "Moreover, the 4x5 entry in USD J**2 USD is "$ write "USD xi(6,5)*xi(4,6);USD hence\\"$ xi(4,6):= 0$ write "\\ USD xi(4,6):=", ws,"USD"$ write "and the 3x5 entry in USD J**2 USD is "$ write "USD xi(6,5)*xi(3,6);USD hence\\"$ xi(3,6):= 0$ write "\\ USD xi(3,6):=", ws,"USD"$ write "and the 1x5 entry in USD J**2 USD is "$ write "USD xi(6,5)*xi(1,6);USD hence\\"$ xi(1,6):= 0$ write "\\ USD xi(1,6):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ Then one gets from equation USD24|1USD :"$ xi(1,4):=0$ write "\\ USD xi(1,4):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 4x4 entry in USD J**2 USD is "$ write "USD {J**2}^4_4=xi(4,4)**2 + xi(4,3)*xi(3,4);USD\\"$ write "Hence USD xi(4,3)xi(3,4)\neq 0 USD"$ write "and USD xi(3,4) =(-1-xi(4,4)**2)/xi(4,3). USD"$ xi(3,4) :=(-1-xi(4,4)**2)/xi(4,3)$ write "\\ USD xi(3,4):=", ws,"USD"$ write "Moreover, the 3x4 entry in USD J**2 USD is "$ write "USD (xi(4,4) + xi(3,3))*xi(3,4);USD hence\\"$ xi(4,4):= -xi(3,3)$ write "\\ USD xi(4,4):=", ws,"USD"$ write "Moreover, the 1x4 entry in USD J**2 USD is "$ write "USD xi(3,4)*xi(1,3);USD hence\\"$ xi(1,3):= 0$ write "\\ USD xi(1,3):=", ws,"USD"$ write "\\ Then one gets from equation USD13|3USD :"$ xi(2,3):=0$ write "\\ USD xi(2,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 1x1 entry in USD J**2 USD is "$ write "USD {J**2}^1_1=xi(1,1)**2 + xi(2,1)*xi(1,2);USD\\"$ write "Hence USD xi(2,1)xi(1,2)\neq 0 USD"$ write "and USD xi(1,2) =(-1-xi(1,1)**2)/xi(2,1). USD"$ xi(1,2) :=(-1-xi(1,1)**2)/xi(2,1)$ write "\\ USD xi(1,2):=", ws,"USD"$ write "Moreover, the 1x2 entry in USD J**2 USD is "$ write "USD (xi(2,2) + xi(1,1))*xi(1,2);USD hence\\"$ xi(2,2):= -xi(1,1)$ write "\\ USD xi(2,2):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Then the 4x1 entry in USD J**2 USD gives "$ %USD {J**2}^4_1=xi(4,3)*xi(3,1) + xi(4,2)*xi(2,1) - xi(4,1)*xi(3,3) + xi(4,1)*xi %(1,1);USD\\$ xi(4,2):=-( xi(4,3)*xi(3,1) %+ xi(4,2)*xi(2,1) - xi(4,1)*xi(3,3) + xi(4,1)*xi(1,1) )/xi(2,1) $ write "\\ USD xi(4,2):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 6x1 entry in USD J**2 USD gives "$ %USD {J**2}^6_1=xi(6,5)*xi(5,1) + xi(6,4)*xi(4,1) + xi(6,3)*xi(3,1) + xi(6,2)*xi %(2,1) - xi(6,1)*xi(5,5) + xi(6,1)*xi(1,1);USD\\$ xi(5,1):=-( %xi(6,5)*xi(5,1) + xi(6,4)*xi(4,1) + xi(6,3)*xi(3,1) + xi(6,2)*xi (2,1) - xi(6,1)*xi(5,5) + xi(6,1)*xi(1,1) )/xi(6,5)$ write "\\ USD xi(5,1):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Then one gets from equation USD14|6USD :"$ %{1,4}|6\\ - xi(6,5)*xi(3,3) + xi(6,5)*xi(1,1) - xi(5,5)*xi(2,1) + xi(3,3)*xi(2,1 %)\\$ xi(1,1):= -( - xi(6,5)*xi(3,3) %+ xi(6,5)*xi(1,1) - xi(5,5)*xi(2,1) + xi(3,3)*xi(2,1))/xi(6,5)$ write "\\ USD xi(1,1):=", ws,"USD"$ *%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, equation USD13|6USD reads :"$ write " USD{1,3}|6\\xi(6,5)*xi(4,3) + xi(6,5)*xi(2,1) - xi(4,3)*xi(2,1) =0 USD\\"$ write "Hence USD xi(4,3) +xi(2,1) \neq 0USD and "$ %{1,3}|6\\xi(6,5)*xi(4,3) + xi(6,5)*xi(2,1) - xi(4,3)*xi(2,1)\\$ xi(6,5):= xi(4,3)*xi(2,1)/(xi(4,3)+xi(2,1))$ write "\\ USD xi(6,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Then the 4x2 entry in USD J**2 