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module lie; % Header module for classification of Lie algebras. % Author: Carsten and Franziska Schoebel. % Copyright (c) 1993 The Leipzig University, Computer Science Dept. % All Rights Reserved. % create!-package('(lie liendmc1 lie1234),nil); switch tr_lie; endmodule; module liendmc1; % N-dimensional Lie algebras with 1-dimensional derived % algebra. % Author: Carsten Schoebel. % e-mail: cschoeb@aix550.informatik.uni-leipzig.de . % Copyright (c) 1993 The Leipzig University, Computer Science Dept. % All Rights Reserved. algebraic; operator heisenberg,commutative,lie_algebra; algebraic procedure liendimcom1(n); begin if (not(symbolic fixp(n)) or n<2) then symbolic rederr "dimension out of range"; symbolic (if gettype 'lienstrucin neq 'ARRAY then rederr "lienstrucin not ARRAY"); if length lienstrucin neq {n+1,n+1,n+1} then symbolic rederr "dimension of lienstrucin out of range"; matrix lientrans(n,n); array lie_cc(n,n,n); lieninstruc(n); lienjactest(n);if lie_jtest neq 0 then <<clear lie_cc,lie_jtest;symbolic rederr "not a Lie algebra">>; <<liendimcom(n); if lie_dim=0 then <<if symbolic !*tr_lie then write "The given Lie algebra is commutative"; lientrans:=lientrans**0;lie_list:={commutative(n)}>> else if lie_dim=1 then <<if lie_help=0 then liencentincom(n,lie_tt,lie_p,lie_q) else liencentoutcom(n,lie_tt,lie_s); if symbolic !*tr_lie then lienoutform(lientrans,n,lie_help,2*lie_kk!*+1); if lie_help=1 then lie_list:={lie_algebra(2),commutative(n-2)} else lie_list:={heisenberg(2*lie_kk!*+1),commutative(n-2*lie_kk!*-1)} >>else <<clear lie_dim,lie_help,lie_p,lie_q,lie_tt,lie_s,lie_kk!*, lie_jtest,lie_cc; symbolic rederr "dimension of derived algebra out of range">>; clear lie_dim,lie_help,lie_p,lie_q,lie_tt,lie_s,lie_kk!*,lie_control>>; clear lie_jtest,lie_cc;return lie_list end; algebraic procedure lieninstruc(n); begin for i:=1:n-1 do for j:=i+1:n do for k:=1:n do <<lie_cc(i,j,k):=lienstrucin(i,j,k); lie_cc(j,i,k):=-lienstrucin(i,j,k)>> end; algebraic procedure lienjactest(n); begin lie_jtest:=0; for i:=1:n-2 do for j:=i+1:n-1 do for k:=j+1:n do for l:=1:n do if (for r:=1:n sum lie_cc(j,k,r)*lie_cc(i,r,l)+lie_cc(i,j,r)*lie_cc(k,r,l)+ lie_cc(k,i,r)*lie_cc(j,r,l)) neq 0 then <<lie_jtest:=1; i:=n-1;j:=n;k:=n+1;l:=n+1>> end; algebraic procedure liendimcom(n); begin integer r; scalar he; lie_dim:=0; for i:=1:n-1 do for j:=i:n do for k:=1:n do if lie_cc(i,j,k) neq 0 then <<lie_dim:=1;lie_p:=i;lie_q:=j;r:=k;i:=n;j:=k:=n+1>>; if lie_dim neq 0 then <<for i:=1:n-1 do for j:=1:n do <<he:=lie_cc(i,j,r)/lie_cc(lie_p,lie_q,r); for k:=1:n do if lie_cc(i,j,k) neq (he*lie_cc(lie_p,lie_q,k)) then <<lie_dim:=2;i:=n;j:=n+1;k:=n+1>>>>; if lie_dim=1 then <<lie_help:=0; for i:=1:n do for j:=1:n do if (for k:=1:n sum (lie_cc(lie_p,lie_q,k)*lie_cc(k,i,j))) neq 0 then <<lie_help:=1;lie_s:=i;r:=j;i:=j:=n+1>>; for i:=1:n do lientrans(1,i):=lie_cc(lie_p,lie_q,i); if lie_help=0 then <<lientrans(2,lie_p):=lientrans(3,lie_q):=1;lie_kk!*:=1; for i:=1:n do <<if (lie_cc(lie_p,lie_q,i) neq 0 and i neq lie_p and i neq lie_q) then <<lie_tt:=i;i:=n+1>>>>>> else <<lientrans(2,lie_s):= lie_cc(lie_p,lie_q,r)/(for k:=1:n sum (lie_cc(lie_p,lie_q,k)*lie_cc(k,lie_s,r))); for i:=1:n do <<if (lie_cc(lie_p,lie_q,i) neq 0 and i neq lie_s) then <<lie_tt:=i;i:=n+1>>>>>>>>>>; end; algebraic procedure liencentincom(n,tt,p,q); begin integer con1,con2; matrix lie_lamb(n,n); lie_control:=0; con1:=con2:=0; for i:=4:n do if (i neq tt and i neq p and i neq q) then lientrans(i,i):=1 else if (tt neq 1 and p neq 1 and q neq 1 and con1 neq 1) then <<lientrans(i,1):=1;con1:=1>> else if (tt neq 2 and p neq 2 and q neq 2 and con2 neq 1) then <<lientrans(i,2):=1;con2:=1>> else lientrans(i,3):=1; if n>3 then <<liennewstruc(n,2,tt); if n>4 then for i:=4 step 2 until n do if (i+1)=n then <<lienfindpair(n,i); if lie_control=1 then lie_kk!*:=lie_kk!*+1>> else if i+1<n then <<lienfindpair(n,i);if lie_control=1 then <<liennewstruc(n,i,tt),lie_kk!*:=lie_kk!*+1>>else i:=n+1>>>> end; algebraic procedure lienfindpair(n,m); begin scalar he; matrix lie_a(n,n); lie_control:=0; for i:=m:n-1 do for j:=i+1:n do <<if lie_lamb(i,j) neq 0 then <<lie_control:=1; lie_a(i,m):=lie_a(m+1,j):=lie_a(j,m+1):=1; lie_a(m,i):=1/lie_lamb(i,j); for k:=1:n do if (k neq i and k neq j and k neq m and k neq (m+1)) then lie_a(k,k):=1; lientrans:=lie_a*lientrans;i:=n;j:=n+1>>>>;clear lie_a end; algebraic procedure liennewstruc(n,m,tt); begin matrix lie_a(n,n); lie_a:=lie_a**0; for i:=m:n-1 do for j:=i+1:n do lie_lamb(i,j):=(for k:=1:n sum for l:=1:n sum lientrans(i,k)*lientrans(j,l)*lie_cc(k,l,tt))/lientrans(1,tt); for i:=m+2:n do <<lie_a(i,m+1):=-lie_lamb(m,i);lie_a(i,m):=lie_lamb(m+1,i)>>; lientrans:=lie_a*lientrans; for i:=m+2:n-1 do for j:=i+1:n do lie_lamb(i,j):=(for k:=1:n sum for l:=1:n sum lientrans(i,k)*lientrans(j,l)*lie_cc(k,l,tt))/lientrans(1,tt); clear lie_a end; algebraic procedure liencentoutcom(n,tt,s); begin integer pp,qq; matrix lie_lamb(2,n),lie_a(n,n); for i:=3:n do <<lientrans(i,i):=1;lie_lamb(1,i):=(for j:=1:n sum lientrans(1,j)*lie_cc(j,i,tt))/lientrans(1,tt); lie_lamb(2,i):=lie_cc(s,i,tt)*lientrans(2,s)/lientrans(1,tt)>>; if (tt>2 and s>2) then <<lientrans(tt,tt):=lientrans(s,s):=0; lientrans(tt,1):=lientrans(s,2):=1; lie_lamb(1,tt):=(for j:=1:n sum lientrans(1,j)*lie_cc(j,1,tt)/lientrans(1,tt)); lie_lamb(1,s):=(for j:=1:n sum lientrans(1,j)*lie_cc(j,2,tt)/lientrans(1,tt)); lie_lamb(2,tt):=lie_cc(s,1,tt)*lientrans(2,s)/lientrans(1,tt); lie_lamb(2,s):=lie_cc(s,2,tt)*lientrans(2,s)/lientrans(1,tt) >> else if (tt>2 or s>2) then <<if tt>2 then <<pp:=3-s;qq:=tt>> else <<pp:=3-tt;qq:=s>>; lientrans(qq,qq):=0;lientrans(qq,pp):=1; lie_lamb(1,qq):=(for j:=1:n sum lientrans(1,j)*lie_cc(j,pp,tt))/lientrans(1,tt); lie_lamb(2,qq):=lie_cc(s,pp,tt)*lientrans(2,s)/lientrans(1,tt)>>; lie_a:=lie_a**0; for i:=3:n do <<lie_a(i,2):=-lie_lamb(1,i);lie_a(i,1):=lie_lamb(2,i)>>; lientrans:=lie_a*lientrans;clear lie_lamb,lie_a end; algebraic procedure lienoutform(at,n,help,kk); begin operator y; lie_a:=at; if help=1 then <<write "Your Lie algebra is the direct sum of the Lie algebra L(2) and"; write "the ",n-2,"-dimensional commutative Lie algebra, where L(2) is"; write "2-dimensional and there exists a basis {X(1),X(2)} in L(2) with"; write "[X(1),X(2)]=X(1).">>else <<write "Your Lie algebra is the direct sum of the Lie algebra H(",kk,")"; write "and the ",n-kk,"-dimensional commutative Lie algebra, where"; write "H(",kk,") is ",kk,"-dimensional and there exists a basis"; write "{X(1),...,X(",kk,")} in H(",kk,") with:"; write "[X(2),X(3)]=[X(2*i),X(2*i+1)]=...