off echo,nat$ out "rcalculderivgtildedelta6_7XVIII.r"$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % introduction of the generic derivation delta of the 6-dimensional algebra g %as computed by %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for all i,j let ksi(i,j)=xi(i,j)$ matrix delta(6,6)$ delta:= mat((ksi(1,1),0,0,0,0,0),(ksi(2,1),ksi(2,2),0,0,0,0),(ksi(3,1),ksi(3,2),2*ksi(1, 1),0,0,0),(ksi(4,1),ksi(4,2),ksi(4,3),ksi(2,2) + ksi(1,1),0,0),(ksi(5,1),ksi(5,2 ),ksi(5,3),ksi(3,2),3*ksi(1,1),0),(ksi(6,1),ksi(6,2),ksi(6,3),ksi(4,2) + ksi(3,1 ),ksi(4,3) - ksi(2,1),ksi(2,2) + 2*ksi(1,1)))$ shortformdelta:={xi(2,1),SS,xi(3,1),xi(3,2),SS,xi(4,3),SS,xi(5,2),xi(5,3),SS,xi(6,2),xi(6,3)}$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % restriction on the parameters for nilpotent derivation and mod(ad g). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xi(1,1):=0$ xi(2,2):=0$ %impose par nilpotence de delta xi(4,2):=0$ xi(4,1):=0$ xi(5,1):=0$ xi(6,1):=0$ %reduction par adjoints %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %initial conditions for specific delta xi(2,1):=0$ xi(3,1):=1$ xi(3,2):=0$ xi(4,3):=0$ xi(5,2):=a$ a:=1$ xi(5,3):=0$ xi(6,2):=0$ xi(6,3):=b$ %b:=1$ write "delta:=",delta$ write "shortformdelta:=",shortformdelta$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % in "the commut relations file of g" % extended to the commut. relations of $\tilde{g}_delta %by the commutation relations involving X(0)=delta % and the projector V(0) on X(0) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% in "6nilp/6nilp.7"$ for j:=0:6 do x(0)*x(j):= if j neq 0 then for k:=1:6 sum delta(k,j)*x(k) else 0 for j:=1:6 do x(j)*x(0):= - x(0)*x(j)$ FOR j:=0:DIM DO V(0)*x(j):= IF j=0 THEN 1 ELSE 0; FOR i:=1:DIM DO V(i)*x(0):=0$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %MATRIXD will denote the generic derivation of the 7-dimensional {\tilde{g}}_{delta} % Let MATRIXD=matrix(D(i,j)). Collecting the equations {i,j} 1\leqslant i < i \leqslant 6 % -D [X(i),X(j)] + [DX(i), X(j)] + [X(i), DX(j)]. MATRIX MATRIXD(7,7)$ for i:=1:7 DO for j:=1:7 DO MATRIXD(i,j):=D(i-1,j-1)$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COLLECT_DERIVGTILDEDELTA:= for i:=0:DIM-1 JOIN for j:=i+1:DIM JOIN { {{i,j}, -(WS:=for k:=0:DIM sum (WS:=V(k)*(WS:=X(i)*X(j))) * (for m:=0:DIM sum D(m,k)*X(m)) ) + (WS:=for k:=0:DIM sum D(k,i)*(WS:=X(k)*X(j))) + (WS:= for k:=0:DIM sum D(k,j)*(WS:=X(i)*X(k))) }}$ %COLLECT_DERIVGTILDEDELTA; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Collecting the nonzero equations {i,j}|k obtained by projecting %on X(k) the derivation equation -D [X(i),X(j)] + [DX(i), X(j)] + [X(i), DX(j)]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %off exp$ %on factor$ %la liste des équations non nulles %write "list of the nonzero derivation equations"$ COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_DERIVGTILDEDELTA) JOIN FOR j2:=0:DIM JOIN IF V(j2)*PART(PART(COLLECT_DERIVGTILDEDELTA,j1),2) NEQ 0 THEN {{{PART(PART(COLLECT_DERIVGTILDEDELTA,j1),1),j2}, V(j2)*PART(PART(COLLECT_DERIVGTILDEDELTA,j1),2)} } ELSE {}$ COMMENT WRITE "Derivation equations to cancel (Reduce output) : \\", COLLECT_EQ$ % WRITE "Torsion equations to cancel (Latex output) : \\USD"$ % for each A in COLLECT_EQ do % if PART(A,2) neq 0 then % <> $ % %<> $ % write "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 <> ELSE UU:= ZZZ. UU $ P: IF j> ELSE <>$ 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 <> ELSE <>$ S2: CALLLET(z(k,j),0) $ S3: IF j

