{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(linalg, exponen tial):" }{TEXT -1 43 " This gives us useful linear algebra stuff." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eye:=Matrix(8,8,shape=identi ty): " }{TEXT -1 15 "Identity matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "one:=Column(eye,1): " }{TEXT -1 67 "We choose the ord ered basis \{1,i,j,k,l,il,jl,kl\} for the octonions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "i:=Column(eye,2):" }{TEXT -1 28 " It is g enerated by \{i,j,l\}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "j :=Column(eye,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "l:=Colu mn(eye,5):" }{TEXT -1 100 " We define multiplication according to Baez , with the unusual correspondence i = e4, j = e1, l = -e7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "mu:=(x,y)->Vector([\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 649 "x[1]*y[1]-x[2]*y[2]-x[3]*y[3]-x[4]*y[4]-x[5]*y[5 ]-x[6]*y[6]-x[7]*y[7]-x[8]*y[8],\nx[1]*y[2]+x[2]*y[1]+x[3]*y[4]-x[4]*y [3]+x[5]*y[6]-x[6]*y[5]-x[7]*y[8]+x[8]*y[7],\nx[1]*y[3]+x[3]*y[1]-x[2] *y[4]+x[4]*y[2]+x[5]*y[7]-x[7]*y[5]+x[6]*y[8]-x[8]*y[6],\nx[1]*y[4]+x[ 4]*y[1]+x[2]*y[3]-x[3]*y[2]+x[5]*y[8]-x[8]*y[5]-x[6]*y[7]+x[7]*y[6],\n x[1]*y[5]+x[5]*y[1]-x[2]*y[6]+x[6]*y[2]-x[3]*y[7]+x[7]*y[3]-x[4]*y[8]+ x[8]*y[4],\nx[1]*y[6]+x[6]*y[1]+x[2]*y[5]-x[5]*y[2]-x[3]*y[8]+x[8]*y[3 ]+x[4]*y[7]-x[7]*y[4],\nx[1]*y[7]+x[7]*y[1]+x[2]*y[8]-x[8]*y[2]+x[3]*y [5]-x[5]*y[3]-x[4]*y[6]+x[6]*y[4],\nx[1]*y[8]+x[8]*y[1]-x[2]*y[7]+x[7] *y[2]+x[3]*y[6]-x[6]*y[3]+x[4]*y[5]-x[5]*y[4]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "k:=mu(i,j):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "il:=mu(i,l):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "jl:=mu(j,l):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "kl:=m u(k,l):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ell:=a->curry(mu ,a);" }{TEXT -1 76 " Now we define l_a(x), r_a(x), and the bracket ope rator to match Fitzgerald." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ellGf *6#%\"aG6\"6$%)operatorG%&arrowGF(-%&curryG6$%#muG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r:=a->rcurry(mu,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%'rcurry G6$%#muG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "b:=(x,y )->x@y-y@x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6$%\"xG%\"yG6\" 6$%)operatorG%&arrowGF),&-%\"@G6$9$9%\"\"\"-F/6$F2F1!\"\"F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Der:=(x,y,z)->b(ell(x),ell(y ))(z)+ b(ell(x),r(y))(z)+ b(r(x),r(y))(z);" }{TEXT -1 19 "Assume g2 = \+ Der(O)." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DerGf*6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF*,(--%\"bG6$-%$ellG6#9$-F46#9%6#9&\"\"\"--F16$ F3-%\"rGF8F:F<--F16$-FAF5F@F:F " 0 "" {MPLTEXT 1 0 106 "g2:=(a,b)->;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2Gf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%4r table/ConstructRowG6*-%$DerG6%9$9%%$oneG-F16%F3F4%\"iG-F16%F3F4%\"jG-F 16%F3F4%\"kG-F16%F3F4%\"lG-F16%F3F4%#ilG-F16%F3F4%#jlG-F16%F3F4%#klGF) F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "h1:=ScalarMultiply( g2(j,il)+g2(i,jl),1/(-6));" }{TEXT -1 47 "g2 has a 2-dimensional torus so we find a basis" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h1G-%'RTABLE G6$\")w\\67-%'MATRIXG6#7*7*\"\"!F.F.F.F.F.F.F.7*F.F.F.F.F.F.