=[X(",kk-1,"),X(",kk,")]=X(1)" >>; write "The transformation into this form is:"; for i:=1:n do write "X(",i,"):=",for j:=1:n sum lie_a(i,j)*y(j);clear y,lie_a end; endmodule; module lie1234; % n-dimensional Lie algebras up to n=4. % Author: Carsten and Franziska Schoebel. % e-mail: cschoeb@aix550.informatik.uni-leipzig.de . % Copyright (c) 1993 The Leipzig University, Computer Science Dept. % All Rights Reserved. algebraic; operator liealg,comtab; algebraic procedure lieclass(dim); begin if not(dim=1 or dim=2 or dim=3 or dim=4) then symbolic rederr "dimension out of range"; symbolic(if gettype 'liestrin neq 'ARRAY then rederr "liestrin not ARRAY"); if length liestrin neq {dim+1,dim+1,dim+1} then symbolic rederr "dimension of liestrin out of range"; if dim=1 then <<if symbolic !*tr_lie then write "one-dimensional Lie algebra"; lie_class:={liealg(1),comtab(0)}>> else if dim=2 then lie2(liestrin(1,2,1),liestrin(1,2,2)) else if dim=3 then <<matrix lie3_ff(3,3); for i:=1:3 do <<lie3_ff(1,i):=liestrin(1,2,i); lie3_ff(2,i):=liestrin(1,3,i); lie3_ff(3,i):=liestrin(2,3,i)>>; lie3(lie3_ff);clear lie3_ff>> else <<array cc(4,4,4); for i:=1:4 do for j:=1:4 do for k:=1:4 do cc(i,j,k):=liestrin(i,j,k); lie4();clear cc>>;return lie_class end; algebraic procedure lie2(f,g); BEGIN IF G=0 THEN IF F=0 THEN liemat:=MAT((1,0),(0,1)) ELSE liemat:=MAT((0,-1/F),(F,0)) ELSE liemat:=MAT((1/G,0),(F,G)); IF (F=0 AND G=0) THEN <<if symbolic !*tr_lie then WRITE "The given Lie algebra is commutative"; lie_class:={liealg(2),comtab(0)}>> ELSE <<if symbolic !*tr_lie then write "[X,Y]=Y";lie_class:={liealg(2),comtab(1)}>> END; algebraic procedure lie3(ff); BEGIN MATRIX liemat(3,3),l_f(3,3); ARRAY l_jj(3); l_f:=ff; FOR N:=1:3 DO l_jj(N):=l_f(1,N)*(-l_f(2,1)-l_f(3,2))+ l_f(2,N)*(l_f(1,1)-l_f(3,3))+ l_f(3,N)*(l_f(1,2)+l_f(2,3)); IF NOT(l_jj(1)=0 AND l_jj(2)=0 AND l_jj(3)=0) THEN <<clear lie3_ff,liemat,l_f,l_jj; symbolic rederr "not a Lie algebra">>; IF l_f=MAT((0,0,0),(0,0,0),(0,0,0)) THEN <<if symbolic !*tr_lie then WRITE "Your Lie algebra is commutative"; lie_class:={liealg(3),comtab(0)};liemat:=liemat**0>> ELSE IF DET(l_f) NEQ 0 THEN com3(ff) ELSE IF independ(1,2,ff)=1 THEN com2(ff,1,2) ELSE IF independ(1,3,ff)=1 THEN com2(ff,1,3) ELSE IF independ(2,3,ff)=1 THEN com2(ff,2,3) ELSE com1(ff); CLEAR l_jj,l_f END; algebraic procedure independ(I,J,F0); BEGIN MATRIX F1(3,3); F1:=F0; IF (F1(I,1)*F1(J,2)-F1(I,2)*F1(J,1)=0 AND F1(I,2)*F1(J,3)-F1(I,3)*F1(J,2)=0 AND F1(I,1)*F1(J,3)-F1(I,3)*F1(J,1)=0) THEN RETURN 0 ELSE RETURN 1 END; algebraic procedure com1(F2); BEGIN SCALAR ALPHA,AA,BB; INTEGER R,I,J,M,N,Z1; MATRIX F3(3,3); ARRAY l_C(3,3,3); F3:=F2; FOR M:=3 STEP -1 UNTIL 1 DO FOR N:=3 STEP -1 UNTIL 1 DO IF F3(M,N) NEQ 0 THEN I:=M; IF I=1 THEN <<I:=1;J:=2>> ELSE IF I=2 THEN <<I:=1;J:=3>> ELSE <<I:=2;J:=3>>; FOR K:=1:3 DO <<l_C(1,2,K):=F3(1,K);l_C(2,1,K):=-F3(1,K); l_C(1,3,K):=F3(2,K);l_C(3,1,K):=-F3(2,K); l_C(2,3,K):=F3(3,K);l_C(3,2,K):=-F3(3,K)>>; Z1:=0; FOR U:=3 STEP -1 UNTIL 1 DO FOR V:=3 STEP -1 UNTIL 1 DO IF l_C(I,J,1)*l_C(V,1,U)+l_C(I,J,2)*l_C(V,2,U)+ l_C(I,J,3)*l_C(V,3,U) NEQ 0 THEN <<M:=U;N:=V;Z1:=1>>; IF Z1=0 THEN <<A1:=MAT((1,0,0),(0,1,0),(l_C(1,2,1),l_C(1,2,2),l_C(1,2,3))); A2:=MAT((1,0,0),(0,0,1),(l_C(1,3,1),l_C(1,3,2),l_C(1,3,3))); A3:=MAT((0,1,0),(0,0,1),(l_C(2,3,1),l_C(2,3,2),l_C(2,3,3))); IF DET(A1) NEQ 0 THEN liemat:=A1 ELSE IF DET(A2) NEQ 0 THEN liemat:=A2 ELSE liemat:=A3; if symbolic !*tr_lie then WRITE "[X,Y]=Z";lie_class:={liealg(3),comtab(1)}>> ELSE <<ALPHA:=(l_C(I,J,1)*l_C(N,1,M)+l_C(I,J,2)*l_C(N,2,M)+ l_C(I,J,3)*l_C(N,3,M))/l_C(I,J,M); A1:=MAT((0,0,0),(0,0,0),(l_C(I,J,1),l_C(I,J,2),l_C(I,J,3))); A1(1,N):=1/ALPHA;A1(2,1):=1; IF DET(A1) NEQ 0 THEN R:=1 ELSE <<A1(2,1):=0;A1(2,2):=1; IF DET(A1) NEQ 0 THEN R:=2 ELSE <<A1(2,2):=0;A1(2,3):=1;R:=3>>>>; AA:=l_C(N,R,M)/(ALPHA*l_C(I,J,M)); BB:=(l_C(I,J,1)*l_C(R,1,M)+l_C(I,J,2)*l_C(R,2,M)+ l_C(I,J,3)*l_C(R,3,M))/l_C(I,J,M); IF AA=0 THEN liemat:=MAT((1,0,0),(-BB,1,0),(0,0,1))*A1 ELSE liemat:=MAT((1,0,0),(BB/AA,-1/AA,1),(0,0,1))*A1; if symbolic !*tr_lie then WRITE "[X,Z]=Z";lie_class:={liealg(3),comtab(2)}>>; CLEAR A1,A2,A3,l_C,F3 END; algebraic procedure com2(F2,M,N); BEGIN SCALAR Z1,ALPHA,ALPHA1,ALPHA2,BETA,BETA1,BETA2; MATRIX F3(3,3); F3:=F2; A1:=MAT((F3(M,1),F3(M,2),F3(M,3)), (F3(N,1),F3(N,2),F3(N,3)),(0,0,0)); A1(3,1):=1;Z1:=DET(A1); IF Z1 NEQ 0 THEN <<ALPHA1:=(-F3(N,3)*(F3(M,2)*F3(1,2)+F3(M,3)*F3(2,2))+ F3(N,2)*(F3(M,2)*F3(1,3)+F3(M,3)*F3(2,3)))/Z1; ALPHA2:=(-F3(N,3)*(F3(N,2)*F3(1,2)+F3(N,3)*F3(2,2))+ F3(N,2)*(F3(N,2)*F3(1,3)+F3(N,3)*F3(2,3)))/Z1; BETA1:=(F3(M,3)*(F3(M,2)*F3(1,2)+F3(M,3)*F3(2,2))- F3(M,2)*(F3(M,2)*F3(1,3)+F3(M,3)*F3(2,3)))/Z1; BETA2:=(F3(M,3)*(F3(N,2)*F3(1,2)+F3(N,3)*F3(2,2))- F3(M,2)*(F3(N,2)*F3(1,3)+F3(N,3)*F3(2,3)))/Z1>> ELSE <<A1(3,1):=0;A1(3,2):=1;Z1:=DET(A1); IF Z1 NEQ 0 THEN <<ALPHA1:=(-F3(N,3)*(F3(M,1)*F3(1,1)-F3(M,3)*F3(3,1))+ F3(N,1)*(F3(M,1)*F3(1,3)-F3(M,3)*F3(3,3)))/Z1; ALPHA2:=(-F3(N,3)*(F3(N,1)*F3(1,1)-F3(N,3)*F3(3,1))+ F3(N,1)*(F3(N,1)*F3(1,3)-F3(N,3)*F3(3,3)))/Z1; BETA1:=(F3(M,3)*(F3(M,1)*F3(1,1)-F3(M,3)*F3(3,1))- F3(M,1)*(F3(M,1)*F3(1,3)-F3(M,3)*F3(3,3)))/Z1; BETA2:=(F3(M,3)*(F3(N,1)*F3(1,1)-F3(N,3)*F3(3,1))- F3(M,1)*(F3(N,1)*F3(1,3)-F3(N,3)*F3(3,3)))/Z1>> ELSE <<A1(3,2):=0;A1(3,3):=1;Z1:=DET(A1); ALPHA1:=(F3(N,2)*(F3(M,1)*F3(2,1)+F3(M,2)*F3(3,1))- F3(N,1)*(F3(M,1)*F3(2,2)+F3(M,2)*F3(3,2)))/Z1; ALPHA2:=(F3(N,2)*(F3(N,1)*F3(2,1)+F3(N,2)*F3(3,1))- F3(N,1)*(F3(N,1)*F3(2,2)+F3(N,2)*F3(3,2)))/Z1; BETA1:=(-F3(M,2)*(F3(M,1)*F3(2,1)+F3(M,2)*F3(3,1))+ F3(M,1)*(F3(M,1)*F3(2,2)+F3(M,2)*F3(3,2)))/Z1; BETA2:=(-F3(M,2)*(F3(N,1)*F3(2,1)+F3(N,2)*F3(3,1))+ F3(M,1)*(F3(N,1)*F3(2,2)+F3(N,2)*F3(3,2)))/Z1>>>>; IF (ALPHA2=0 AND BETA1=0 AND ALPHA1=BETA2) THEN <<liemat:=MAT((1,0,0),(0,1,0),(0,0,1/ALPHA1))*A1; if symbolic !*tr_lie then WRITE "[X,Z]=X, [Y,Z]=Y";lie_class:={liealg(3),comtab(3)}>> ELSE <<IF ALPHA2 NEQ 0 THEN <<ALPHA:=ALPHA1+BETA2;BETA:=ALPHA2*BETA1-ALPHA1*BETA2; A2:=MAT((0,BETA1-ALPHA1*BETA2/ALPHA2,0), (1,-ALPHA1/ALPHA2,0),(0,0,1))>> ELSE IF BETA1 NEQ 0 THEN <<ALPHA:=1+ALPHA1/BETA2;BETA:=-ALPHA1/BETA2; A2:=MAT((-ALPHA1*BETA2/BETA1,0,0), (-(BETA2**2)/BETA1,BETA2,0),(0,0,1/BETA2))>> ELSE <<ALPHA:=ALPHA1+BETA2;BETA:=-ALPHA1*BETA2; A2:=MAT((1,1,0),(1/ALPHA1,1/BETA2,0),(0,0,1))>>; IF ALPHA=0 THEN <<liemat:=MAT((1,0,0),(0,SQRT(ABS(BETA)),0), (0,0,1/SQRT(ABS(BETA))))*A2*A1; if symbolic !