%This is "program11_-1.red" (formerly "structcomplM14-1.tex") : %computes the complex structures on the Lie algebra G_{M,14-1} %and leads to impossibility %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% PRECISER SORTIE RESULTATS %%%%%%%%%%%%%%%%%%%%% %ON FACTOR$ off echo$ off nat$ OUT "rprogram11_-1.tex"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ON RAT$ OFF MSG$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% ENTRER LE FICHIER DE L'ALGEBRE %%%%%%%%%%%%%%%%%%%%% %DIM:=(dimension de l'algebre)$ DIM:= 6$ in "6nilp/m.14-1"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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{program11_-1.tex}.\\"$ WRITE "Computation of all complex structures on the real Lie Algebra"$ write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$ %write "\\ Case USD xi(4,3) =xi(2,1) USD."$ %WRITE "textbf{up to equivalence}"$ 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))$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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 USD46|1USD :"$ xi(1,6):=0$ write "\\ USD xi(1,6):=", ws,"USD"$ write "\\ and from equation USD56|2USD :"$ xi(2,6):=0$ write "\\ USD xi(2,6):=", ws,"USD"$ write "\\Then, if USD xi(3,6)=0 USD "$ xi(3,6):=0$ write "\\ USD xi(3,6):=", ws,"USD"$ write "\\ one gets from equation USD26|5USD :"$ xi(5,6):=0$ write "\\ USD xi(5,6):=", ws,"USD"$ write "\\ and from equation USD16|4USD :"$ xi(4,6):=0$ write "\\ USD xi(4,6):=", ws,"USD"$ write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 6x6 entry in USD J**2 USD is USD xi(6,6)**2 USD"$ write "\\ hence, one gets a contradiction "$ write "\\ Hence USD xi(3,6)USD has to be USD \neq 0 .USD"$ clear xi(3,6),xi(4,6),xi(5,6)$ write "\\ clear USD xi(3,6),xi(4,6),xi(5,6) USD"$ write "\\ USD xi(3,6):=", xi(3,6), "USD"$ write "\\ USD xi(5,6):=", xi(5,6),"USD"$ write "\\ USD xi(4,6):=", xi(4,6),"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then, one gets from equation USD16|1USD :"$ xi(1,4):=0$ write "\\ USD xi(1,4):=", ws,"USD"$ write "\\ and from equation USD16|2USD :"$ xi(2,4):=0$ write "\\ USD xi(2,4):=", ws,"USD"$ write "\\ and from equation USD26|1USD :"$ xi(1,5):=0$ write "\\ USD xi(1,5):=", ws,"USD"$ write "\\ and from equation USD26|2USD :"$ xi(2,5):=0$ write "\\ USD xi(2,5):=", ws,"USD"$ write "\\ and from equation USD36|4USD :"$ xi(1,3):=0$ write "\\ USD xi(1,3):=", ws,"USD"$ write "\\ and from equation USD36|5USD :"$ xi(2,3):=0$ write "\\ USD xi(2,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then, one gets from equation USD16|3USD :"$ xi(4,6):=-xi(3,4)$ write "\\ USD xi(4,6):=", ws,"USD"$ write "\\ and from equation USD26|3USD :"$ xi(5,6):=xi(3,5)$ write "\\ USD xi(5,6):=", 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 "\\Suppose here USD xi(1,1) \neq 0 USD "$ write "\\ Then, one gets from equation USD14|4USD :"$ xi(3,4):=0$ write "\\ USD xi(3,4):=", ws,"USD"$ write "\\ and from equation USD25|5USD :"$ xi(3,5):=0$ write "\\ USD xi(3,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\\ Then, one gets from equation USD13|3USD :"$ xi(4,3):=0$ write "\\ USD xi(4,3):=", ws,"USD"$ write "\\ and from equation USD16|6USD :"$ xi(6,4):=0$ write "\\ USD xi(6,4):=", ws,"USD"$ write "\\ and from equation USD23|3USD :"$ xi(5,3):=0$ write "\\ USD xi(5,3):=", ws,"USD"$ write "\\ and from equation USD26|6USD :"$ xi(6,5):=0$ write "\\ USD xi(6,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD14|3USD :"$ write "USD xi(4,4)=-xi(1,1)"$ write "\\ and from equation USD16|4USD :"$ write "USD xi(4,4)=xi(1,1)"$ write "\\ hence USD xi(1,1):=0. USD a contradiction"$ write "\\ Hence one cannot have USD