\"\"\"F.7* F.F.F.F.F.F0F.F.F-F-7*F.F.!\"\"F.F.F.F.F.7*F.F3F.F.F.F.F.F.F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "h2:=ScalarMultiply(g2(j,il)+ g2(i,jl)-3*(g2(j,il)-g2(i,jl)),1/(-12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h2G-%'RTABLEG6$\")%e(>8-%'MATRIXG6#7*7*\"\"!F.F.F.F.F.F.F.7*F .F.F.F.F.F.\"\"\"F.F-7*F.F.F.F.F0F.F.F.7*F.F.F.!\"\"F.F.F.F.F-7*F.F3F. F.F.F.F.F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "h:=(t1,t2)- >ScalarMultiply(h1,t1)+ScalarMultiply(h2,t2);" }{TEXT -1 38 "a map to \+ a general toral element of g2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"h Gf*6$%#t1G%#t2G6\"6$%)operatorG%&arrowGF),&-_%.LinearAlgebraG%/ScalarM ultiplyG6$%#h1G9$\"\"\"-F/6$%#h2G9%F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Q:=JordanForm(h(t1,t2),output='Q');" }{TEXT -1 80 "We will need a maximal toral element of G2. To get such we will use t he exp map." }}{PARA 0 "" 0 "" {TEXT -1 70 "Since h is semisimple, we \+ diagonalize to make the computations easier." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'RTABLEG6$\"(O#QK-%'MATRIXG6#7*7*\"\"\"\"\"!F/F /F/F/F/F/7*F/F/F/F/F/#F.\"\"#F1F/7*F/F/F/^##!\"\"F2^#F1F/F/F/7*F/F1F1F /F/F/F/F/7*F/F7F4F/F/F/F/F/7*F/F/F/F1F1F/F/F/7*F/F/F/F/F/F7F4F/7*F.F/F /F/F/F/F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Ad:=(T,G) -> T.G.T^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AdGf*6$%\"TG%\"GG6 \"6$%)operatorG%&arrowGF)-%\".G6%9$9%*&\"\"\"F3F0!\"\"F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g2C:=(a,b)->Ad(Q^(-1),g2(a,b ));" }{TEXT -1 25 "transformed g2 lives in C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g2CGf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%#AdG6$ *&\"\"\"F1%\"QG!\"\"-%#g2G6$9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "hC:=(t1,t2)->simplify(Ad(Q^(-1),h(t1,t2)));" }{TEXT -1 39 "the corresponding general toral element" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hCGf*6$%#t1G%#t2G6\"6$%)operatorG%&arrowGF)-%)simpli fyG6#-%#AdG6$*&\"\"\"F4%\"QG!\"\"-%\"hG6$9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "hC(t1,t2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")cp&f\"-%'MATRIXG6#7*7*\"\"!F,F,F,F,F,F,F,7*F,*&^# \"\"\"F0%#t2GF0F,F,F,F,F,F,7*F,F,*&^#!\"\"F0F1F0F,F,F,F,F,7*F,F,F,*&F/ F0%#t1GF0F,F,F,F,7*F,F,F,F,*&F4F0F8F0F,F,F,7*F,F,F,F,F,,&F7F0F.F0F,F,7 *F,F,F,F,F,F,,&F:F0F3F0F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "TC:=(t1,t2)->simplify(Matrix(convert(exponential(hC(-I*ln(t1),-I*l n(t2))),'exp')));" }{TEXT -1 40 "corresponding maximal torus in the gr oup" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TCGf*6$%#t1G%#t2G6\"6$%)oper atorG%&arrowGF)-%)simplifyG6#-%'MatrixG6#-%(convertG6$-%,exponentialG6 #-%#hCG6$*&^#!