*tr_lie then WRITE "[X,Z]=",BETA/ABS(BETA),"Y, [Y,Z]=X"; if BETA>0 then lie_class:={liealg(3),comtab(4)} else lie_class:={liealg(3),comtab(5)}>> ELSE <<liemat:=MAT((1,0,0),(0,-ALPHA,0),(0,0,-1/ALPHA))*A2*A1; if symbolic !*tr_lie then WRITE "[X,Z]=-X+",BETA/(ALPHA**2),"Y, [Y,Z]=X"; lie_class:={liealg(3),comtab(6),BETA/(ALPHA**2)}>>>>; CLEAR A1,A2,F3 END; algebraic procedure com3(F2); BEGIN MATRIX l_K(3,3),F3(3,3); F3:=F2; l_K(1,1):=F3(1,2)**2+2*F3(1,3)*F3(2,2)+F3(2,3)**2; l_K(1,2):=-F3(1,1)*F3(1,2)+F3(1,3)*F3(3,2)- F3(2,1)*F3(1,3)+F3(2,3)*F3(3,3); l_K(1,3):=-F3(1,1)*F3(2,2)-F3(1,2)*F3(3,2)- F3(2,1)*F3(2,3)-F3(2,2)*F3(3,3); l_K(2,1):=l_K(1,2); l_K(2,2):=F3(1,1)**2-2*F3(1,3)*F3(3,1)+F3(3,3)**2; l_K(2,3):=F3(1,1)*F3(2,1)+F3(1,2)*F3(3,1)- F3(3,1)*F3(2,3)-F3(3,2)*F3(3,3); l_K(3,1):=l_K(1,3); l_K(3,2):=l_K(2,3); l_K(3,3):=F3(2,1)**2+2*F3(2,2)*F3(3,1)+F3(3,2)**2; IF NOT(NUMBERP(l_K(1,1)) AND NUMBERP(l_K(1,1)*l_K(2,2)-l_K(1,2)*l_K(2,1)) AND NUMBERP(DET(l_K))) THEN <<WRITE "Is ",-l_K(1,1),">0 and ", l_K(1,1)*l_K(2,2)-l_K(1,2)*l_K(2,1),">0 and ", -DET(l_K),">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN so3(F2) ELSE so21(F2)>> ELSE IF (-l_K(1,1)>0 AND l_K(1,1)*l_K(2,2)-l_K(1,2)*l_K(2,1)>0 AND -DET(l_K)>0) THEN so3(F2) ELSE so21(F2); CLEAR l_K,F3 END; algebraic procedure so3(F4); BEGIN SCALAR S,TT,Q,R,ALPHA; MATRIX F5(3,3); F5:=F4; S:=F5(2,2)/ABS(F5(2,2)); TT:=ABS(F5(1,2)**2+F5(1,3)*F5(2,2)); R:=F5(1,1)-F5(1,2)*F5(2,1)/F5(2,2); ALPHA:=TT*(-R*R-((F5(2,1)/F5(2,2))**2+F5(3,1)/F5(2,2))*TT); Q:=1/SQRT(ALPHA); liemat(1,1):=1/(S*SQRT(TT)); liemat(1,2):=0; liemat(1,3):=0; liemat(2,1):=Q*R; liemat(2,2):=0; liemat(2,3):=-Q*TT/F5(2,2); liemat(3,1):=-Q*S*SQRT(TT)*F5(2,1)/F5(2,2); liemat(3,2):=-Q*S*SQRT(TT); liemat(3,3):=Q*S*SQRT(TT)*F5(1,2)/F5(2,2); if symbolic !*tr_lie then WRITE "[X,Y]=Z, [X,Z]=-Y, [Y,Z]=X";lie_class:={liealg(3),comtab(7)}; CLEAR F5; END; algebraic procedure so21(F4); BEGIN SCALAR GAM,EPS,S,TT,Q,R,ALPHA; MATRIX l_G(3,3),F5(3,3); F5:=F4; liemat:=MAT((1,0,0),(0,1,0),(0,0,1)); IF F5(2,2)=0 THEN IF F5(1,3) NEQ 0 THEN <<liemat:=MAT((1,0,0),(0,0,1),(0,1,0)); l_G(1,1):=F5(2,1);l_G(1,2):=F5(2,3);l_G(1,3):=F5(2,2); l_G(2,1):=F5(1,1);l_G(2,2):=F5(1,3);l_G(2,3):=F5(1,2); l_G(3,1):=-F5(3,1);l_G(3,2):=-F5(3,3);l_G(3,3):=-F5(3,2); F5:=l_G>> ELSE IF F5(3,1) NEQ 0 THEN <<liemat:=MAT((0,1,0),(1,0,0),(0,0,1)); l_G(1,1):=-F5(1,2);l_G(1,2):=-F5(1,1);l_G(1,3):=-F5(1,3); l_G(2,1):=F5(3,2);l_G(2,2):=F5(3,1);l_G(2,3):=F5(3,3); l_G(3,1):=F5(2,2);l_G(3,2):=F5(2,1);l_G(3,3):=F5(2,3); F5:=l_G>> ELSE <<liemat:=MAT((1,0,1),(1,0,0),(0,1,0)); l_G(1,1):=-F5(2,3);l_G(1,2):=F5(2,3)-F5(2,1);l_G(1,3):=0; l_G(2,1):=-F5(3,3);l_G(2,2):=2*F5(1,1); l_G(2,3):=F5(1,2)-F5(3,2); l_G(3,1):=0;l_G(3,2):=F5(1,1);l_G(3,3):=F5(1,2); F5:=l_G>>; IF F5(1,2)**2+F5(1,3)*F5(2,2)=0 THEN <<GAM:=-F5(1,2)/F5(2,2);EPS:=F5(1,1)-F5(1,2)*F5(2,1)/F5(2,2); IF 1/4*(F5(3,2)**2+F5(3,1)*F5(2,2))-EPS*F5(2,2)/2=0 THEN <<liemat:=MAT((0,0,1),(0,2/EPS,2*GAM/EPS),(1,0,0))*liemat; l_G(1,1):=2*GAM*F5(3,2)/EPS-F5(3,3); l_G(1,2):=-F5(3,2);l_G(1,3):=-2*F5(3,1)/EPS; l_G(2,1):=0;l_G(2,2):=-EPS*F5(2,2)/2;l_G(2,3):=-F5(2,1); l_G(3,1):=0;l_G(3,2):=0;l_G(3,3):=-2;F5:=l_G>> ELSE <<liemat:=MAT((1/2,0,1/2),(0,1/EPS,GAM/EPS),(-1/2,0,1/2))*liemat; l_G(1,1):=-F5(3,1)/(2*EPS);l_G(1,2):=-F5(3,2)/2; l_G(1,3):=F5(3,1)/(2*EPS)-1; l_G(2,1):=F5(2,1)/2;l_G(2,2):=F5(2,2)*EPS/2; l_G(2,3):=-F5(2,1)/2;l_G(3,1):=F5(3,1)/(2*EPS)+1; l_G(3,2):=F5(3,2)/2;l_G(3,3):=-F5(3,1)/(2*EPS);F5:=l_G>>>>; IF NOT(NUMBERP(F5(1,2)**2+F5(1,3)*F5(2,2))) THEN <<WRITE "Is ",F5(1,2)**2+F5(1,3)*F5(2,2), "<0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN <<S:=F5(2,2)/ABS(F5(2,2)); TT:=ABS(F5(1,2)**2+F5(1,3)*F5(2,2)); R:=F5(1,1)-F5(1,2)*F5(2,1)/F5(2,2); ALPHA:=TT*(-R*R-((F5(2,1)/F5(2,2))**2+F5(3,1)/F5(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); l_G(1,1):=-Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(1,2):=-Q*S*SQRT(TT); l_G(1,3):=Q*S*SQRT(TT)*F5(1,2)/F5(2,2); l_G(2,1):=1/(S*SQRT(TT)); l_G(2,2):=0; l_G(2,3):=0; l_G(3,1):=Q*R; l_G(3,2):=0; l_G(3,3):=-Q*TT/F5(2,2); liemat:=l_G*liemat>> ELSE <<S:=F5(2,2)/ABS(F5(2,2)); TT:=F5(1,2)**2+F5(1,3)*F5(2,2); R:=F5(1,1)-F5(1,2)*F5(2,1)/F5(2,2); ALPHA:=TT*(R*R-((F5(2,1)/F5(2,2))**2+F5(3,1)/F5(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); IF NOT(NUMBERP(ALPHA)) THEN <<WRITE "Is ",ALPHA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*R; l_G(2,2):=0; l_G(2,3):=Q*TT/F5(2,2); l_G(3,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); liemat:=l_G*liemat>> ELSE <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); l_G(3,1):=Q*R; l_G(3,2):=0; l_G(3,3):=Q*TT/F5(2,2); liemat:=l_G*liemat>>>> ELSE IF ALPHA>0 THEN <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*R; l_G(2,2):=0; l_G(2,3):=Q*TT/F5(2,2); l_G(3,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); liemat:=l_G*liemat>> ELSE <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); l_G(3,1):=Q*R; l_G(3,2):=0; l_G(3,3):=Q*TT/F5(2,2); liemat:=l_G*liemat>>>>>> ELSE IF F5(1,2)**2+F5(1,3)*F5(2,2)<0 THEN <<S:=F5(2,2)/ABS(F5(2,2)); TT:=ABS(F5(1,2)**2+F5(1,3)*F5(2,2)); R:=F5(1,1)-F5(1,2)*F5(2,1)/F5(2,2); ALPHA:=TT*(-R*R-((F5(2,1)/F5(2,2))**2+F5(3,1)/F5(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); l_G(1,1):=-Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(1,2):=-Q*S*SQRT(TT); l_G(1,3):=Q*S*SQRT(TT)*F5(1,2)/F5(2,2); l_G(2,1):=1/(S*SQRT(TT)); l_G(2,2):=0; l_G(2,3):=0; l_G(3,1):=Q*R; l_G(3,2):=0; l_G(3,3):=-Q*TT/F5(2,2); liemat:=l_G*liemat>> ELSE <<S:=F5(2,2)/ABS(F5(2,2)); TT:=F5(1,2)**2+F5(1,3)*F5(2,2); R:=F5(1,1)-F5(1,2)*F5(2,1)/F5(2,2); ALPHA:=TT*(R*R-((F5(2,1)/F5(2,2))**2+F5(3,1)/F5(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); IF NOT(NUMBERP(ALPHA)) THEN <<WRITE "Is ",ALPHA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*R; l_G(2,2):=0; l_G(2,3):=Q*TT/F5(2,2); l_G(3,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); liemat:=l_G*liemat>> ELSE <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); l_G(3,1):=Q*R; l_G(3,2):=0; l_G(3,3):=Q*TT/F5(2,2); liemat:=l_G*liemat>>>> ELSE IF ALPHA>0 THEN <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*R; l_G(2,2):=0; l_G(2,3):=Q*TT/F5(2,2); l_G(3,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); liemat:=l_G*liemat>> ELSE <<l_G(1,1):=1/(S*SQRT(TT)); l_G(1,2):=0; l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*F5(2,1)/F5(2,2); l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*F5(1,2)/F5(2,2); l_G(3,1):=Q*R; l_G(3,2):=0; l_G(3,3):=Q*TT/F5(2,2); liemat:=l_G*liemat>>>>; if symbolic !