USD gives "$ %USD {J**2}^4_2=(xi(5,5)*xi(4,3)*xi(3,1) + xi(5,5)*xi(3,1)*xi(2,1) + xi(4,3)*xi( %3,3)*xi(3,1) + xi(4,3)*xi(3,2)*xi(2,1) - xi(4,1)*xi(3,3)**2 - xi(4,1) - xi(3,3)* %xi(3,1)*xi(2,1))/xi(2,1);USD\\$ xi(4,1):=(xi(5,5)*xi(4,3)*xi(3,1) + xi(5,5)*xi(3,1)*xi(2,1) + xi(4,3)*xi( 3,3)*xi(3,1) + xi(4,3)*xi(3,2)*xi(2,1) %- xi(4,1)*xi(3,3)**2 - xi(4,1) - xi(3,3)*xi(3,1)*xi(2,1))/(xi(3,3)**2+1)$ write "\\ USD xi(4,1):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Then the 6x3 entry in USD J**2 USD gives "$ %USD {J**2}^6_3=(xi(6,4)*xi(4,3)**2 + xi(6,4)*xi(4,3)*xi(2,1) - xi(6,3)*xi(5,5)* %xi(4,3) - xi(6,3)*xi(5,5)*xi(2,1) + xi(6,3)*xi(4,3)*xi(3,3) + xi(6,3)*xi(3,3)*xi %(2,1) + xi(5,3)*xi(4,3)*xi(2,1))/(xi(4,3) + xi(2,1));USD\\$ xi(5,3):=- (xi(6,4)*xi(4,3)**2 + xi(6,4)*xi(4,3)*xi(2,1) - xi(6,3)*xi(5,5)* xi(4,3) - xi(6,3)*xi(5,5)*xi(2,1) + xi(6,3)*xi(4,3)*xi(3,3) + xi(6,3)*xi(3,3)*xi (2,1) %+ xi(5,3)*xi(4,3)*xi(2,1) )/(xi(4,3) * xi(2,1))$ write "\\ USD xi(5,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Then the 6x4 entry in USD J**2 USD gives "$ %USD {J**2}^6_4=( - xi(6,4)*xi(5,5)*xi(4,3)**2 - xi(6,4)*xi(5,5)*xi(4,3)*xi(2,1) % - xi(6,4)*xi(4,3)**2*xi(3,3) - xi(6,4)*xi(4,3)*xi(3,3)*xi(2,1) - xi(6,3)*xi(4,3 %)*xi(3,3)**2 - xi(6,3)*xi(4,3) - xi(6,3)*xi(3,3)**2*xi(2,1) - xi(6,3)*xi(2,1) + %xi(5,4)*xi(4,3)**2*xi(2,1))/(xi(4,3)*(xi(4,3) + xi(2,1)));USD\\$ xi(5,4):=- ( - xi(6,4)*xi(5,5)*xi(4,3)**2 - xi(6,4)*xi(5,5)*xi(4,3)*xi(2,1) - xi(6,4)*xi(4,3)**2*xi(3,3) - xi(6,4)*xi(4,3)*xi(3,3)*xi(2,1) - xi(6,3)*xi(4,3 )*xi(3,3)**2 - xi(6,3)*xi(4,3) - xi(6,3)*xi(3,3)**2*xi(2,1) - xi(6,3)*xi(2,1) %+xi(5,4)*xi(4,3)**2*xi(2,1) )/(xi(4,3)**2 *xi(2,1))$ write "\\ USD xi(5,4):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Then the 5x1 entry in USD J**2 USD gives "$ %USD {J**2}^5_1=( - xi(6,4)*xi(5,5)**2*xi(4,3)**4*xi(3,1) - 3*xi(6,4)*xi(5,5)**2 xi(5,2):=-( - xi(6,4)*xi(5,5)**2*xi(4,3)**4*xi(3,1) - 3*xi(6,4)*xi(5,5)**2 *xi(4,3)**3*xi(3,1)*xi(2,1) - 3*xi(6,4)*xi(5,5)**2*xi(4,3)**2*xi(3,1)*xi(2,1)**2 - xi(6,4)*xi(5,5)**2*xi(4,3)*xi(3,1)*xi(2,1)**3 - xi(6,4)*xi(5,5)*xi(4,3)**4*xi (3,2)*xi(2,1) + 2*xi(6,4)*xi(5,5)*xi(4,3)**3*xi(3,3)*xi(3,1)*xi(2,1) - 2*xi(6,4) *xi(5,5)*xi(4,3)**3*xi(3,2)*xi(2,1)**2 + 4*xi(6,4)*xi(5,5)*xi(4,3)**2*xi(3,3)*xi (3,1)*xi(2,1)**2 - xi(6,4)*xi(5,5)*xi(4,3)**2*xi(3,2)*xi(2,1)**3 + 2*xi(6,4)*xi( 5,5)*xi(4,3)*xi(3,3)*xi(3,1)*xi(2,1)**3 + xi(6,4)*xi(4,3)**4*xi(3,3)*xi(3,2)*xi( 2,1) - xi(6,4)*xi(4,3)**4*xi(3,1) + 2*xi(6,4)*xi(4,3)**3*xi(3,3)*xi(3,2)*xi(2,1) **2 - xi(6,4)*xi(4,3)**3*xi(3,1)*xi(2,1) - xi(6,4)*xi(4,3)**2*xi(3,3)**2*xi(3,1) *xi(2,1)**2 + xi(6,4)*xi(4,3)**2*xi(3,3)*xi(3,2)*xi(2,1)**3 - xi(6,4)*xi(4,3)*xi (3,3)**2*xi(3,1)*xi(2,1)**3 + xi(6,3)*xi(4,3)**3*xi(3,3)**2*xi(3,2)*xi(2,1) + xi (6,3)*xi(4,3)**3*xi(3,2)*xi(2,1) + xi(6,3)*xi(4,3)**2*xi(3,3)**2*xi(3,2)*xi(2,1) **2 + xi(6,3)*xi(4,3)**2*xi(3,2)*xi(2,1)**2 - 2*xi(6,2)*xi(5,5)*xi(4,3)**3*xi(3, 3)**2*xi(2,1) - 2*xi(6,2)*xi(5,5)*xi(4,3)**3*xi(2,1) - 3*xi(6,2)*xi(5,5)*xi(4,3) **2*xi(3,3)**2*xi(2,1)**2 - 3*xi(6,2)*xi(5,5)*xi(4,3)**2*xi(2,1)**2 - xi(6,2)*xi (5,5)*xi(4,3)*xi(3,3)**2*xi(2,1)**3 - xi(6,2)*xi(5,5)*xi(4,3)*xi(2,1)**3 + xi(6, 2)*xi(4,3)**2*xi(3,3)**3*xi(2,1)**2 + xi(6,2)*xi(4,3)**2*xi(3,3)*xi(2,1)**2 + xi (6,2)*xi(4,3)*xi(3,3)**3*xi(2,1)**3 + xi(6,2)*xi(4,3)*xi(3,3)*xi(2,1)**3 - xi(6, 1)*xi(5,5)**2*xi(4,3)**3*xi(3,3)**2 - xi(6,1)*xi(5,5)**2*xi(4,3)**3 - 3*xi(6,1)* xi(5,5)**2*xi(4,3)**2*xi(3,3)**2*xi(2,1) - 3*xi(6,1)*xi(5,5)**2*xi(4,3)**2*xi(2, 1) - 3*xi(6,1)*xi(5,5)**2*xi(4,3)*xi(3,3)**2*xi(2,1)**2 - 3*xi(6,1)*xi(5,5)**2* xi(4,3)*xi(2,1)**2 - xi(6,1)*xi(5,5)**2*xi(3,3)**2*xi(2,1)**3 - xi(6,1)*xi(5,5) **2*xi(2,1)**3 + 2*xi(6,1)*xi(5,5)*xi(4,3)**2*xi(3,3)**3*xi(2,1) + 2*xi(6,1)*xi( 5,5)*xi(4,3)**2*xi(3,3)*xi(2,1) + 4*xi(6,1)*xi(5,5)*xi(4,3)*xi(3,3)**3*xi(2,1)** 2 + 4*xi(6,1)*xi(5,5)*xi(4,3)*xi(3,3)*xi(2,1)**2 + 2*xi(6,1)*xi(5,5)*xi(3,3)**3* xi(2,1)**3 + 2*xi(6,1)*xi(5,5)*xi(3,3)*xi(2,1)**3 - xi(6,1)*xi(4,3)**3*xi(3,3)** 2 - xi(6,1)*xi(4,3)**3 - xi(6,1)*xi(4,3)**2*xi(3,3)**2*xi(2,1) - xi(6,1)*xi(4,3) **2*xi(2,1) - xi(6,1)*xi(4,3)*xi(3,3)**4*xi(2,1)**2 - xi(6,1)*xi(4,3)*xi(3,3)**2 *xi(2,1)**2 - xi(6,1)*xi(3,3)**4*xi(2,1)**3 - xi(6,1)*xi(3,3)**2*xi(2,1)**3 %+ xi(5,2)*xi(4,3)**3*xi(3,3)**2*xi(2,1)**2 + xi(5,2)*xi(4,3)**3*xi(2,1)**2 )/(xi(4,3)**3*xi(2,1)**2*(xi(3,3)**2 + 1))$ write "\\ USD xi(5,2):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %write "*****************************************************************"$ write "\\ \starline"$ write "At the present stage, we have : "$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off exp$ on factor$ %WRITE " ****Now the nonzero torsion equations left are :"$ WRITE " \\ \4stars 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 "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$ write "The matrix USD J USD is given by the following formula,"$ write "with USD xi(2,1)xi(4,3) \neq 0 USD and "$ write " USD xi(4,3)+xi(2,1) \neq 0 : USD"$ m off exp$ on factor$ FOR i:=1:DIM DO FOR j:=1:DIM DO <>$ %write "matJ:=",matj$ %on exp$ %off factor$ %%%%%%%%%%%%%%%%% MATJCARRE:=(MATJ)**2$ %WRITE "Matrice de J**2:=",(MATJ)**2$ % FOR i:=1:DIM DO FOR j:=1:DIM DO % <>$ % WRITE "\\$ det J:=",DET(MATJ)$ % WRITE "Trace J:=",TRACE(MATJ)$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off exp$ on factor$ write "\\"$ %write "{\fontsize{8}{10} \selectfont"$ write "\begin{equation} \label{68general} \end{equation}"$ write " J = \begin{pmatrix}"$ for i:=1:dim do if i neq dim then for j:=1:dim do if j neq dim then <> else <> else for j:=1:dim do if j neq dim then <> else <>$ write "\end{equation}"$ % write "}"$ %%%%%%%%%%%%%%%%% 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ WRITE "\\USD det J:=",DET(MATJ)," USD"$ WRITE "USD Trace J:=",TRACE(MATJ)," USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %write "where \\"$ %write "USD a:=xi(3,1)", xi(3,1), ",\\USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% writing the specific file containing the reduce data for J %%%%%%%%%%%%% out "matJ9.red"$ WRITE "% This file contains the REDUCE data for the general complex structures"$ write "% on the Lie algebra g_{", dim, ",", PART(REFALGTEX,1), "}"$ 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 "USD "$ for each A in