xi(1,1) \neq 0. USD "$ xi(1,1):=0$ write "\\ USD xi(1,1):=", ws,"USD"$ clear xi(3,4),xi(3,5),xi(5,3),xi(4,3),xi(6,4),xi(6,5)$ write "\\ Clear USDxi(3,4),xi(3,5),xi(5,3),xi(4,3),xi(6,4),xi(6,5) USD"$ write "\\ USD xi(3,4):=", xi(3,4),"USD"$ write "\\ USD xi(3,5):=", xi(3,5),"USD"$ write "\\ USD xi(4,3):=", xi(4,3),"USD"$ write "\\ USD xi(6,4):=", xi(6,4),"USD"$ write "\\ USD xi(5,3):=", xi(5,3),"USD"$ write "\\ USD xi(6,5):=", xi(6,5),"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD14|3USD :"$ xi(4,4):=-xi(3,4)**2/xi(3,6)$ write "\\ USD xi(4,4):=", ws,"USD"$ write "\\ and from equation USD25|3USD :"$ xi(5,5):=xi(3,5)**2/xi(3,6)$ write "\\ USD xi(5,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD23|3USD :"$ %2,3}|3\\(xi(3,5)*xi(3,3)*xi(2,1) - xi(3,4) - xi(5,3)*xi(3,6)*xi(2,1))/xi(2,1)\\ xi(3,4):= xi(3,5)*xi(3,3)*xi(2,1) - xi(5,3)*xi(3,6)*xi(2,1)$ write "\\ USD xi(3,4):=", ws,"USD"$ write "\\ and from equation USD23|6USD :"$ %{2,3}|6\\( - (xi(6,4) - xi(4,3) - xi(6,5)*xi(3,3)*xi(2,1)) - xi(6,6)*xi(5,3)*xi( %2,1))/xi(2,1)\\$ xi(4,3):=-( - xi(6,4) + xi(6,5)*xi(3,3)*xi(2,1) - xi(6,6)*xi(5,3)*xi(2,1))$ write "\\ USD xi(4,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD26|6USD :"$ %{2,6}|6\\xi(5,3)*xi(3,6) - xi(3,5)*xi(3,3) + xi(6,5)*xi(3,6) - xi(6,6)*xi(3,5)\\ xi(5,3):=-( - xi(3,5)*xi(3,3) + xi(6,5)*xi(3,6) - xi(6,6)*xi(3,5))/xi(3,6)$ write "\\ USD xi(5,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD12|3USD :"$ %{1,2}|3\\xi(4,2)*xi(3,6) - xi(3,5)*xi(3,1) + xi(5,1)*xi(3,6) + xi(6,5)*xi(3,6)* %xi(3,2)*xi(2,1) - xi(6,6)*xi(3,5)*xi(3,2)*xi(2,1)\\$ xi(4,2):=-( - xi(3,5)*xi(3,1) + xi(5,1)*xi(3,6) + xi(6,5)*xi(3,6)* xi(3,2)*xi(2,1) - xi(6,6)*xi(3,5)*xi(3,2)*xi(2,1))/xi(3,6)$ write "\\ USD xi(4,2):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD16|6USD :"$ %{1,6}|6\\(xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(6,6)*xi(2,1) + xi(6,4)*xi(3,6) + % xi(3,5)*xi(2,1)\\$ xi(6,4):=-((xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(6,6)*xi(2,1) %+ xi(6,4)*xi(3,6) + xi(3,5)*xi(2,1))/xi(3,6)$ write "\\ USD xi(6,4):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 3x1 entry in USD J**2 USD gives "$ %USD {J**2}^3_1=xi(3,3)*xi(3,1) + xi(3,2)*xi(2,1) + xi(5,1)*xi(3,5) + xi(6,1)*xi %(3,6) - (xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(4,1)*xi(2,1);USD\\$ xi(6,1):=-(xi(3,3)*xi(3,1) + xi(3,2)*xi(2,1) + xi(5,1)*xi(3,5) %+ xi(6,1)*xi(3,6) - (xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(4,1)*xi(2,1))/xi(3,6)$ write "\\ USD xi(6,1):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "Then the 3x6 entry in USD J**2 USD gives "$ %USD {J**2}^3_6=xi(3,6)*xi(3,3) + xi(3,5)**2 + xi(6,6)*xi(3,6) - (xi(6,6)*xi(3,5 %) - xi(6,5)*xi(3,6))**2*xi(2,1)**2;USD\\$ xi(3,3):=-(xi(3,5)**2 + xi(6,6)*xi(3,6) - (xi(6,6)*xi(3,5 ) - xi(6,5)*xi(3,6))**2*xi(2,1)**2)/xi(3,6)$ write "\\ USD xi(3,3):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, one gets from equation USD15|3USD :"$ %{1,5}|3\\(xi(4,5) - xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5) %**2*xi(2,1)\\$ xi(4,5):= -(( - xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5) **2*xi(2,1))/xi(3,6)$ write "\\ USD xi(4,5):=", ws,"USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\par With these values, \textit{Reduce} computes again all equations."$ write "\\ Then, equation USD26|4USD reads :"$ write " USD {2,6}|4\\(xi(3,6)*xi(2,1)**2 + xi(3,6))/xi(2,1)\\"$ write "Hence USD xi(3,6) must be zero, a contradiction"$ write "HENCE THERE IS NO COMPLEX STRUCTURE ON THE ALGEBRA"$ write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "\end{document}"$ bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%