*tr_lie then WRITE "[X,Y]=Z, [X,Z]=Y, [Y,Z]=X";lie_class:={liealg(3),comtab(8)}; CLEAR l_G,F5 END; algebraic procedure lie4(); BEGIN SCALAR LAM,JAC1,JAC2,JAC3,JAC4; INTEGER P1,M1,M2,M3,DIML1; MATRIX l_F(6,4); ARRAY ORDV(12); ORDV(1):=ORDV(3):=ORDV(7):=1;ORDV(2):=ORDV(5):=ORDV(9):=2; ORDV(4):=ORDV(6):=ORDV(11):=3;ORDV(8):=ORDV(10):=ORDV(12):=4; FOR I:=1:4 DO <<l_F(1,I):=CC(1,2,I);l_F(2,I):=CC(1,3,I);l_F(3,I):=CC(2,3,I); l_F(4,I):=CC(1,4,I);l_F(5,I):=CC(2,4,I);l_F(6,I):=CC(3,4,I); CC(1,1,I):=CC(2,2,I):=CC(3,3,I):=CC(4,4,I):=0; CC(2,1,I):=-l_F(1,I);CC(3,1,I):=-l_F(2,I);CC(3,2,I):=-l_F(3,I); CC(4,1,I):=-l_F(4,I);CC(4,2,I):=-l_F(5,I);CC(4,3,I):=-l_F(6,I)>>; FOR S:=1:4 DO <<JAC1:=FOR R:=1:4 SUM CC(1,2,R)*CC(R,3,S)+CC(2,3,R)*CC(R,1,S)+CC(3,1,R)*CC(R,2,S); JAC2:=FOR R:=1:4 SUM CC(1,2,R)*CC(R,4,S)+CC(2,4,R)*CC(R,1,S)+CC(4,1,R)*CC(R,2,S); JAC3:=FOR R:=1:4 SUM CC(1,3,R)*CC(R,4,S)+CC(3,4,R)*CC(R,1,S)+CC(4,1,R)*CC(R,3,S); JAC4:=FOR R:=1:4 SUM CC(2,3,R)*CC(R,4,S)+CC(3,4,R)*CC(R,2,S)+CC(4,2,R)*CC(R,3,S); IF (JAC1 NEQ 0 OR JAC2 NEQ 0 OR JAC3 NEQ 0 OR JAC4 NEQ 0 ) THEN S:=4>>; IF (JAC1 NEQ 0 OR JAC2 NEQ 0 OR JAC3 NEQ 0 OR JAC4 NEQ 0 )THEN <<clear l_F,ORDV,CC;symbolic rederr "not a Lie algebra">>; M1:=0; FOR S:=1:6 DO FOR TT:=1:4 DO IF l_F(S,TT) NEQ 0 THEN <<M1:=S;P1:=TT;S:=6;TT:=4>>; IF M1=0 THEN DIML1:=0 ELSE IF M1=6 THEN DIML1:=1 ELSE <<M2:=0; FOR S:=M1+1:6 DO <<LAM:=l_F(S,P1)/l_F(M1,P1); FOR TT:=1:4 DO IF l_F(S,TT) NEQ LAM*l_F(M1,TT) THEN <<M2:=S;S:=6;TT:=4>>>>; IF M2=0 THEN DIML1:=1 ELSE IF M2=6 THEN DIML1:=2 ELSE <<M3:=0; FOR S:=M2+1:6 DO IF NOT(DET(MAT((l_F(M1,2),l_F(M1,3),l_F(M1,4)), (l_F(M2,2),l_F(M2,3),l_F(M2,4)), (l_F(S,2),l_F(S,3),l_F(S,4))))=0 AND DET(MAT((l_F(M1,1),l_F(M1,3),l_F(M1,4)), (l_F(M2,1),l_F(M2,3),l_F(M2,4)), (l_F(S,1),l_F(S,3),l_F(S,4))))=0 AND DET(MAT((l_F(M1,1),l_F(M1,2),l_F(M1,4)), (l_F(M2,1),l_F(M2,2),l_F(M2,4)), (l_F(S,1),l_F(S,2),l_F(S,4))))=0 AND DET(MAT((l_F(M1,1),l_F(M1,2),l_F(M1,3)), (l_F(M2,1),l_F(M2,2),l_F(M2,3)), (l_F(S,1),l_F(S,2),l_F(S,3))))=0) THEN <<M3:=S;S:=6>>; IF M3=0 THEN DIML1:=2 ELSE DIML1:=3>>>>; IF DIML1=0 THEN <<if symbolic !*tr_lie then WRITE "Your Lie algebra is commutative"; lie_class:={liealg(4),comtab(0)}; liemat:=mat((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1))>> ELSE IF DIML1=3 THEN com43(ORDV(2*M1-1),ORDV(2*M1),ORDV(2*M2-1),ORDV(2*M2), ORDV(2*M3-1),ORDV(2*M3)) ELSE IF DIML1=1 THEN com41(ORDV(2*M1-1),ORDV(2*M1),P1) ELSE com42(ORDV(2*M1-1),ORDV(2*M1),ORDV(2*M2-1),ORDV(2*M2)); CLEAR ORDV,l_F END; algebraic procedure com41(I1,J1,P1); BEGIN SCALAR Y1,Y2,Y3,BETA1,BETA2,BETA3,BETA4,BETA5,BETA6; MATRIX liemat(4,4); FOR I:=1:4 DO liemat(1,I):=CC(I1,J1,I); IF P1=1 THEN <<Y1:=2;Y2:=3;Y3:=4>> ELSE IF P1=2 THEN <<Y1:=1;Y2:=3;Y3:=4>> ELSE IF P1=3 THEN <<Y1:=1;Y2:=2;Y3:=4>> ELSE <<Y1:=1;Y2:=2;Y3:=3>>; liemat(2,Y1):=liemat(3,Y2):=liemat(4,Y3):=1; BETA1:=(FOR L:=1:4 SUM CC(I1,J1,L)*CC(L,Y1,P1))/CC(I1,J1,P1); BETA2:=(FOR L:=1:4 SUM CC(I1,J1,L)*CC(L,Y2,P1))/CC(I1,J1,P1); BETA3:=CC(Y1,Y2,P1)/CC(I1,J1,P1); BETA4:=(FOR L:=1:4 SUM CC(I1,J1,L)*CC(L,Y3,P1))/CC(I1,J1,P1); BETA5:=CC(Y1,Y3,P1)/CC(I1,J1,P1); BETA6:=CC(Y2,Y3,P1)/CC(I1,J1,P1); IF (BETA1=0 AND BETA2=0 AND BETA3=0 AND BETA4=0 AND BETA5=0) THEN <<liemat:=MAT((1,0,0,0),(0,0,0,1),(0,0,1,0),(0,1,0,0))*liemat; BETA3:=-BETA6;BETA6:=0>> ELSE IF (BETA1=0 AND BETA2=0 AND BETA3=0) THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,0,1),(0,0,1,0))*liemat; BETA2:=BETA4;BETA3:=BETA5;BETA4:=BETA5:=0;BETA6:=-BETA6>>; IF (BETA1=0 AND BETA2=0) THEN <<liemat:=MAT((BETA3,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; Y1:=BETA4;Y2:=BETA5/BETA3;Y3:=BETA6/BETA3>> ELSE IF BETA1=0 THEN <<liemat:=MAT((1,0,0,0),(-BETA3/BETA2,1,0,0),(0,0,1/BETA2,0), (0,0,0,1))*liemat;Y1:=BETA4; Y2:=BETA5-BETA3*BETA4/BETA2;Y3:=BETA6/BETA2>> ELSE <<liemat:=MAT((1,0,0,0),(BETA3/BETA1,-BETA2/BETA1,1,0), (0,1/BETA1,0,0),(0,0,0,1))*liemat; Y1:=BETA4;Y2:=(BETA3*BETA4-BETA2*BETA5)/BETA1; Y3:=BETA5/BETA1>>; IF (BETA1=0 AND BETA2=0) THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,Y3,-Y2,1),(0,0,1,0))*liemat; if symbolic !*tr_lie then WRITE "[X,Z]=W";lie_class:={liealg(4),comtab(2)}>> ELSE <<IF Y1=0 THEN liemat:=MAT((1,0,0,0),(0,1,0,0),(-Y3,0,0,-1),(0,0,1,1))*liemat ELSE liemat:=MAT((1,0,0,0),(0,1,0,0),(-Y3/Y1,0,1,-1/Y1),(0,0,0,1/Y1))* liemat; if symbolic !*tr_lie then WRITE "[W,Z]=W";lie_class:={liealg(4),comtab(1)}>> END; algebraic procedure com42(I1,J1,I2,J2); BEGIN SCALAR D,D1,D2,D3,D4,A1,A2,A3,A4,A5,B1,B2,B3,B4,B5; MATRIX liemat(4,4); ARRAY SOL(1,4); FOR I:=1:4 DO <<liemat(1,I):=CC(I1,J1,I);liemat(2,I):=CC(I2,J2,I)>>; liemat(3,1):=liemat(4,2):=1;IF (D:=DET(liemat)) NEQ 0 THEN <<D1:=1;D2:=2;D3:=3;D4:=4>> ELSE <<liemat(4,2):=0;liemat(4,3):=1;IF (D:=DET(liemat)) NEQ 0 THEN <<D1:=1;D2:=3;D3:=2;D4:=4;D:=-D>> ELSE <<liemat(3,1):=0;liemat(3,2):=1;IF (D:=DET(liemat)) NEQ 0 THEN <<D1:=2;D2:=3;D3:=1;D4:=4>> ELSE <<liemat(3,2):=liemat(4,3):=0;liemat(3,1):=liemat(4,4):=1; IF (D:=DET(liemat)) NEQ 0 THEN <<D1:=1;D2:=4;D3:=2;D4:=3>> ELSE <<liemat(3,1):=0;liemat(3,2):=1;IF (D:=DET(liemat)) NEQ 0 THEN <<D1:=2;D2:=4;D3:=1;D4:=3;D:=-D>> ELSE <<liemat(3,2):=0;liemat(3,3):=1;D:=DET(liemat); D1:=3;D2:=4;D3:=1;D4:=2>> >>>>>>>>; A1:=FOR R:=1:4 SUM ( CC(I1,J1,R)*CC(R,D1,D3)*CC(I2,J2,D4)- CC(I1,J1,R)*CC(R,D1,D4)*CC(I2,J2,D3))/D; B1:=FOR R:=1:4 SUM (-CC(I1,J1,R)*CC(R,D1,D3)*CC(I1,J1,D4)+ CC(I1,J1,R)*CC(R,D1,D4)*CC(I1,J1,D3))/D; A2:=FOR R:=1:4 SUM ( CC(I2,J2,R)*CC(R,D1,D3)*CC(I2,J2,D4)- CC(I2,J2,R)*CC(R,D1,D4)*CC(I2,J2,D3))/D; B2:=FOR R:=1:4 SUM (-CC(I2,J2,R)*CC(R,D1,D3)*CC(I1,J1,D4)+ CC(I2,J2,R)*CC(R,D1,D4)*CC(I1,J1,D3))/D; A3:=FOR R:=1:4 SUM ( CC(I1,J1,R)*CC(R,D2,D3)*CC(I2,J2,D4)- CC(I1,J1,R)*CC(R,D2,D4)*CC(I2,J2,D3))/D; B3:=FOR R:=1:4 SUM (-CC(I1,J1,R)*CC(R,D2,D3)*CC(I1,J1,D4)+ CC(I1,J1,R)*CC(R,D2,D4)*CC(I1,J1,D3))/D; A4:=FOR R:=1:4 SUM ( CC(I2,J2,R)*CC(R,D2,D3)*CC(I2,J2,D4)- CC(I2,J2,R)*CC(R,D2,D4)*CC(I2,J2,D3))/D; B4:=FOR R:=1:4 SUM (-CC(I2,J2,R)*CC(R,D2,D3)*CC(I1,J1,D4)+ CC(I2,J2,R)*CC(R,D2,D4)*CC(I1,J1,D3))/D; A5:=( CC(D1,D2,D3)*CC(I2,J2,D4)-CC(D1,D2,D4)*CC(I2,J2,D3))/D; B5:=(-CC(D1,D2,D3)*CC(I1,J1,D4)+CC(D1,D2,D4)*CC(I1,J1,D3))/D; findcentre(A1,A2,A3,A4,A5,B1,B2,B3,B4,B5); IF NOTTRIV=0 THEN trivcent(A1,A2,A3,A4,A5,B1,B2,B3,B4,B5) ELSE IF (SOL(1,3)=0 AND SOL(1,4)=0) THEN IF SOL(1,1)=0 THEN <<liemat:=MAT((0,1,0,0),(1,0,0,0),(0,0,1,0),(0,0,0,1))*liemat; centincom(B1,B3,B5,A1,A3,A5)>> ELSE <<liemat:=MAT((1,SOL(1,2)/SOL(1,1),0,0),(0,1,0,0),(0,0,1,0), (0,0,0,1))*liemat;centincom(A2,A4,A5, B2-SOL(1,2)/SOL(1,1)*A2,B4-SOL(1,2)/SOL(1,1)*A4, B5-SOL(1,2)/SOL(1,1)*A5)>> ELSE IF