\"\"\"\"\"-%#lnG6#9$F?*&F=F?-FA6#9%F?.%$expGF)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "TC(t1,t2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(+(RK-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F- F-F-F-7*F-%#t2GF-F-F-F-F-F-7*F-F-*&F,F,F/!\"\"F-F-F-F-F-7*F-F-F-%#t1GF -F-F-F-7*F-F-F-F-*&F,F,F4F2F-F-F-7*F-F-F-F-F-*&F4F,F/F,F-F-7*F-F-F-F-F -F-*&F,F,*&F4F,F/F,F2F-7*F-F-F-F-F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "f:=(X,i)->X[floor((i-1)/8)+1,(i-1 mod 8)+1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"XG%\"iG6\"6$%)operatorG%& arrowGF)&9$6$,&-%&floorG6#,&9%#\"\"\"\"\")#F7F8!\"\"F7F7F7,&-%$modG6$, &F5F7F7F:F8F7F7F7F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " toVector:=X->Vector(64,curry(f,X));" }{TEXT -1 73 "Maple 7 does not kn ow how to interpret 2 dimensional arrays as vectors..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)toVectorGf*6#%\"XG6\"6$%)operatorG%&arrowGF(-%' VectorG6$\"#k-%&curryG6$%\"fG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bv:=(a,b)->toVector(g2C(a,b));" }{TEXT -1 28 "We woul d like a basis of g2." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bvGf*6$%\" aG%\"bG6\"6$%)operatorG%&arrowGF)-%)toVectorG6#-%$g2CG6$9$9%F)F)F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bi:=curry(bv,i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#biGf*6\"F&6$%)operatorG%&arrowGF&-T$6$X*% )anythingG6\"F&[gl!#%!!!\")\")\"\"!\"\"\"F0F0F0F0F0F09\"F&F&6$%\"pG%#b vG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bj:=curry(bv,j);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bjGf*6\"F&6$%)operatorG%&arrowGF&-T $6$X*%)anythingG6\"F&[gl!#%!!!\")\")\"\"!F0\"\"\"F0F0F0F0F09\"F&F&6$% \"pG%#bvG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bk:=curry(bv,k );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bkGf*6\"F&6$%)operatorG%&arro wGF&-T$6$X*%)anythingG6\"F&[gl!#%!!!\")\")\"\"!F0F0\"\"\"F0F0F0F09\"F& F&6$%\"pG%#bvG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bl:=curry (bv,l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#blGf*6\"F&6$%)operatorG% &arrowGF&-T$6$X*%)anythingG6\"F&[gl!#%!!!\")\")\"\"!F0F0F0\"\"\"F0F0F0 9\"F&F&6$%\"pG%#bvG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "bil: =curry(bv,il);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$bilGf*6\"F&6$%)op eratorG%&arrowGF&-T$6$X*%)anythingG6\"F&[gl!#%!!!\")\")\"\"!F0F0F0F0\" \"\"F0F09\"F&F&6$%\"pG%#bvG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "bjl:=curry(bv,jl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$bjlGf*6\" F&6$%)operatorG%&arrowGF&-T$6$X*%)anythingG6\"F&[gl!#%!!!\")\")\"\"!F0 F0F0F0F0\"\"\"F09\"F&F&6$%\"pG%#bvG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "B:=Basis([\nbi(j),bi(k),bi(l),bi(il),bi(kl),\nbj(k), bj(l),bj(jl),bj(kl),\nbk(il),bk(jl),bk(kl),\nbl(il),bl(jl),bl(kl),\nbi l(jl),bil(kl),\nbjl(kl)]);" }{TEXT -1 59 "This is a good sign since we were hoping for 12 dimensions." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% \"BG7.