> 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 <> ELSE IF j>0 THEN <> ELSE IF i>0 THEN <> ELSE <>$ 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 <>$ write "on resout l'equation " , PART(PART(L,k),1)," qui est maintenant AA:=",AA$ W:= SELECTVAR(AA)$ if W=0 then <>$ 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 <>$ clear AA,W,WW$ END$ write "phase 1 de la resolution des equations"$ resol(COLLECT_EQ)$ on factor$ WRITE "Derivation equations to cancel (Reduce output) : \\", COLLECT_EQ$ % bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "Il y a une phase 2"$ %write "il n'y a pas de phase 2"$ write "Pour a neq 0, on doit avoir"$ %{{{0,2},5},(2*(d(0,1) - d(0,0)) + d(2,2))*a}, d(2,2):=-2*(d(0,1) - d(0,0))$ write "d(2,2):=", d(2,2)$ %on factor$ write "collect_eq:=",collect_eq$ %bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% matrix MATD(7,7)$ for i:=1:7 do for j:=1:7 do MATD(i,j):=D(i-1,j-1)$ off nat$ write "derivation generique de gtildedelta:"$ write "MATD:=",MATD$ on nat$ write "pour delta:=",delta$ write "pour shortformdelta:=",shortformdelta$ off nat$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$ listparametresMATDincluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$ listparametresMATD:= FOR EACH U IN listparametresMATDincluantparamalgebre_a join if U neq a and U neq b then {U} ELSE {}$ write "listeparametresMATD",WS$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LISTparametrestemoinMATD:= for each h in listparametresMATD collect dd(part(h,1),part(h,2))$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write "dim Der(gtildedelta):=",LENGTH(listparametresMATD)$ % bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% ecriture du premier membre du tore %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %ecrit le tore t1=D(i,j) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEDURE CALCUL_t1(i,j)$ %fait d(i,j)=1 autres nuls dans la matrice MATD BEGIN$ integer i,j$ WS:=for each U in listparametresMATD JOIN IF U neq d(i,j) and U neq a then {U} else {}$ for each U in WS DO <>$ d(i,j):=1$ off nat$ WS:=MATD; for each U in listparametrestemoinMATD do <>$ RETURN WS$ END$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %introduction de la matrice ID identite matrix ID(7,7)$ for i:=1:7 do for j:=1:7 do <>$ write "un element t1 d'un tore "$ %write "seul candidat a etre un element t1 d'un tore "$ write "t1:=D(0,0)"$ matrix t1(7,7)$ t1:=calcul_t1(0,0)$ on nat$ write "t1:=", t1$ write "MATD:=", MATD$ %bye$ %% %off nat$ %on factor$ %det(t1-x*ID); %off factor$ mateigen(t1,x); %bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEDURE COMMUTANT(t1)$ BEGIN$ Z:=MATD*t1-t1*MATD$ SSOLVE(7,7)$ WS1:=MATD$ listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$ listparametresCOMMUTANT_t1_incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$ listparametresCOMMUTANT_t1:= FOR EACH U IN listparametresCOMMUTANT_t1_incluantparamalgebre_a join if U neq a then {U} ELSE {}$ listparametrestemoinsresolusdanscalculCOMMUTANT_t1 := for each h in LISTparametrestemoinMATD join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1 THEN {} ELSE {dd(part(h,1),part(h,2))} $ for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1 do <>$ write "commutant de t1 dans der(gtildedelta):"$ RETURN WS1$ END$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% write COMMUTANT(t1)$ %bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %write "Mais t1 n'est pas semisimple "$ %write "gtildedelta est caracteristiquement nilpotente"$ %for j:=1:7 do <>$ d(i,j):=1$ off nat$ WS:=MATD; for each U in listparametrestemoinMATD do <>$ %restaure les parametres de MATD RETURN WS$ END$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEDURE COMMUTANT2(t1,t2)$ BEGIN$ Z:=MATD*t1-t1*MATD$ SSOLVE(7,7)$ Z:=MATD*t2-t2*MATD$ SSOLVE(7,7)$ WS1:=MATD$ listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$ listparametresCOMMUTANT_t1_t2_incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$ listparametresCOMMUTANT_t1_t2:= FOR EACH U IN listparametresCOMMUTANT_t1_t2_incluantparamalgebre_a join if U neq a then {U} ELSE {}$ listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2 := for each h in LISTparametrestemoinMATD join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1_t2 THEN {} ELSE {dd(part(h,1),part(h,2))} $ for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2 do <>$ write "commutant simultane de t1,t2 