DET(MAT((1,0,0,0),(0,1,0,0), (SOL(1,1),SOL(1,2),SOL(1,3),SOL(1,4)),(0,0,0,1)))=0 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0), (SOL(1,1),SOL(1,2),SOL(1,3),SOL(1,4)),(0,0,1,0))*liemat; centoutcom(A1,A2,B1,B2)>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0), (SOL(1,1),SOL(1,2),SOL(1,3),SOL(1,4)),(0,0,0,1))*liemat; centoutcom(A3,A4,B3,B4)>>; CLEAR SOL,NOTTRIV END; algebraic procedure findcentre(A1,A2,A3,A4,A5,B1,B2,B3,B4,B5); BEGIN INTEGER FLAG; SCALAR HELP; NOTTRIV:=0;FLAG:=0; CENT:=MAT((A1,A2,0,-A5),(A3,A4,A5,0),(B1,B2,0,-B5), (B3,B4,B5,0),(0,0,A1,A3),(0,0,A2,A4), (0,0,B1,B3),(0,0,B2,B4)); FOR I:=1:4 DO IF (CENT(I,1) NEQ 0 AND FLAG=0) THEN <<FLAG:=1;FOR J:=1:4 DO <<HELP:=CENT(1,J);CENT(1,J):=CENT(I,J);CENT(I,J):=HELP>>>>; IF FLAG=0 THEN <<NOTTRIV:=1;SOL(1,1):=1>> ELSE <<FOR I:=2:4 DO <<HELP:=CENT(I,1)/CENT(1,1); FOR J:=1:4 DO CENT(I,J):=CENT(I,J)-HELP*CENT(1,J)>>; FLAG:=0; FOR I:=2:4 DO IF (CENT(I,2) NEQ 0 AND FLAG=0) THEN <<FLAG:=1;FOR J:=2:4 DO <<HELP:=CENT(2,J);CENT(2,J):=CENT(I,J);CENT(I,J):=HELP>>>>; IF FLAG=0 THEN <<NOTTRIV:=1;SOL(1,1):=-CENT(1,2); SOL(1,2):=CENT(1,1)>> ELSE <<FOR I:=3:4 DO <<HELP:=CENT(I,2)/CENT(2,2); FOR J:=2:4 DO CENT(I,J):=CENT(I,J)-HELP*CENT(2,J)>>; FLAG:=0; FOR I:=3:8 DO IF (CENT(I,3) NEQ 0 AND FLAG=0) THEN <<FLAG:=1;FOR J:=3:4 DO <<HELP:=CENT(3,J);CENT(3,J):=CENT(I,J);CENT(I,J):=HELP>>>>; IF FLAG=0 THEN <<NOTTRIV:=1; SOL(1,1):=(CENT(1,2)*CENT(2,3)/CENT(2,2)-CENT(1,3))/CENT(1,1); SOL(1,2):=-CENT(2,3)/CENT(2,2);SOL(1,3):=1>> ELSE <<FOR I:=4:8 DO <<HELP:=CENT(I,3)/CENT(3,3); FOR J:=3:4 DO CENT(I,J):=CENT(I,J)-HELP*CENT(3,J)>>; FLAG:=0; FOR I:=4:8 DO IF (CENT(I,4) NEQ 0 AND FLAG=0) THEN <<FLAG:=1;CENT(4,4):=CENT(I,4)>>; IF FLAG=0 THEN <<NOTTRIV:=1; SOL(1,1):=(-(CENT(2,3)*CENT(3,4)/CENT(3,3)-CENT(2,4))* CENT(1,2)/CENT(2,2)+CENT(3,4)*CENT(1,3)/ CENT(3,3)-CENT(1,4))/CENT(1,1); SOL(1,2):=(CENT(2,3)*CENT(3,4)/CENT(3,3)-CENT(2,4))/ CENT(2,2); SOL(1,3):=-CENT(3,4)/CENT(3,3);SOL(1,4):=1>> >>>>>>; CLEAR CENT END; algebraic procedure centincom(A,C,E,B,D,F); BEGIN SCALAR V1,W1,V2,W2; IF C=0 THEN IF D=0 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,0,1),(0,0,1,0))*liemat; V1:=A;V2:=-E;W1:=B;W2:=-F>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,-B/D),(0,0,0,1))*liemat; V1:=C;V2:=E;W1:=D;W2:=F>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,-A/C),(0,0,0,1))*liemat; V1:=C;V2:=E;W1:=D;W2:=F>>; IF W1=0 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,-V2/W2,V1/W2,0),(0,0,0,1/V1))* liemat; if symbolic !*tr_lie then WRITE "[X,Z]=W, [Y,Z]=X";lie_class:={liealg(4),comtab(6)}>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,-W2/(W1*V2-W2*V1), W1*W1/(W1*V2-W2*V1),0),(0,0,0,1/W1))* MAT((1,0,0,0),(V1,W1,0,0),(0,0,1,0),(0,0,0,1))*liemat; if symbolic !*tr_lie then WRITE "[X,Z]=X, [Y,Z]=W";lie_class:={liealg(4),comtab(7)}>> END; algebraic procedure centoutcom(A,C,B,D); BEGIN INTEGER FLAG; SCALAR ALPHA,BETA; FLAG:=0; IF C NEQ 0 THEN <<liemat:=MAT((0,B-A*D/C,0,0),(1,-A/C,0,0),(0,0,1,0),(0,0,0,1))*liemat; ALPHA:=A+D;BETA:=B*C-A*D>> ELSE IF B NEQ 0 THEN <<liemat:=MAT((-A*D/B,0,0,0),(-D*D/B,D,0,0),(0,0,1,0),(0,0,0,1/D))* liemat; ALPHA:=1+A/D;BETA:=-A/D>> ELSE IF A NEQ D THEN <<liemat:=MAT((1,1,0,0),(1/A,1/D,0,0),(0,0,1,0),(0,0,0,1))*liemat; ALPHA:=A+D;BETA:=-A*D>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1/A))*liemat; FLAG:=1>>; IF FLAG=1 THEN <<if symbolic !*tr_lie then WRITE "[W,Z]=W, [X,Z]=X";lie_class:={liealg(4),comtab(10)}>> ELSE IF ALPHA=0 THEN <<liemat:=MAT((1,0,0,0),(0,SQRT(ABS(BETA)),0,0),(0,0,1,0), (0,0,0,1/SQRT(ABS(BETA))))*liemat; if symbolic !*tr_lie then WRITE "[W,Z]=",BETA/ABS(BETA),"X, [X,Z]=W"; if BETA>0 then lie_class:={liealg(4),comtab(11)} else lie_class:={liealg(4),comtab(8)}>> ELSE <<liemat:=MAT((1,0,0,0),(0,-ALPHA,0,0),(0,0,1,0), (0,0,0,-1/ALPHA))*liemat; if symbolic !*tr_lie then WRITE "[W,Z]=-W+",BETA/(ALPHA**2),"X, [X,Z]=W"; lie_class:={liealg(4),comtab(9),BETA/(ALPHA**2)}>> END; algebraic procedure trivcent(A1,A2,A3,A4,A5,B1,B2,B3,B4,B5); BEGIN INTEGER FLAG; SCALAR HE,HELP,ALPHA,BETA,C1,C2,C3,C4,C5, D1,D2,D3,D4,D5,P,E1,E2,E3,E4,E5,E6; IF (A1*B2-A2*B1)=0 THEN IF (A3*B4-A4*B3)=0 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,1),(0,0,0,1))*liemat; A1:=A1+A3;B1:=B1+B3;A2:=A2+A4;B2:=B2+B4>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,0,1),(0,0,1,0))*liemat; HELP:=A1;A1:=A3;A3:=HELP;HELP:=A2;A2:=A4;A4:=HELP; HELP:=B1;B1:=B3;B3:=HELP;HELP:=B2;B2:=B4;B4:=HELP; A5:=-A5;B5:=-B5>>; IF A2 NEQ 0 THEN <<ALPHA:=A1+B2;BETA:=A2*B1-A1*B2; IF ALPHA=0 THEN <<C1:=0;C2:=B1-A1*B2/A2;C3:=SQRT(ABS(BETA));C4:=-C3*A1/A2; C5:=1/C3;D1:=A1/(A2*C2);D2:=C5;D3:=1/C2;D4:=0;D5:=C3; IF NOT(NUMBERP(BETA)) THEN <<WRITE "Is ",BETA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN FLAG:=2 ELSE FLAG:=3>> ELSE IF BETA>0 THEN FLAG:=2 ELSE FLAG:=3>> ELSE <<C1:=0;C2:=B1-A1*B2/A2;C3:=-ALPHA;C4:=ALPHA*A1/A2; C5:=1/C3;D1:=A1/(A2*C2);D2:=C5;D3:=1/C2;D4:=0;D5:=C3; FLAG:=4;P:=BETA/(ALPHA*ALPHA)>>>> ELSE IF B1 NEQ 0 THEN <<ALPHA:=1+A1/B2;BETA:=-A1/B2; IF ALPHA=0 THEN <<C1:=-A1*B2/B1;C2:=0;C3:=-SQRT(ABS(BETA))*B2/B1;C4:=-C3*B1; C5:=1/C4;D1:=1/C1;D2:=0;D3:=-1/(A1*B2);D4:=C5;D5:=C4; IF NOT(NUMBERP(BETA)) THEN <<WRITE "Is ",BETA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN FLAG:=2 ELSE FLAG:=3>> ELSE IF BETA>0 THEN FLAG:=2 ELSE FLAG:=3>> ELSE <<C1:=-A1*B2/B1;C2:=0;C3:=ALPHA*B2/B1;C4:=-ALPHA*B2; C5:=1/C4;D1:=1/C1;D2:=0;D3:=-1/(A1*B2);D4:=C5;D5:=C4; FLAG:=4;P:=BETA/(ALPHA*ALPHA)>>>> ELSE IF A1 NEQ B2 THEN <<ALPHA:=A1+B2;BETA:=-A1*B2; IF ALPHA=0 THEN <<C1:=1;C2:=1;C3:=SQRT(ABS(BETA))/A1;C4:=SQRT(ABS(BETA))/B2; C5:=1/SQRT(ABS(BETA));HELP:=1/B2-1/A1;D1:=1/(B2*HELP); D2:=-C5/HELP;D3:=-1/(A1*HELP);D4:=-D2;D5:=1/C5; IF NOT(NUMBERP(BETA)) THEN <<WRITE "Is ",BETA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN FLAG:=2 ELSE FLAG:=3>> ELSE IF BETA>0 THEN FLAG:=2 ELSE FLAG:=3>> ELSE <<C1:=1;C2:=1;C3:=-ALPHA/A1;C4:=-ALPHA/B2;C5:=-1/ALPHA; HELP:=1/B2-1/A1;D1:=1/(B2*HELP);D2:=1/(ALPHA*HELP); D3:=-1/(A1*HELP);D4:=-D2;D5:=-ALPHA; FLAG:=4;P:=BETA/(ALPHA*ALPHA)>>>> ELSE <<C1:=1;C2:=0;C3:=0;C4:=1;C5:=1/A1; D1:=1;D2:=0;D3:=0;D4:=1;D5:=A1;FLAG:=1>>; liemat:=MAT((C1,C2,0,0),(C3,C4,0,0),(0,0,C5,0),(0,0,0,1))*liemat; E1:=D1*(C1*A3+C2*A4)+D3*(C1*B3+C2*B4); E2:=D2*(C1*A3+C2*A4)+D4*(C1*B3+C2*B4); E3:=D1*(C3*A3+C4*A4)+D3*(C3*B3+C4*B4); E4:=D2*(C3*A3+C4*A4)+D4*(C3*B3+C4*B4); E5:=C5*A5*D1+C5*B5*D3; E6:=C5*A5*D2+C5*B5*D4; IF FLAG=4 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,E1+E4,1),(0,0,1,0))*liemat; A1:=-E4;A2:=E1+E3+E4;A3:=-1;A4:=1;A5:=-E5; B1:=P*(E1+E4)+E2;B2:=E4;B3:=P;B4:=0;B5:=-E6>> ELSE