-%'RTABLEG6*\")?Ck6%)anythingG&%'VectorG6#%'columnG%,rectangular G%.Fortran_orderG7\"\"\"\";F2\"#k-F'6*\")7[j7F*F+F/F0F1F2F3-F'6*\")#\\ \"*H\"F*F+F/F0F1F2F3-F'6*\"(?*[GF*F+F/F0F1F2F3-F'6*\")O%=W\"F*F+F/F0F1 F2F3-F'6*\")[+[6F*F+F/F0F1F2F3-F'6*\"(C/+$F*F+F/F0F1F2F3-F'6*\"(Okd#F* F+F/F0F1F2F3-F'6*\")!)4*G\"F*F+F/F0F1F2F3-F'6*\")g+U6F*F+F/F0F1F2F3-F' 6*\")!o,K\"F*F+F/F0F1F2F3-F'6*\")WY@7F*F+F/F0F1F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f2:=(X,i,j)->(X[(i-1)*8+j]);" }{TEXT -1 72 "We have to convert back to 2 dimensional arrays to interpret the r esult." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6%%\"XG%\"iG%\"jG6\" 6$%)operatorG%&arrowGF*&9$6#,(9%\"\")F3!\"\"9&\"\"\"F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "toMatrix:=X->Matrix(8,curry(f2,X)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)toMatrixGf*6#%\"XG6\"6$%)operat orG%&arrowGF(-%'MatrixG6$\"\")-%&curryG6$%#f2G9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for count from 1 to 12 do X[count]: =(Ad(TC(t1,t2),toMatrix(B[count]))) end do:" }{TEXT -1 47 "Act on the \+ basis elements by the maximal torus." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "SUM:=(X[1]+X[2]+X[3]+X[4]+X[5]+X[6]+X[7]+X[8]+X[9]+X[ 10]+X[11]+X[12]);" }{TEXT -1 34 "This indicates that the roots are:" } }{PARA 0 "" 0 "" {TEXT -1 78 "t2,t2/t1,t1t2,t1t2^2,1/t2,1/(t1t2),t1/t2 ,1/(t1t2^2),1/(t1^2t2),t1^2t2,1/t1,t1." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SUMG-%'RTABLEG6$\")wT-7-%'MATRIXG6#7*7*\"\"!F.F.F.F.F.F.F.7*, &*&^#\"\"#\"\"\"%$t2|irGF4F4*&\"\"%F4F5F4F4F.F.,$*&F5F4%$t1|irG!\"\"! \"$,&*&F:F4F5F4F;*(F2F4F:F4F5F4F4*&^#FF4*(FJF4F:F4F5F4F4F.F.,&*&^# F;F4F5F;F4*&F3F4F5F;F4*(FWF4)F:F3F4F5F4Fen7**&^#FEF4F:F;,&FNF4*(F2F4F: F;F5F;F4,$F9FXF.F.*(FAF4F:FKF5F;,&*&^#F4F4F5F4F4*&F3F4F5F4F4Fbo7*,&F>F K*(^#F7F4F:F4F5F4F4*&FAF4F:F4,&FCFgn*(FWF4F:F4FDF4F4,&*&F]oF4F5F4F4*&F 3F4F5F4F;*(FAF4F`oF4F5F4F.F.F]p7*,&FNFK*(^#!\"%F4F:F;F5F;F4,&FTFgn*(FA F4F:F;F5FKF4*&FWF4F:F;*(FWF4F:FKF5F;,&*&FjoF4F5F;F4*&F3F4F5F;F;F.F.Fhp 7*F.F`qFcpF@FYFdoFinF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "r ootSubGroup:=(a,b)-> Matrix(8,curry(rsghelper,toVector(SUM),a,b));" } {TEXT -1 49 "Despite the name, these are actually subalgebras." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-rootSubGroupGf*6$%\"aG%\"bG6\"6$%)o peratorG%&arrowGF)-%'MatrixG6$\"\")-%&curryG6&%*rsghelperG-%)toVectorG 6#%$SUMG9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "rsghe lper:=(X,a,b,i,j)->coeff(coeff((X[(i-1)*8+j]),t1,a),t2,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*rsghelperGf*6'%\"XG%\"aG%\"bG%\"iG%\"jG6 \"6$%)operatorG%&arrowGF,-%&coeffG6%-F16%&9$6#,(9'\"\")F:!\"\"9(\"\"\" %#t1G9%%#t2G9&F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p01 :=Ad(Q,rootSubGroup(0,1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "n10:=Ad(Q,rootSubGroup(-1,1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p12:=Ad(Q,rootSubGroup(1,1)):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "p13:=Ad(Q,rootSubGroup(1,2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "n01:=Ad(Q,rootSubGroup(0,-1)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "n12:=Ad(Q,rootSubGroup(-1,-1 )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p10:=Ad(Q,rootSubGro up(1,-1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "n13:=Ad(Q,roo tSubGroup(-1,-2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "n23:= Ad(Q,rootSubGroup(-2,-1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p23:=Ad(Q,rootSubGroup(2,1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "n11:=Ad(Q,rootSubGroup(-1,0)):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "p11:=Ad(Q,rootSubGroup(1,0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ad(Q,TC(t1,t2));" }{TEXT -1 67 "Thi s is the maximal torus of the real compact form of the group G2." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")Ca@<-%'MATRIXG6#7*7*\" \"\"\"\"!F-F-F-F-F-F-7*F-,&*&%$t1|irGF,%$t2|irGF,#F,\"\"#*(F3F,F1!\"\" F2F6F,F-F-F-F-,&*(^##F6F4F,F1F,F2F,F,*(^#F3F,F1F6F2F6F,F-7*F-F-,&F1F3* &F3F,F1F6F,F-F-,&*&F9F,F1F,F,*&FF-F-7*F-,&*(F " 0 "" {MPLTEXT 1 0 53 "Exp:=(x)->IdentityMatrix(8)+x+(1/2)*x.x+(1/6)*x. x.x:\n" }{TEXT -1 116 "Since each nXX and pXX are nilpotent, we check \+ that X^4 vanishes for all of them. Thus, we can simplify the exp map. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "chi:=(x,r)->Exp(ScalarM ultiply(x,r)):" }{TEXT -1 38 "These are the 1-psgs for the group G2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(p01,1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")sf5;-%'MATRIXG6#7*7*\"\"\"\"\"!F- F-F-F-F-F-7*F-F,^$#F,\"\"#!\"\"F-F-^$F2#F2F1F-F-7*F-^$F4F,F,F-F-F-F3F- 7*F-F-F-^$F4!\"#^$F1#!\"$F1F-F-^$F1F,7*F-F-F-F:^$#\"\"&F1F1F-F-^$F2F17 *F-^$F,F0F-F-F-F,F6F-7*F-F-FDF-F-F/F,F-7*F-F-F-^$F9F2^$F,F9F-F-F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(p10,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")7'4D\"-%'MATRIXG6#7*7*\"\"\"\"\"!F- F-F-F-F-F-7*F-F,F-F-F-F-F-F-7*F-F-F,^##!\"$\"\"#F1F-F-F-7*F-F-^##\"\"$ F3F,F-F1F-F-7*F-F-F6F-F,F5F-F-7*F-F-F-F6F0F,F-F-7*F-F-F-F-F-F-F,F-7*F- F-F-F-F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(p11,1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")#\\dc\"-%'MATRIXG 6#7*7*\"\"\"\"\"!F-F-F-F-F-F-7*F-F,F-^##!\"$\"\"#F0F-F-F-7*F-F-#!\"(F2 F-F-^##!\"*F2F-F17*F-^##\"\"$F2F-F,F-F-F0F-7*F-F;F-F-F,F-F:F-7*F-F-F6F -F-#\"#6F2F-^#F17*F-F-F-F;F/F-F,F-7*F-F-F " 0 "" {MPLTEXT 1 0 11 "chi(p12,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(SP]#-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F-F -F-7*F-^$#\"\"&\"\"#F2F-F-F-F-^$!\"##\"\"$F2^$!\"\"F27*F-F-F,^$F8#F8F2 ^$#F,F2F8F-F-F-7*F-F-^$F,F=F,F-^$F;F,F-F-7*F-F-F@F-F,F:F-F-7*F-F-F-F " 0 "" {MPLTEXT 1 0 11 "chi(p13,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")/=N7-%'MATRIXG6#7*7*\"\"\"\"\"!F-F- F-F-F-F-7*F-F,F-^$!\"$#\"\"$\"\"#^$#F0F3F0F-F-F-7*F-F-F,F-F-F-F-F-7*F- ^$F2F5F-F,F-F-^$F1F2F-7*F-F9F-F-F,F-F/F-7*F-F-F-F-F-F,F-F-7*F-F-F-F4F8 F-F,F-7*F-F-F-F-F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(p23,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")#p,i\"- %'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F-F-F-7*F-F,#!