dans der(gtildedelta):"$ RETURN WS1$ END$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% matrix t2(7,7)$ t2:=calcul_t2(0,1)$ on nat$ write "t2:=D(0,1):=",t2$ %load_package normform$ %Jordansymbolic(t2); %bye$ %%off nat$ %on factor$ %det(t2-x*ID); %off factor$ mateigen(t2,x); %bye$ write COMMUTANT2(t1,t2)$ %bye$ % %write "le calcul du commutant simultane de t1 et t2 dans der(gtildedelta):"$ %write "commutant de t1 et t2 dans der(gtildedelta):"$ %Z:=MATD*t1-t1*MATD$ %SSOLVE(7,7)$ %Z:=MATD*t2-t2*MATD$ %SSOLVE(7,7)$ %write "commutant de t1 et t2:=",MATD; %bye$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% ecriture du troiseme membre du tore %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEDURE COMMUTANT3(t1,t2,t3)$ BEGIN$ Z:=MATD*t1-t1*MATD$ SSOLVE(7,7)$ Z:=MATD*t2-t2*MATD$ SSOLVE(7,7)$ Z:=MATD*t3-t3*MATD$ SSOLVE(7,7)$ WS1:=MATD$ listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$ listparametresCOMMUTANT_t1_t2_t3_incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$ listparametresCOMMUTANT_t1_t2_t3:= FOR EACH U IN listparametresCOMMUTANT_t1_t2_t3_incluantparamalgebre_a join if U neq a then {U} ELSE {}$ listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2_t3 := for each h in LISTparametrestemoinMATD join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1_t2_t3 THEN {} ELSE {dd(part(h,1),part(h,2))} $ for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2_t3 do <>$ write "commutant simultane de t1,t2,t3 dans der(gtildedelta):"$ RETURN WS1$ END$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEDURE CALCUL_t3(i,j)$ %fait d(i,j)=1 autres nuls dans la matrice restante MATD dans le commutant de t1 et t2 BEGIN$ integer i,j$ % calcule d'abord le matrices MATD du commutant de t1 et t2 Z:=MATD*t1-t1*MATD$ SSOLVE(7,7)$ Z:=MATD*t2-t2*MATD$ SSOLVE(7,7)$ ws:=UNKNOWNSINLIST(listmatD)$ ws:= FOR EACH U IN ws join if U neq a then {U} ELSE {}$ for each U in WS DO <>$ d(i,j):=1$ off nat$ WS:=MATD; for each U in listparametrestemoinMATD do <>$ %restaure les parametres de MATD RETURN WS$ END$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %t3:=calcul_t3(2,2)$ %on nat$ %write "t3:=D(2,2):=",t3$ %%off nat$ %on factor$ %det(t3-x*ID); %off factor$ %mateigen(t3,x); %bye$ %write COMMUTANT3(t1,t2,t3)$ % %write "commutant de t1 et t2 et t3 dans der(gtildedelta):"$ %Z:=MATD*t3-t3*MATD$ %SSOLVE(7,7)$ %write "commutant de t1,t2,t3:=",MATD$ %write "le calcul du commutant de t1 et t2 et t3 dans der(gtildedelta) montre que t1,t2,t3 est un tore maximal."$ write " t1,t2 est un tore maximal."$ %write " t1 est un tore maximal."$ %write "seul tore maximal: {0}."$ %bye$ write "matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base diagonalisant le tore maximal: on peut prendre"$ matrix P(7,7)$ %P:=ID$ for i:=1:7 do <>$ P(7,1):=a**2*b**3/48$ P(5,1):=-6*a*b**2/48$ P(3,1):=24*b/48$ P(7,3):=a**2*b**2/8$ P(5,3):=-4*a*b/8$ P(3,4):=-1/a$ P(7,5):=-a*b/2$ P(5,6):=-1/a$ on nat$ write "P:=",P$ write "P**(-1)*t1*P:=", P**(-1)*t1*P; write "P**(-1)*t2*P:=", P**(-1)*t2*P; %write "P**(-1)*t3*P:=", P**(-1)*t3*P; write "matrice des derivations dans cette base diagonalisante Y(1),...,Y(7):"$ off nat$ write "P**(-1)*MATD*P:=",P**(-1)*MATD*P$ % Si besoin seconde diagonalisation matrix Q(7,7)$ %Q:=ID$ %si pas besoin de deuxieme diagonalisation for i:=1:7 do <>$ Q(1,2):=-1$ Q(3,4):=1/a$ Q(5,6):=1/a$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PP:=P*Q$ write "PP:=",PP$ write "PP**(-1)*t1*PP:=", PP**(-1)*t1*PP; write "PP**(-1)*t2*PP:=", PP**(-1)*t2*PP; %write "P**(-1)*t3*P:=", P**(-1)*t3*P; %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 <>$ %bye$ for j:=1:7 do <>$ write "on pose :"$ write "avec comme matrice de changemment de base :"$ matrix isom(7,7)$ isom(1,2):=1$ isom(2,1):=-1$ isom(3,3):=-1$ isom(4,4):=1$ isom(5,5):=1$ isom(6,5):=-1$ isom(6,6):=-1$ isom(7,7):=1$ on nat$ write isom$ write "det(isom):= ", det(isom)$ For j:=1:7 do ZZ(j):=for k:=1:7 sum isom(k,j)*diaY(k)$ For i:=1:7 do <