IF FLAG=1 THEN IF (E1+E4=0) THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,0,1),(0,0,1,0))*liemat; A1:=E1;A2:=E3;A3:=1;A4:=0;A5:=-E5; B1:=E2;B2:=E4;B3:=0;B4:=1;B5:=-E6>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,E1+E4,-2),(0,0,0,1))*liemat; A1:=E4-E1;A2:=-2*E3;A3:=E1;A4:=E3;A5:=E5*(E1+E4); B1:=-2*E2;B2:=E1-E4;B3:=E2;B4:=E4;B5:=E6*(E1+E4)>>; IF (FLAG=1 OR FLAG=4) THEN IF A1*B2-A2*B1=0 THEN IF B1=0 THEN <<liemat:=MAT((A2,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; FLAG:=5;E1:=A3;E2:=B3*A2;E3:=A4/A2;E4:=B4;E5:=A5/A2; E6:=B5>> ELSE <<liemat:=MAT((A1,B1,0,0),(1,0,0,0),(0,0,1,0),(0,0,0,1))*liemat; FLAG:=5;E1:=(A1*B3+B1*B4)/B1; E2:=A1*A3+B1*A4-A1*(A1*B3+B1*B4)/B1;E3:=B3/B1; E4:=A3-A1*B3/B1;E5:=B5/B1;E6:=A5-B5*A1/B1>> ELSE <<IF A2 NEQ 0 THEN <<BETA:=A2*B1-A1*B2;C1:=0;C2:=B1-A1*B2/A2; C3:=SQRT(ABS(BETA));C4:=-C3*A1/A2;C5:=1/C3; D1:=A1/(A2*C2);D2:=C5;D3:=1/C2;D4:=0;D5:=C3>> ELSE IF B1 NEQ 0 THEN <<BETA:=-A1/B2;C1:=-A1*B2/B1;C2:=0; C3:=-SQRT(ABS(BETA))*B2/B1;C4:=-C3*B1;C5:=1/C4; D1:=1/C1;D2:=0;D3:=-1/(A1*B2);D4:=C5;D5:=C4>> ELSE <<BETA:=-A1*B2;C1:=1;C2:=1;C3:=SQRT(ABS(BETA))/A1; C4:=SQRT(ABS(BETA))/B2;C5:=1/SQRT(ABS(BETA)); HELP:=1/B2-1/A1;D1:=1/(B2*HELP);D2:=-C5/HELP; D3:=-1/(A1*HELP);D4:=-D2;D5:=1/C5>>; IF NOT(NUMBERP(BETA)) THEN <<WRITE "Is ",BETA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN FLAG:=2 ELSE FLAG:=3>> ELSE IF BETA>0 THEN FLAG:=2 ELSE FLAG:=3; liemat:=MAT((C1,C2,0,0),(C3,C4,0,0),(0,0,C5,0),(0,0,0,1))*liemat; E1:=D1*(C1*A3+C2*A4)+D3*(C1*B3+C2*B4); E2:=D2*(C1*A3+C2*A4)+D4*(C1*B3+C2*B4); E3:=D1*(C3*A3+C4*A4)+D3*(C3*B3+C4*B4); E4:=D2*(C3*A3+C4*A4)+D4*(C3*B3+C4*B4); E5:=C5*A5*D1+C5*B5*D3; E6:=C5*A5*D2+C5*B5*D4>>; IF FLAG=2 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(-E5/E1,-E6/E1,1,0), (0,0,-E2/E1,1/E1))*liemat; liemat:=MAT((1/2,1/2,0,0),(1/2,-1/2,0,0),(0,0,1/2,1/2), (0,0,-1/2,1/2))*liemat; if symbolic !*tr_lie then WRITE "[W,Y]=W, [X,Z]=X";lie_class:={liealg(4),comtab(3)}>> ELSE IF FLAG=3 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(-E5/E1,-E6/E1,1,0), (0,0,E2/E1,1/E1))*liemat; if symbolic !*tr_lie then WRITE "-[W,Y]=[X,Z]=X, [X,Y]=[W,Z]=W"; lie_class:={liealg(4),comtab(4)}>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(-E5/E1,-E6/E1,1,0), (0,0,-E3/E1,1/E1))*liemat; if symbolic !*tr_lie then WRITE "[X,Y]=[W,Z]=W, [X,Z]=X";lie_class:={liealg(4),comtab(5)}>>; END; algebraic procedure com43(I1,J1,I2,J2,I3,J3); BEGIN INTEGER LL; MATRIX liemat(4,4),BB(4,4),FF(3,3); ARRAY l_Z(4,4,3); FOR I:=1:4 DO <<CC(2,1,I):=-CC(1,2,I);CC(3,1,I):=-CC(1,3,I); CC(3,2,I):=-CC(2,3,I);CC(4,1,I):=-CC(1,4,I); CC(4,2,I):=-CC(2,4,I);CC(4,3,I):=-CC(3,4,I); CC(1,1,I):=CC(2,2,I):=CC(3,3,I):=CC(4,4,I):=0; liemat(1,I):=CC(I1,J1,I);liemat(2,I):=CC(I2,J2,I); liemat(3,I):=CC(I3,J3,I)>>; liemat(4,1):=1;IF DET(liemat) NEQ 0 THEN LL:=1 ELSE FOR J:=2:4 DO <<liemat(4,J-1):=0;liemat(4,J):=1; IF DET(liemat) NEQ 0 THEN <<LL:=J;J:=4>>>>; BB:=1/liemat; FOR I:=1:3 DO <<l_Z(1,2,I):=FOR R:=1:4 SUM FOR S:=1:4 SUM FOR TT:=1:4 SUM liemat(1,R)*liemat(2,S)*CC(R,S,TT)*BB(TT,I); l_Z(1,3,I):=FOR R:=1:4 SUM FOR S:=1:4 SUM FOR TT:=1:4 SUM liemat(1,R)*liemat(3,S)*CC(R,S,TT)*BB(TT,I); l_Z(2,3,I):=FOR R:=1:4 SUM FOR S:=1:4 SUM FOR TT:=1:4 SUM liemat(2,R)*liemat(3,S)*CC(R,S,TT)*BB(TT,I); l_Z(1,4,I):=FOR R:=1:4 SUM FOR TT:=1:4 SUM liemat(1,R)*CC(R,LL,TT)*BB(TT,I); l_Z(2,4,I):=FOR R:=1:4 SUM FOR TT:=1:4 SUM liemat(2,R)*CC(R,LL,TT)*BB(TT,I); l_Z(3,4,I):=FOR R:=1:4 SUM FOR TT:=1:4 SUM liemat(3,R)*CC(R,LL,TT)*BB(TT,I)>>; FOR I:=1:3 DO <<FF(1,I):=l_Z(1,2,I);FF(2,I):=l_Z(1,3,I);FF(3,I):=l_Z(2,3,I)>>; LL:=0; FOR I:=1:3 DO FOR J:=1:3 DO IF FF(I,J) NEQ 0 THEN <<LL:=1;I:=3;J:=3>>; IF LL=0 THEN comcom0() ELSE IF DET(FF)=0 THEN comcom1() ELSE comcom3(); CLEAR BB,FF,l_Z END; algebraic procedure comcom0(); BEGIN SCALAR HE,A1,B1,C1,A2,B2,C2,A3,B3,C3,AA1,BB1,CC1, AA2,BB2,CC2,AL1,BE1,GA1,AL2,BE2,GA2,R,S,P,Q; A1:=l_Z(1,4,1);B1:=l_Z(1,4,2);C1:=l_Z(1,4,3); A2:=l_Z(2,4,1);B2:=l_Z(2,4,2);C2:=l_Z(2,4,3); A3:=l_Z(3,4,1);B3:=l_Z(3,4,2);C3:=l_Z(3,4,3); IF (A3=0 AND B3=0) THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1/C3))*liemat; AL1:=A1/C3;BE1:=B1/C3;GA1:=C1/C3; AL2:=A2/C3;BE2:=B2/C3;GA2:=C2/C3>> ELSE <<IF (A3=0 AND B3 NEQ 0) THEN <<liemat:=MAT((0,B3,C3,0),(1,0,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA1:=B2+C3;BB1:=B3*A2;CC1:=B3*C2-B2*C3; AA2:=B1/B3;BB2:=A1;CC2:=C1-B1*C3/B3>> ELSE <<liemat:=MAT((A3,B3,C3,0),(0,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA1:=A1+B3*A2/A3+C3;BB1:=A3*B1-A1*B3-B3*B3*A2/A3+B3*B2; CC1:=A3*C1-A1*C3-B3*A2*C3/A3+B3*C2; AA2:=A2/A3;BB2:=B2-A2*B3/A3;CC2:=C2-A2*C3/A3>>; <<liemat:=MAT((1,0,0,0),(0,1,-AA2,0),(0,0,1,0),(0,0,0,1))*liemat; CC1:=CC1+BB1*AA2;CC2:=CC2+BB2*AA2;AA2:=0>>; IF (BB1=0 AND AA1=BB2 AND CC2 NEQ 0) THEN <<liemat:=MAT((0,0,1,0),(0,1,0,0),(1,-CC1/CC2,0,0),(0,0,0,1/AA1))* liemat; AL1:=0;BE1:=CC1/(AA1*CC2);GA1:=1/AA1; AL2:=CC2/AA1;BE2:=1;GA2:=0>> ELSE IF (BB1=0 AND AA1 NEQ BB2 AND CC2 NEQ 0) THEN <<A1:=1/(BB2-AA1);B1:=(BB2*AA1-BB2*BB2+CC1)/(CC2*(AA1-BB2)); liemat:=MAT((1,0,0,0),(0,1,0,0),(A1,B1,1,0),(0,0,0,1/BB2))*liemat; AL1:=(AA1-CC1*A1)/BB2;BE1:=-B1*CC1/BB2;GA1:=CC1/BB2; AL2:=-CC2*A1/BB2;BE2:=1-B1*CC2/BB2;GA2:=CC2/BB2>>ELSE IF(BB1=0 AND CC2=0) THEN <<liemat:=MAT((1,0,0,0),(0,0,1,0),(0,1,0,0),(0,0,0,1/BB2))*liemat; AL1:=AA1/BB2;BE1:=CC1/BB2;AL2:=1/BB2;GA1:=BE2:=GA2:=0>> ELSE <<R:=-AA1-BB2;S:=AA1*BB2-CC1;P:=S-R*R/3; Q:=2*R*R*R/27-S*R/3+BB2*CC1-BB1*CC2; C1:=(-Q/2+SQRT(Q*Q/4+P*P*P/27))**(1/3)+ (-Q/2-SQRT(Q*Q/4+P*P*P/27))**(1/3)-R/3; A1:=(C1-BB2)/BB1;B1:=(C1-BB2)*(C1-AA1)/BB1; liemat:=MAT((1,0,0,0),(0,0,1,0),(A1,1,B1,0),(0,0,0,1/C1))*liemat; AL1:=(AA1-A1*BB1)/C1;BE1:=(CC1-B1*BB1)/C1; GA1:=BB1/C1;AL2:=1/C1;BE2:=GA2:=0>>>>; IF GA2 NEQ 0 THEN <<liemat:=MAT((1,-GA1/GA2,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA1:=AL1-GA1*AL2/GA2;BB1:=BE1+AL1*GA1/GA2-AL2*GA1*GA1/ (GA2*GA2)-GA1*BE2/GA2;AA2:=AL2;BB2:=BE2+AL2*GA1/GA2;CC2:=GA2>> ELSE <<liemat:=MAT((0,1,0,0),(1,0,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA1:=BE2;BB1:=AL2;AA2:=BE1;BB2:=AL1;CC2:=GA1>>; IF (AA2=0 AND AA1-BB1-BB2=0 AND -AA1-BB1+BB2=0 AND CC2=0) THEN c0111(AA1,AA1) ELSE <<IF AA2=0 THEN IF (AA1-BB1-BB2) NEQ 0 THEN <<liemat:=MAT((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA2:=AA1-BB1-BB2;BB2:=BB1+BB2;AA1:=AA1-BB1>> ELSE IF (-AA1-BB1+BB2) NEQ 0 THEN <<liemat:=MAT((1,0,0,0),(-1,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA2:=-AA1-BB1+BB2;BB2:=BB2-BB1;AA1:=AA1+BB1>> ELSE <<liemat:=MAT((0,0,1,0),(0,1,0,0),(1,0,0,0),(0,0,0,1/AA1))*liemat; AA2:=CC2/AA1;BB2:=1;CC2:=0;AA1:=1/AA1>>; liemat:=MAT((1,-AA1/AA2,AA1*CC2/AA2,0),(0,1,0,0),(0,0,1,0), (0,0,0,1))*liemat; BE1:=BB1-AA1*BB2/AA2; AL2:=AA2;BE2:=AA1+BB2;GA2:=CC2-AA1*CC2; liemat:=MAT((1,0,0,0),(-BE2,BE1,0,0),(0,0,1,0),(0,0,0,1))*liemat; AA1:=BE2; AA2:=AL2*BE1;CC2:=GA2*BE1; IF (CC2 NEQ 0 AND AA2=(1-AA1)) THEN <<liemat:=MAT((1,0,0,0),(1,1,0,0),(0,0,CC2,0),(0,0,0,1))*liemat; AL1:=AA1-1; IF AL1=1 THEN <<if symbolic !