COLLECT_EQ do if part(A,2) neq 0 then <>$ for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$ if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off exp$ on allfac$ WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$ FOR i:=1:DIM DO <>$ %factor x(1),x(2),x(3),x(4),x(5),x(6)$ 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 DO <>$ 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 <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 BB(j,k) = - B1(j,k) + F(x(j))*x(k)$ COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM JOIN FOR j2:=1:DIM JOIN IF BB(j1,j2) NEQ 0 THEN {{{j1,j2},BB(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"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %off exp$ %on factor$ write "\par Now we'll use equivalence by automorphisms."$ write "All automorphisms of"$ write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$ write "are of the following form :"$ matrix phi(6,6)$ for i:=1:6 do for j:=1:6 do <>$ %for i:=1:4 do for j:=5:6 do <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %b(1,1):=b(1,1)$ %b(1,2):=b(1,2)$ b(1,3):=0$ b(1,4):=0$ b(1,5):=0$ b(1,6):=0$ b(2,1):=0$ %b(2,2):=b(2,2)$ b(2,3):=0$ b(2,4):=0$ b(2,5):=0$ b(2,6):=0$ %b(3,1):=b(3,1)$ %b(3,2):=b(3,2)$ b(3,3):=b(1,1)**2$ b(3,4):=0$ b(3,5):=0$ b(3,6):=0$ %b(4,1):=b(4,1)$ %b(4,2):=b(4,2)$ b(4,3):=0$ b(4,4):=b(2,2)*b(1,1)$ b(4,5):=0$ b(4,6):=0$ %b(5,1):=b(5,1)$ %b(5,2):=b(5,2)$ %b(5,3):=b(5,3)$ b(5,4):=b(4,2)*b(1,1) - b(4,1)*b(1,2) - b(3,1)*b(2,2)$ b(5,5):=b(2,2)*b(1,1)**2$ b(5,6):=b(2,2)*b(1,2)*b(1,1)$ %b(6,1):=b(6,1)$ %b(6,2):=b(6,2)$ %b(6,3):=b(6,3)$ b(6,4):= - b(4,1)*b(2,2)$ b(6,5):=0$ b(6,6):=b(2,2)**2*b(1,1)$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "where USD \det \Phi:=b(2,2)**5 b(1,1)**7 \neq 0 USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% on exp$ off factor$ write "\textit{Reduce} here checks that USD \Phi USD is indeed an automorphism:\\"$ for all i let phi_x(i) = for j:=1:6 sum phi(j,i)*x(j)$ WS:={ {{1,2},phi_x(1)*phi_x(2)-phi_x(4)}, {{1,3},phi_x(1)*phi_x(3)}, {{1,4},phi_x(1)*phi_x(4)-phi_x(5)}, {{1,5},phi_x(1)*phi_x(5)}, {{1,6},phi_x(1)*phi_x(6)}, {{2,3},phi_x(2)*phi_x(3)-phi_x(5)}, {{2,4},phi_x(2)*phi_x(4)-phi_x(6)}, {{2,5},phi_x(2)*phi_x(5)}, {{2,6},phi_x(2)*phi_x(6)}, {{3,4},phi_x(3)*phi_x(4)}, {{3,5},phi_x(3)*phi_x(5)}, {{3,6},phi_x(3)*phi_x(6)}, {{4,5},phi_x(4)*phi_x(5)}, {{4,6},phi_x(4)*phi_x(6)}, {{5,6},phi_x(5)*phi_x(6)}} $ COLLECT_aut:=FOR each A in ws JOIN IF PART(A,2) NEQ 0 THEN {PART(A,1),PART(A,2)} ELSE {}$ IF LENGTH(COLLECT_aut) NEQ 0 THEN <> else <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %on factor$ %off exp$ off factor$ on exp$ write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has the following selected entries :"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% matrix J2(6,6)$ r1:=phi**(-1)$ r2:=r1*MATJ$ J2:=r2*phi$ %J2:=phi**(-1)*MATJ*phi$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %on factor$ %off exp$ off factor$ on exp$ write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$ for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$ write "USD det \Phi:=",det(phi),"USD"$ off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ \4stars Take the following values :"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\\USD J2(2,1):=(b(1,1)*xi(2,1))/b(2,2)USD\\$ b(2,2):=b(1,1)*xi(2,1)$ write "\\ USD b(2,2):=", ws,"USD"$ %\\USD J2(3,1):=( - b(4,1)*b(2,2)*xi(3,3)**2 - b(4,1)*b(2,2) - b(3,2)*b(1,1)*xi(4 %,3)*xi(2,1) - b(3,1)*b(2,2)*xi(5,5)*xi(4,3) - b(3,1)*b(2,2)*xi(5,5)*xi(2,1) + b( %3,1)*b(2,2)*xi(4,3)*xi(3,3) + b(3,1)*b(2,2)*xi(3,3)*xi(2,1) + b(3,1)*b(1,2)*xi(4 %,3)*xi(2,1) + b(2,2)*b(1,1)*xi(4,3)*xi(3,1))/(b(2,2)*b(1,1)**2*xi(4,3))USD\\$ b(3,2):= ( - b(4,1)*b(2,2)*xi(3,3)**2 - b(4,1)*b(2,2) %- b(3,2)*b(1,1)*xi(4,3)*xi(2,1) - b(3,1)*b(2,2)*xi(5,5)*xi(4,3) - b(3,1)*b(2,2)*xi(5,5)*xi(2,1) + b( 3,1)*b(2,2)*xi(4,3)*xi(3,3) + b(3,1)*b(2,2)*xi(3,3)*xi(2,1) + b(3,1)*b(1,2)*xi(4 ,3)*xi(2,1) + b(2,2)*b(1,1)*xi(4,3)*xi(3,1))/(b(1,1)*xi(4,3)*xi(2,1))$ write "\\ USD b(3,2):=", ws,"USD"$ %\\USD J2(5,5):=(b(2,2)*xi(5,5)*xi(4,3) + b(2,2)*xi(5,5)*xi(2,1) - b(1,2)*xi(4,3) %*xi(2,1))/(b(2,2)*(xi(4,3) + xi(2,1)))USD\\$ b(1,2):=(b(2,2)*xi(5,5)*xi(4,3) + b(2,2)*xi(5,5)*xi(2,1) %- b(1,2)*xi(4,3)*xi(2,1) )/(xi(4,3) *xi(2,1))$ write "\\ USD b(1,2):=", ws,"USD"$ %\\USD J2(6,3):=( - b(6,3)*xi(5,5)*xi(4,3) - b(6,3)*xi(5,5)*xi(2,1) - b(6,3)*xi(4 b(5,3):=-( - b(6,3)*xi(5,5)*xi(4,3) - b(6,3)*xi(5,5)*xi(2,1) - b(6,3)*xi(4 ,3)*xi(3,3) - b(6,3)*xi(3,3)*xi(2,1) %+ b(5,3)*xi(4,3)*xi(2,1) + b(4,1)*b(1,1)*xi (4,3)**2 + b(4,1)*b(1,1)*xi(4,3)*xi(2,1) + b(1,1)**2*xi(6,3)*xi(4,3) + b(1,1)**2 *xi(6,3)*xi(2,1))/(xi(4,3) * xi(2,1))$ write "\\ USD b(5,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\\USD J2(3,2):=( - b(4,2)*xi(4,3)*xi(3,3)**2 - b(4,2)*xi(4,3) - b(4,1)*xi(4,3)* %xi(3,3)**3 - b(4,1)*xi(4,3)*xi(3,3) + b(4,1)*xi(3,3)**3*xi(2,1) + b(4,1)*xi(3,3) %*xi(2,1) + b(3,1)*xi(4,3)**2*xi(3,3)**2 + b(3,1)*xi(4,3)**2 + b(1,1)*xi(5,5)*xi( %4,3)**2*xi(3,1) + b(1,1)*xi(5,5)*xi(4,3)*xi(3,1)*xi(2,1) + b(1,1)*xi(4,3)**2*xi( %3,3)*xi(3,1) + b(1,1)*xi(4,3)**2*xi(3,2)*xi(2,1) - b(1,1)*xi(4,3)*xi(3,3)*xi(3,1 %)*xi(2,1))/(b(1,1)**2*xi(4,3)**2)USD\\$ b(4,2):= ( %- b(4,2)*xi(4,3)*xi(3,3)**2 - b(4,2)*xi(4,3) - b(4,1)*xi(4,3)* xi(3,3)**3 - b(4,1)*xi(4,3)*xi(3,3) + b(4,1)*xi(3,3)**3*xi(2,1) + b(4,1)*xi(3,3) *xi(2,1) + b(3,1)*xi(4,3)**2*xi(3,3)**2 + b(3,1)*xi(4,3)**2 + b(1,1)*xi(5,5)*xi( 4,3)**2*xi(3,1) + b(1,1)*xi(5,5)*xi(4,3)*xi(3,1)*xi(2,1) + b(1,1)*xi(4,3)**2*xi( 3,3)*xi(3,1) + b(1,1)*xi(4,3)**2*xi(3,2)*xi(2,1) - b(1,1)*xi(4,3)*xi(3,3)*xi(3,1 )*xi(2,1))/((xi(3,3)**2+1)*xi(4,3))$ write "\\ USD b(4,2):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\\USD J2(6,1):=( - b(6,2)*xi(4,3)**2 - b(6,2)*xi(4,3)*xi(2,1) - b(6,1)*xi(5,5)* %xi(4,3)**2 - b(6,1)*xi(5,5)*xi(4,3)*xi(2,1) + b(6,1)*xi(4,3)*xi(3,3)*xi(2,1) + b %(6,1)*xi(3,3)*xi(2,1)**2 + b(5,1)*xi(4,3)**2*xi(2,1) + b(4,1)*xi(6,4)*xi(4,3)**2 % + b(4,1)*xi(6,4)*xi(4,3)*xi(2,1) + b(3,1)*xi(6,3)*xi(4,3)**2 + b(3,1)*xi(6,3)* %xi(4,3)*xi(2,1) + b(1,1)*xi(6,1)*xi(4,3)**2 + b(1,1)*xi(6,1)*xi(4,3)*xi(2,1))/(b %(1,1)**3*xi(4,3)*xi(2,1)**2*(xi(4,3) + xi(2,1)))USD\\$ b(5,1):=-(- b(6,2)*xi(4,3)**2 - b(6,2)*xi(4,3)*xi(2,1)- b(6,1)*xi(5,5)* xi(4,3)**2 - b(6,1)*xi(5,5)*xi(4,3)*xi(2,1) + b(6,1)*xi(4,3)*xi(3,3)*xi(2,1) + b (6,1)*xi(3,3)*xi(2,1)**2 %+ b(5,1)*xi(4,3)**2*xi(2,1) + b(4,1)*xi(6,4)*xi(4,3)**2 + b(4,1)*xi(6,4)*xi(4,3)*xi(2,1) + b(3,1)*xi(6,3)*xi(4,3)**2 + b(3,1)*xi(6,3)* xi(4,3)*xi(2,1) + b(1,1)*xi(6,1)*xi(4,3)**2 + b(1,1)*xi(6,1)*xi(4,3)*xi(2,1) )/(xi(4,3)**2*xi(2,1))$ write "\\ USD b(5,1):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$ %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 <> % else <> % else % for j:=1:dim do % if j neq dim then <> % else <>$ % % FOR i:=1:DIM DO FOR j:=1:DIM DO % <>$ %write "phi:=",phi$ off exp$ on factor$ write "USD det \Phi:=",det(phi),"USD"$ on exp$ off factor$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% matrix J2(6,6)$ r1:=phi**(-1)$ r2:=r1*MATJ$ J2:=r2*phi$ %J2:=phi**(-1)*MATJ*phi$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %on factor$ %off exp$ off factor$ on exp$ write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$ for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$ write "USD det \Phi:=",det(phi),"USD"$ off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ \4stars Hence, we are led to the case where"$ xi(2,1):=1$ write "\\ USD xi(2,1):=", ws,"USD"$ xi(3,1):=0$ write "\\ USD xi(3,1):=", ws,"USD"$ xi(3,2):=0$ write "\\ USD xi(3,2):=", ws,"USD"$ xi(5,5):=0$ write "\\ USD xi(5,5):=", ws,"USD"$ xi(6,1):=0$ write "\\ USD xi(6,1):=", ws,"USD"$ xi(6,3):=0$ write "\\ USD xi(6,3):=", ws,"USD"$ write "\\clear USD b(2,2),b(1,2),b(3,2),b(4,2),b(5,3),b(5,1)USD"$ clear b(2,2),b(1,2),b(3,2),b(4,2),b(5,3),b(5,1)$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% off exp$ on factor$ % FOR i:=1:DIM DO FOR j:=1:DIM DO % <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% off exp$ on factor$ write "\\"$ %write "{\fontsize{8}{10} \selectfont"$ 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ % write "}"$ %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% matrix J2(6,6)$ r1:=phi**(-1)$ r2:=r1*MATJ$ J2:=r2*phi$ %J2:=phi**(-1)*MATJ*phi$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %on factor$ %off exp$ off factor$ on exp$ write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$ for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$ write "USD det \Phi:=",det(phi),"USD"$ off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ \4stars Take now the following values :"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\USD J2(2,1):=b(1,1)/b(2,2)USD\\$ b(2,2):=b(1,1)$ write "\\ USD b(2,2):=", ws,"USD"$ %\USD J2(3,1):=( - b(4,1)*b(2,2)*xi(3,3)**2 - b(4,1)*b(2,2) - b(3,2)*b(1,1)*xi(4 %3) + b(3,1)*b(2,2)*xi(4,3)*xi(3,3) + b(3,1)*b(2,2)*xi(3,3) + b(3,1)*b(1,2)*xi(4 %,3))/(b(2,2)*b(1,1)**2*xi(4,3))USD\\$ b(4,1):=( %- b(4,1)*b(2,2)*xi(3,3)**2 - b(4,1)*b(2,2) - b(3,2)*b(1,1)*xi(4 ,3) + b(3,1)*b(2,2)*xi(4,3)*xi(3,3) + b(3,1)*b(2,2)*xi(3,3) + b(3,1)*b(1,2)*xi(4 ,3))/(b(2,2)*(xi(3,3)**2 +1))$ write "\\ USD b(4,1):=", ws,"USD"$ %\\USD J2(3,2):=( - b(4,2)*b(2,2)*b(1,1)*xi(4,3)*xi(3,3)**2 - b(4,2)*b(2,2)*b(1,1 b(4,2):=( %- b(4,2)*b(2,2)*b(1,1)*xi(4,3)*xi(3,3)**2 - b(4,2)*b(2,2)*b(1,1)*xi(4,3) + b(3,2)*b(2,2)*b(1,1)*xi(4,3)**2*xi(3,3) - b(3,2)*b(2,2)*b(1,1)*xi(4, 