*tr_lie then WRITE "[W,Z]=W+X, [X,Z]=X+Y, [Y,Z]=Y"; lie_class:={liealg(4),comtab(12)}>> ELSE <<liemat:=MAT((0,0,1,0),(0,1,0,0),(1,1/(AL1-1), 1/((AL1-1)*(AL1-1)),0),(0,0,0,1/AL1))*liemat; liemat:=MAT((0,1,0,0),(1/AL1,0,0,0),(0,0,1,0),(0,0,0,1))*liemat; if symbolic !*tr_lie then WRITE "[W,Z]=",1/AL1,"W+X, [X,Z]=",1/AL1,"X, [Y,Z]=Y"; lie_class:={liealg(4),comtab(15),1/AL1}>>>> ELSE <<IF CC2 NEQ 0 THEN liemat:=MAT((1,0,-CC2/(1-AA2-AA1),0),(0,1,(-1+AA1)* CC2/(1-AA2-AA1),0),(0,0,CC2/(1-AA2-AA1),0), (0,0,0,1))*liemat; liemat:=MAT((1,0,0,0),(AA1/2,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; R:=(AA1*AA1/4+AA2); IF R=0 THEN <<if symbolic !*tr_lie then WRITE "[W,Z]=",AA1/2,"W+X, [X,Z]=",AA1/2,"X, [Y,Z]=Y"; lie_class:={liealg(4),comtab(15),AA1/2}>> ELSE <<liemat:=MAT((SQRT(ABS(R)),0,0,0),(0,1,0,0),(0,0,1,0), (0,0,0,1))*liemat; IF NOT(NUMBERP(R)) THEN <<WRITE "Is ",R,"<0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0), (0,0,0,SQRT(ABS(1/R))))*liemat; S:=AA1/(2*SQRT(ABS(R))); if symbolic !*tr_lie then WRITE "[W,Z]=",S,"W+X, [X,Z]=-W+",S,"X, [Y,Z]=", SQRT(ABS(1/R)),"Y"; lie_class:={liealg(4),comtab(14),S,SQRT(ABS(1/R))}>> ELSE <<liemat:=MAT((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; liemat:=MAT((-2*SQRT(ABS(R)),SQRT(ABS(R)),0,0), (0,SQRT(ABS(R)),0,0),(0,0,1,0),(0,0,0,1))*liemat; <<c0111(AA1/2-SQRT(ABS(R)),AA1/2+SQRT(ABS(R)))>>>>>> ELSE IF R<0 THEN <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0), (0,0,0,SQRT(ABS(1/R))))*liemat; S:=AA1/(2*SQRT(ABS(R))); if symbolic !*tr_lie then WRITE "[W,Z]=",S,"W+X, [X,Z]=-W+",S,"X, [Y,Z]=", SQRT(ABS(1/R)),"Y"; lie_class:={liealg(4),comtab(14),S,SQRT(ABS(1/R))}>> ELSE <<liemat:=MAT((1,0,0,0),(1,1,0,0),(0,0,1,0),(0,0,0,1))*liemat; liemat:=MAT((-2*SQRT(ABS(R)),SQRT(ABS(R)),0,0), (0,SQRT(ABS(R)),0,0),(0,0,1,0),(0,0,0,1))*liemat; c0111(AA1/2-SQRT(ABS(R)),AA1/2+SQRT(ABS(R)))>>>> >>>> END; algebraic procedure c0111(MY,NY); BEGIN liemat:=MAT((0,0,1,0),(1,0,0,0),(0,1,0,0),(0,0,0,1))*liemat; if symbolic !*tr_lie then WRITE "[W,Z]=W, [X,Z]=",MY,"X, [Y,Z]=",NY,"Y"; lie_class:={liealg(4),comtab(13),MY,NY} END; ALGEBRAIC PROCEDURE COMCOM1(); BEGIN INTEGER II; SCALAR HE,A1,A2,A3,B2,B3,C2,C3,HELP; MATRIX A11(4,4),A22(4,4),A33(4,4),CCC(3,3); HELP:=0; FOR M:=1:3 DO FOR N:=1:3 DO IF FF(M,N) NEQ 0 THEN <<II:=M;M:=3;N:=3>>; A11:=MAT((1,0,0,0),(0,1,0,0),(FF(II,1),FF(II,2),FF(II,3),0), (0,0,0,1)); A22:=MAT((1,0,0,0),(0,0,1,0),(FF(II,1),FF(II,2),FF(II,3),0), (0,0,0,1)); A33:=MAT((0,1,0,0),(0,0,1,0),(FF(II,1),FF(II,2),FF(II,3),0), (0,0,0,1)); IF DET(A11) NEQ 0 THEN liemat:=A11*liemat ELSE IF DET(A22) NEQ 0 THEN liemat:=A22*liemat ELSE liemat:=A33*liemat; liemat:=MAT((0,0,1,0),(1,0,0,0),(0,1,0,0),(0,0,0,1))*liemat; A11:=1/liemat; FOR M:=1:3 DO FOR N:=1:3 DO CCC(M,N):=FOR I:=1:4 SUM FOR J:=1:4 SUM FOR K:=1:4 SUM liemat(M,I)*liemat(4,J)*CC(I,J,K)*A11(K,N); A1:=CCC(1,1);A2:=CCC(2,1);A3:=CCC(3,1);B2:=CCC(2,2); B3:=CCC(3,2);C2:=CCC(2,3);C3:=CCC(3,3); IF A1=0 THEN <<IF C2=0 THEN IF B3=0 THEN <<liemat:=MAT((1,0,0,0),(0,1,1,0),(0,0,1,0),(0,0,0,1))*liemat; A2:=A2+A3;C2:=-2*B2>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,B2/B3,0),(0,0,1,0),(0,0,0,1))*liemat; A2:=A2+A3*B2/B3;C2:=-3*B2*B2/B3;B2:=2*B2>>; HELP:=B2*B2+C2*B3;C3:=SQRT(ABS(HELP)); liemat:=MAT((C2/C3,0,0,0),(0,1,0,0),(0,B2/C3,C2/C3,0), (0,A3*C3/HELP,-A2*C3/HELP,C3/HELP))*liemat; if symbolic !*tr_lie then WRITE "[X,Y]=W, [X,Z]=",HELP/ABS(HELP),"Y, [Y,Z]=X"; if HELP>0 then lie_class:={liealg(4),comtab(19)} else lie_class:={liealg(4),comtab(20)}>> ELSE <<liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0), (0,2*A3/A1,-2*A2/A1,2/A1))*liemat; B2:=2*B2/A1;C2:=2*C2/A1;B3:=2*B3/A1;C3:=2*C3/A1; IF B3 NEQ 0 THEN <<liemat:=MAT((1,0,0,0),(0,1,(1-B2)/B3,0),(0,0,1,0),(0,0,0,1))*liemat; C2:=C2+(1-B2)*(C3-1)/B3;B2:=C3:=1; IF C2=0 THEN <<liemat:=MAT((-1,0,0,0),(0,0,1,0),(0,1,0,0),(0,0,0,1))*liemat; C2:=B3>> ELSE <<A1:=B3/ABS(B3);A2:=C2/ABS(C2);A3:=SQRT(ABS(B3*C2)); liemat:=MAT((1,0,0,0),(0,(ABS(B3/C2))**(1/4),0,0), (0,0,(ABS(C2/B3))**(1/4),0),(0,0,0,1))*liemat; IF A1=A2 THEN <<IF NOT(NUMBERP(A1)) THEN <<WRITE "Is ",A1,"<0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN A3:=-A3>> ELSE IF A1<0 THEN A3:=-A3; liemat:=MAT((1,0,0,0),(0,1,0,0),(0,1,1,0),(0,0,0,1))*liemat; B2:=1-A3;C2:=A3;C3:=A3+1>> ELSE <<HELP:=1; IF NOT(NUMBERP(A1)) THEN <<WRITE "Is ",A1,"<0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN liemat:=MAT((-1,0,0,0),(0,0,1,0),(0,1,0,0),(0,0,0,1))* liemat>> ELSE IF A1<0 THEN liemat:=MAT((-1,0,0,0),(0,0,1,0),(0,1,0,0),(0,0,0,1)) *liemat; if symbolic !*tr_lie then WRITE "[W,Z]=2W, [X,Y]=W, [X,Z]=X-",A3,"Y, ", "[Y,Z]=",A3,"X+Y";lie_class:={liealg(4),comtab(17),A3}>>>>>>; IF (HELP NEQ 1) THEN IF (C2=0 OR B2 NEQ C3) THEN <<IF (B2 NEQ C3) THEN liemat:=MAT((1,0,0,0),(0,1,C2/(B2-C3),0),(0,0,1,0),(0,0,0,1))* liemat; IF NOT(NUMBERP(B2)) THEN <<WRITE "Is ",B2,"<1 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN liemat:=MAT((-1,0,0,0),(0,0,1,0),(0,1,0,0),(0,0,0,1))*liemat; HELP:=B2;B2:=C3;C3:=HELP>> ELSE IF B2<1 THEN <<liemat:=MAT((-1,0,0,0),(0,0,1,0),(0,1,0,0),(0,0,0,1))*liemat; HELP:=B2;B2:=C3;C3:=HELP>>; if symbolic !*tr_lie then WRITE "[W,Z]=2W, [X,Y]=W, [X,Z]=",B2,"X, [Y,Z]=",C3,"Y"; lie_class:={liealg(4),comtab(16),B2-1}>> ELSE <<liemat:=MAT((1,0,0,0),(0,1/SQRT(ABS(C2)),0,0), (0,0,SQRT(ABS(C2)),0),(0,0,0,1))*liemat; IF NOT(NUMBERP(C2)) THEN <<WRITE "Is ",C2,"<0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN liemat:=MAT((-1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,1))* liemat>> ELSE IF C2<0 THEN liemat:=MAT((-1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,1))*liemat; if symbolic !*tr_lie then WRITE "[W,Z]=2W, [X,Y]=W, [X,Z]=X+Y, [Y,Z]=Y"; lie_class:={liealg(4),comtab(18)}>>>>; CLEAR A11,A22,A33,CCC END; algebraic procedure comcom3(); BEGIN INTEGER HELP; SCALAR HE,AL,BE,GA; MATRIX l_K(3,3),l_A(3,3); HELP:=0; l_K(1,1):=FF(1,2)**2+2*FF(1,3)*FF(2,2)+FF(2,3)**2; l_K(1,2):=-FF(1,1)*FF(1,2)+FF(1,3)*FF(3,2)- FF(2,1)*FF(1,3)+FF(2,3)*FF(3,3); l_K(1,3):=-FF(1,1)*FF(2,2)-FF(1,2)*FF(3,2)- FF(2,1)*FF(2,3)-FF(2,2)*FF(3,3); l_K(2,1):=l_K(1,2); l_K(2,2):=FF(1,1)**2-2*FF(1,3)*FF(3,1)+FF(3,3)**2; l_K(2,3):=FF(1,1)*FF(2,1)+FF(1,2)*FF(3,1)- FF(3,1)*FF(2,3)-FF(3,2)*FF(3,3); l_K(3,1):=l_K(1,3);l_K(3,2):=l_K(2,3); l_K(3,3):=FF(2,1)**2+2*FF(2,2)*FF(3,1)+FF(3,2)**2; IF NOT(NUMBERP(l_K(1,1)) AND NUMBERP(l_K(1,1)*l_K(2,2)-l_K(1,2)*l_K(2,1)) AND NUMBERP(DET(l_K))) THEN <<WRITE "Is ",-l_K(1,1),">0 and ", l_K(1,1)*l_K(2,2)-l_K(1,2)*l_K(2,1),">0 and ", -DET(l_K),">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN <<HELP:=1;lie4so3()>> ELSE lie4so21()>> ELSE IF (-l_K(1,1)>0 AND l_K(1,1)*l_K(2,2)-l_K(1,2)*l_K(2,1)>0 AND -DET(l_K)>0) THEN <<HELP:=1;lie4so3()>> ELSE lie4so21(); liemat:=MAT((l_A(1,1),l_A(1,2),l_A(1,3),0),(l_A(2,1),l_A(2,2), l_A(2,3),0), (l_A(3,1),l_A(3,2),l_A(3,3),0),(0,0,0,1))*liemat; BB:=1/liemat; AL:=FOR J:=1:4 SUM FOR K:=1:4 SUM FOR L:=1:4 SUM liemat(1,J)*liemat(4,K)*CC(J,K,L)*BB(L,2); BE:=FOR J:=1:4 SUM FOR K:=1:4 SUM FOR L:=1:4 SUM liemat(1,J)*liemat(4,K)*CC(J,K,L)*BB(L,3); GA:=FOR J:=1:4 SUM FOR K:=1:4 SUM FOR L:=1:4 SUM liemat(2,J)*liemat(4,K)*CC(J,K,L)*BB(L,3); IF HELP=1 THEN liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0),(GA,-BE,AL,1))*liemat ELSE liemat:=MAT((1,0,0,0),(0,1,0,0),(0,0,1,0),(GA,-BE,-AL,1))*liemat; IF HELP=1 THEN <<if symbolic !*tr_lie then WRITE "[W,X]=Y, [W,Y]=-X, [X,Y]=W"; lie_class:={liealg(4),comtab(21)}>> ELSE <<if symbolic !*tr_lie then WRITE "[W,X]=Y, [W,Y]=X, [X,Y]=W"; lie_class:={liealg(4),comtab(22)}>>; CLEAR l_K,l_A END; algebraic procedure lie4so3(); BEGIN SCALAR S,TT,Q,R,ALPHA; S:=FF(2,2)/ABS(FF(2,2)); TT:=ABS(FF(1,2)**2+FF(1,3)*FF(2,2)); R:=FF(1,1)-FF(1,2)*FF(2,1)/FF(2,2); ALPHA:=TT*(-R*R-((FF(2,1)/FF(2,2))**2+FF(3,1)/FF(2,2))*TT); Q:=1/SQRT(ALPHA); l_A(1,1):=1/(S*SQRT(TT));l_A(1,2):=l_A(1,3):=l_A(2,2):=0;l_A(2,1):=Q*R; l_A(2,3):=-Q*TT/FF(2,2);l_A(3,1):=-Q*S*SQRT(TT)*FF(2,1)/FF(2,2); l_A(3,2):=-Q*S*SQRT(TT);l_A(3,3):=Q*S*SQRT(TT)*FF(1,2)/FF(2,2) END; algebraic procedure lie4so21(); BEGIN SCALAR GAM,EPS,S,TT,Q,R,ALPHA; MATRIX l_G(3,3); l_A:=MAT((1,0,0),(0,1,0),(0,0,1)); IF FF(2,2)=0 THEN IF FF(1,3) NEQ 0 THEN <<l_A:=MAT((1,0,0),(0,0,1),(0,1,0)); l_G(1,1):=FF(2,1);l_G(1,2):=FF(2,3);l_G(1,3):=FF(2,2); l_G(2,1):=FF(1,1);l_G(2,2):=FF(1,3);l_G(2,3):=FF(1,2); l_G(3,1):=-FF(3,1);l_G(3,2):=-FF(3,3);l_G(3,3):=-FF(3,2);FF:=l_G>> ELSE IF FF(3,1) NEQ 0 THEN <<l_A:=MAT((0,1,0),(1,0,0),(0,0,1)); l_G(1,1):=-FF(1,2);l_G(1,2):=-FF(1,1);l_G(1,3):=-FF(1,3); l_G(2,1):=FF(3,2);l_G(2,2):=FF(3,1);l_G(2,3):=FF(3,3); l_G(3,1):=FF(2,2);l_G(3,2):=FF(2,1);l_G(3,3):=FF(2,3);FF:=l_G>> ELSE <<l_A:=MAT((1,0,1),(1,0,0),(0,1,0)); l_G(1,1):=-FF(2,3);l_G(1,2):=FF(2,3)-FF(2,1);l_G(1,3):=0; l_G(2,1):=-FF(3,3);l_G(2,2):=2*FF(1,1);l_G(2,3):=FF(1,2)-FF(3,2); l_G(3,1):=0;l_G(3,2):=FF(1,1);l_G(3,3):=FF(1,2);FF:=l_G>>; IF FF(1,2)**2+FF(1,3)*FF(2,2)=0 THEN <<GAM:=-FF(1,2)/FF(2,2);EPS:=FF(1,1)-FF(1,2)*FF(2,1)/FF(2,2); IF 1/4*(FF(3,2)**2+FF(3,1)*FF(2,2))-EPS*FF(2,2)/2=0 THEN <<l_A:=MAT((0,0,1),(0,2/EPS,2*GAM/EPS),(1,0,0))*l_A; l_G(1,1):=2*GAM*FF(3,2)/EPS-FF(3,3); l_G(1,2):=-FF(3,2);l_G(1,3):=-2*FF(3,1)/EPS; l_G(2,1):=0;l_G(2,2):=-EPS*FF(2,2)/2;l_G(2,3):=-FF(2,1); l_G(3,1):=l_G(3,2):=0;l_G(3,3):=-2;FF:=l_G>> ELSE <<l_A:=MAT((1/2,0,1/2),(0,1/EPS,GAM/EPS),(-1/2,0,1/2))*l_A; l_G(1,1):=-FF(3,1)/(2*EPS);l_G(1,2):=-FF(3,2)/2; l_G(1,3):=FF(3,1)/(2*EPS)-1; l_G(2,1):=FF(2,1)/2;l_G(2,2):=FF(2,2)*EPS/2; l_G(2,3):=-FF(2,1)/2;l_G(3,1):=FF(3,1)/(2*EPS)+1; l_G(3,2):=FF(3,2)/2;l_G(3,3):=-FF(3,1)/(2*EPS);FF:=l_G>>>>; IF NOT(NUMBERP(FF(1,2)**2+FF(1,3)*FF(2,2))) THEN <<WRITE "Is ",FF(1,2)**2+FF(1,3)*FF(2,2), "<0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE=y THEN <<S:=FF(2,2)/ABS(FF(2,2)); TT:=ABS(FF(1,2)**2+FF(1,3)*FF(2,2)); R:=FF(1,1)-FF(1,2)*FF(2,1)/FF(2,2); ALPHA:=TT*(-R*R-((FF(2,1)/FF(2,2))**2+FF(3,1)/FF(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); l_G(1,1):=-Q*S*SQRT(TT)*FF(2,1)/FF(2,2); l_G(1,2):=-Q*S*SQRT(TT);l_G(1,3):=Q*S*SQRT(TT)*FF(1,2)/FF(2,2); l_G(2,1):=1/(S*SQRT(TT));l_G(2,2):=l_G(2,3):=0; l_G(3,1):=Q*R;l_G(3,2):=0;l_G(3,3):=-Q*TT/FF(2,2);l_A:=l_G*l_A>> ELSE <<S:=FF(2,2)/ABS(FF(2,2)); TT:=FF(1,2)**2+FF(1,3)*FF(2,2); R:=FF(1,1)-FF(1,2)*FF(2,1)/FF(2,2); ALPHA:=TT*(R*R-((FF(2,1)/FF(2,2))**2+FF(3,1)/FF(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); IF NOT(NUMBERP(ALPHA)) THEN <<WRITE "Is ",ALPHA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE =y THEN <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*R;l_G(2,2):=0;l_G(2,3):=Q*TT/FF(2,2); l_G(3,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2);l_A:=l_G*l_A>> ELSE <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2); l_G(3,1):=Q*R;l_G(3,2):=0;l_G(3,3):=Q*TT/FF(2,2); l_A:=l_G*l_A>>>> ELSE IF ALPHA>0 THEN <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*R;l_G(2,2):=0;l_G(2,3):=Q*TT/FF(2,2); l_G(3,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2);l_A:=l_G*l_A>> ELSE <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2); l_G(3,1):=Q*R;l_G(3,2):=0;l_G(3,3):=Q*TT/FF(2,2);l_A:=l_G*l_A>> >>>> ELSE IF FF(1,2)**2+FF(1,3)*FF(2,2)<0 THEN <<S:=FF(2,2)/ABS(FF(2,2)); TT:=ABS(FF(1,2)**2+FF(1,3)*FF(2,2)); R:=FF(1,1)-FF(1,2)*FF(2,1)/FF(2,2); ALPHA:=TT*(-R*R-((FF(2,1)/FF(2,2))**2+FF(3,1)/FF(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); l_G(1,1):=-Q*S*SQRT(TT)*FF(2,1)/FF(2,2); l_G(1,2):=-Q*S*SQRT(TT);l_G(1,3):=Q*S*SQRT(TT)*FF(1,2)/FF(2,2); l_G(2,1):=1/(S*SQRT(TT));l_G(2,2):=l_G(2,3):=0; l_G(3,1):=Q*R;l_G(3,2):=0;l_G(3,3):=-Q*TT/FF(2,2); l_A:=l_G*l_A>> ELSE <<S:=FF(2,2)/ABS(FF(2,2)); TT:=FF(1,2)**2+FF(1,3)*FF(2,2); R:=FF(1,1)-FF(1,2)*FF(2,1)/FF(2,2); ALPHA:=TT*(R*R-((FF(2,1)/FF(2,2))**2+FF(3,1)/FF(2,2))*TT); Q:=1/SQRT(ABS(ALPHA)); IF NOT(NUMBERP(ALPHA)) THEN <<WRITE "Is ",ALPHA,">0 ? (y/n) and press <RETURN>"; HE:=SYMBOLIC READ(); IF HE =y THEN <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*R;l_G(2,2):=0;l_G(2,3):=Q*TT/FF(2,2); l_G(3,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2);l_A:=l_G*l_A>> ELSE <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2); l_G(3,1):=Q*R;l_G(3,2):=0;l_G(3,3):=Q*TT/FF(2,2); l_A:=l_G*l_A>>>> ELSE IF ALPHA>0 THEN <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*R;l_G(2,2):=0;l_G(2,3):=Q*TT/FF(2,2); l_G(3,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(3,2):=Q*S*SQRT(TT); l_G(3,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2);l_A:=l_G*l_A>> ELSE <<l_G(1,1):=1/(S*SQRT(TT));l_G(1,2):=l_G(1,3):=0; l_G(2,1):=Q*S*SQRT(TT)*FF(2,1)/FF(2,2);l_G(2,2):=Q*S*SQRT(TT); l_G(2,3):=-Q*S*SQRT(TT)*FF(1,2)/FF(2,2); l_G(3,1):=Q*R;l_G(3,2):=0;l_G(3,3):=Q*TT/FF(2,2);l_A:=l_G*l_A>>>>; CLEAR l_G END; endmodule; end;