3)*xi(3,3) - b(3,2)*b(1,2)*b(1,1)*xi(4,3)**2 + b(3,1)*b(2,2)**2*xi(4,3)**2 + b(3 ,1)*b(2,2)**2*xi(3,3)**2 + 2*b(3,1)*b(2,2)*b(1,2)*xi(4,3)*xi(3,3) + b(3,1)*b(1,2 )**2*xi(4,3)**2)/(b(2,2)*b(1,1)*xi(4,3)*(xi(3,3)**2+1))$ write "\\ USD b(4,2):=", ws,"USD"$ %\\USD J2(5,5):= - (b(1,2)*xi(4,3))/(b(2,2)*(xi(4,3) + 1))USD\\$ b(1,2):=0$ write "\\ USD b(1,2):=", ws,"USD"$ %\\USD J2(6,3):=( - b(6,3)*xi(4,3)*xi(3,3) - b(6,3)*xi(3,3) b(5,3):=-( - b(6,3)*xi(4,3)*xi(3,3) - b(6,3)*xi(3,3) %+ b(5,3)*xi(4,3) + b( 4,1)*b(1,1)*xi(4,3)**2 + b(4,1)*b(1,1)*xi(4,3))/xi(4,3) $ write "\\ USD b(5,3):=", ws,"USD"$ b(3,1):=0$ write "\\ USD b(3,1):=", ws,"USD"$ %\\USD J2(6,1):=( - b(6,2)*xi(4,3)**2*xi(3,3)**2 - b(6,2)*xi(4,3)**2 - b(6,2)*xi( %4,3)*xi(3,3)**2 - b(6,2)*xi(4,3) + b(6,1)*xi(4,3)*xi(3,3)**3 + b(6,1)*xi(4,3)*xi %(3,3) + b(6,1)*xi(3,3)**3 + b(6,1)*xi(3,3) + b(5,1)*xi(4,3)**2*xi(3,3)**2 + b(5, %1)*xi(4,3)**2 - b(3,2)*xi(6,4)*xi(4,3)**3 - b(3,2)*xi(6,4)*xi(4,3)**2)/(b(1,1)** %3*xi(4,3)*(xi(4,3)*xi(3,3)**2 + xi(4,3) + xi(3,3)**2 + 1))USD\\$ b(5,1):=-( - b(6,2)*xi(4,3)**2*xi(3,3)**2 - b(6,2)*xi(4,3)**2 - b(6,2)*xi( 4,3)*xi(3,3)**2 - b(6,2)*xi(4,3) + b(6,1)*xi(4,3)*xi(3,3)**3 + b(6,1)*xi(4,3)*xi (3,3) + b(6,1)*xi(3,3)**3 + b(6,1)*xi(3,3) %+ b(5,1)*xi(4,3)**2*xi(3,3)**2 + b(5,1)*xi(4,3)**2 - b(3,2)*xi(6,4)*xi(4,3)**3 - b(3,2)*xi(6,4)*xi(4,3)**2)/ (xi(4,3)**2 *(xi(3,3)**2 + 1))$ write "\\ USD b(5,1):=", ws,"USD"$ %\\USD J2(6,2):=( - b(6,2)*xi(4,3)**2*xi(3,3)**3 - b(6,2)*xi(4,3)**2*xi(3,3) - b( b(5,2):=-( - b(6,2)*xi(4,3)**2*xi(3,3)**3 - b(6,2)*xi(4,3)**2*xi(3,3) - b( 6,2)*xi(4,3)*xi(3,3)**3 - b(6,2)*xi(4,3)*xi(3,3) + b(6,1)*xi(4,3)**3*xi(3,3)**2 + b(6,1)*xi(4,3)**3 + b(6,1)*xi(4,3)**2*xi(3,3)**2 + b(6,1)*xi(4,3)**2 + b(6,1)* xi(4,3)*xi(3,3)**4 + b(6,1)*xi(4,3)*xi(3,3)**2 + b(6,1)*xi(3,3)**4 + b(6,1)*xi(3 ,3)**2 %+ b(5,2)*xi(4,3)**3*xi(3,3)**2 + b(5,2)*xi(4,3)**3 + b(3,2)*xi(6,4)*xi(4, 3)**4*xi(3,3) - b(3,2)*xi(6,4)*xi(4,3)**2*xi(3,3) + b(1,1)*xi(6,2)*xi(4,3)**3*xi (3,3)**2 + b(1,1)*xi(6,2)*xi(4,3)**3 + b(1,1)*xi(6,2)*xi(4,3)**2*xi(3,3)**2 + b( 1,1)*xi(6,2)*xi(4,3)**2)/(xi(4,3)**3*(xi(3,3)**2 +1))$ write "\\ USD b(5,2):=", ws,"USD"$ %\\USD J2(6,4):=(b(6,3)*xi(4,3)*xi(3,3)**4 + 2*b(6,3)*xi(4,3)*xi(3,3)**2 + b(6,3) %*xi(4,3) + b(6,3)*xi(3,3)**4 + 2*b(6,3)*xi(3,3)**2 + b(6,3) + 2*b(3,2)*b(1,1)*xi %(4,3)**3*xi(3,3) + b(1,1)**2*xi(6,4)*xi(4,3)**2*xi(3,3)**2 + b(1,1)**2*xi(6,4)* %xi(4,3)**2 + b(1,1)**2*xi(6,4)*xi(4,3)*xi(3,3)**2 + b(1,1)**2*xi(6,4)*xi(4,3))/( %b(1,1)**3*xi(4,3)*(xi(4,3)*xi(3,3)**2 + xi(4,3) + xi(3,3)**2 + 1))USD\\$ b(6,3):=-( %b(6,3)*xi(4,3)*xi(3,3)**4 + 2*b(6,3)*xi(4,3)*xi(3,3)**2 + b(6,3) %*xi(4,3) + b(6,3)*xi(3,3)**4 + 2*b(6,3)*xi(3,3)**2 + b(6,3) + 2*b(3,2)*b(1,1)*xi (4,3)**3*xi(3,3) + b(1,1)**2*xi(6,4)*xi(4,3)**2*xi(3,3)**2 + b(1,1)**2*xi(6,4)* xi(4,3)**2 + b(1,1)**2*xi(6,4)*xi(4,3)*xi(3,3)**2 + b(1,1)**2*xi(6,4)*xi(4,3))/( (xi(4,3)+1)* ( xi(3,3)**2 + 1)**2)$ write "\\ USD b(6,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off exp$ on factor$ write "USD det \Phi:=",det(phi),"USD"$ on exp$ off factor$ r1:=phi**(-1)$ r2:=r1*MATJ$ J2:=r2*phi$ %J2:=phi**(-1)*MATJ*phi$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %on factor$ %off exp$ off factor$ on exp$ write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$ for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$ write "USD det \Phi:=",det(phi),"USD"$ off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ \4stars Hence, we are led to the case where moreover "$ xi(6,2):=0$ write "\\ USD xi(6,2):=", ws,"USD"$ xi(6,4):=0$ write "\\ USD xi(6,4):=", ws,"USD"$ write "\\clear USD b(2,2),b(1,2),b(3,1),b(4,1),b(4,2),b(5,2),b(5,1),b(5,3),b(6,3)USD"$ clear b(2,2),b(1,2),b(3,1),b(4,1),b(4,2),b(5,2),b(5,1),b(5,3),b(6,3)$ write "Then we have"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% off exp$ on factor$ write "\\"$ %write "{\fontsize{8}{10} \selectfont"$ 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ % write "}"$ %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% off exp$ on factor$ write "USD det \Phi:=",det(phi),"USD"$ on exp$ off factor$ r1:=phi**(-1)$ r2:=r1*MATJ$ J2:=r2*phi$ %J2:=phi**(-1)*MATJ*phi$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %on factor$ %off exp$ off factor$ on exp$ write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$ for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$ write "USD det \Phi:=",det(phi),"USD"$ off factor$ on exp$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Hence any USD J USD in \ref{68general} "$ write "is equivalent to "$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% off exp$ on factor$ write "\\"$ write "%{\fontsize{8}{10} \selectfont"$ write "\begin{equation}"$ write "\label{68final}"$ % write "USDUSD J = \begin{pmatrix}"$ write " J(xi(3,3),xi(4,3)) = \begin{pmatrix}"$ for i:=1:dim do if i neq dim then for j:=1:dim do if j neq dim then <> else <> else for j:=1:dim do if j neq dim then <> else <>$ write "\end{equation}"$ write "%}"$ write " where USD xi(4,3) \neq 0,- 1.\\USD"$ write " USD J(xi(3,3),xi(4,3))\cong J(\eta^3_3,\eta^4_3)USD"$ write "if and only if USD xi(3,3) = \eta^3_3USD and USDxi(4,3)=\eta^4_3.USD"$ %%%%%%%%%%%%%%%%% 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 <> else <> else for j:=1:dim do if j neq dim then <> else <>$ WRITE "\\USD det J:=",DET(MATJ)," USD"$ WRITE "USD Trace J:=",TRACE(MATJ)," USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 "USD "$ for each A in COLLECT_EQ do if part(A,2) neq 0 then <>$ for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$ if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % off exp$ % on allfac$ on exp$ off factor$ WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$ FOR i:=1:DIM DO <>$ %factor x(1),x(2),x(3),x(4),x(5),x(6)$ 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 DO <>$ 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 <>$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 BB(j,k) = - B1(j,k) + F(x(j))*x(k)$ COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM JOIN FOR j2:=1:DIM JOIN IF BB(j1,j2) NEQ 0 THEN {{{j1,j2},BB(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 "\\ \starline"$ write "\end{document}"$ bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%