\"$\"\"#F-F-^#F/F-F-7*F- #\"\"$F1F,F-F-F-^#F4F-7*F-F-F-F,F-F-F-F-7*F-F-F-F-F,F-F-F-7*F-F6F-F-F- F,F/F-7*F-F-F2F-F-F4F,F-7*F-F-F-F-F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "chi(n10,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'R TABLEG6$\");k98-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F-F-F-7*F-F,F-F-F-F-F- F-7*F-F-F,^##\"\"$\"\"##!\"$F3F-F-F-7*F-F-^#F4F,F-F4F-F-7*F-F-F1F-F,F7 F-F-7*F-F-F-F1F0F,F-F-7*F-F-F-F-F-F-F,F-7*F-F-F-F-F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(n11,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")O**\\;-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F -F-F-7*F-F,F-^##\"\"$\"\"##!\"$F2F-F-F-7*F-F-#!\"(F2F-F-^##\"\"*F2F-F4 7*F-^#F3F-F,F-F-F3F-7*F-F0F-F-F,F-F " 0 "" {MPLTEXT 1 0 11 "chi(n12,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"( Gf\"G-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F-F-F-7*F-^$#\"\"&\"\"#!\"#F-F-F -F-^$F3#!\"$F2^$!\"\"F37*F-F-F,^$F8#F,F2^$F;F,F-F-F-7*F-F-^$F,#F8F2F,F -^$F?F8F-F-7*F-F-F@F-F,F:F-F-7*F-F-F-FF,F-F-7*F-F4F-F-F-F-^$F?F2^$F 3F,7*F-^$F,F2F-F-F-F-^$F2F8F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(n13,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")sH -;-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F-F-F-7*F-F,F-^$!\"$#F0\"\"#^$F1\" \"$F-F-F-7*F-F-F,F-F-F-F-F-7*F-^$F4#F4F2F-F,F-F-^$F8F0F-7*F-F9F-F-F,F- F/F-7*F-F-F-F-F-F,F-F-7*F-F-F-F3F7F-F,F-7*F-F-F-F-F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(n23,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")[(e@\"-%'MATRIXG6#7*7*\"\"\"\"\"!F-F-F-F -F-F-7*F-F,#!\"$\"\"#F-F-^##\"\"$F1F-F-7*F-F3F,F-F-F-^#F/F-7*F-F-F-F,F -F-F-F-7*F-F-F-F-F,F-F-F-7*F-F6F-F-F-F,F/F-7*F-F-F2F-F-F3F,F-7*F-F-F-F -F-F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "chi(n01,1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")oaV9-%'MATRIXG6#7*7*\" \"\"\"\"!F-F-F-F-F-F-7*F-F,^$#F,\"\"#F,F-F-^$!\"\"F0F-F-7*F-^$#F3F1F3F ,F-F-F-F2F-7*F-F-F-^$F6F1^$F1#\"\"$F1F-F-^$F1F37*F-F-F-F9^$#\"\"&F1!\" #F-F-^$F3FA7*F-^$F,F6F-F-F-F,F5F-7*F-F-FDF-F-F/F,F-7*F-F-F-^$FAF,^$F,F 1F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "70 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12114976 13197584 3238236 15956956 3239700 11642420 12634812 12991492 2848920 14418436 11480048 3000424 2576436 12890980 11420060 13201680 12214644 12024176 17215424 16105972 12509612 15657492 2503740 12351804 16201692 13146416 16499936 2815928 16022972 12158748 14435468 }{RTABLE M7R0 I5RTABLE_SAVE/12114976X,%)anythingG6"6"[gl!"%!!!#[o")")""!F'F'F'F'F'F'F'F'F'F'F 'F'F'!""F'F'F'F'F'F'F(F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'"""F'F'F'F'F'F'F) F'F'F'F'F'F'F'F'F'F'F'F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/13197584X,%)anythingG6"6"[gl!"%!!!#[o")")""!F'F'F'F'F'F'F'F'F'F'F 'F'F'!""F'F'F'F'F'F'F'F'F'F'F'F'F'F(F'F'F'F'F'F'"""F'F'F'F'F'F'F'F'F'F'F'F'F'F) F'F'F'F'F'F'F'F'F'F'F'F'F'F'F& } {RTABLE M7R0 I4RTABLE_SAVE/3238236X,%)anythingG6"6"[gl!"%!!!#[o")")"""""!F(F(F(F(F(F'F(F(F(# F'""#^#F)F(F(F(F(F(F(F)^##!""F*F(F(F(F(F(F,F(F(F)F(F(F(F(F+F(F(F)F(F(F(F)F(F(F( F(F+F(F(F)F(F(F(F(F,F(F(F(F(F(F(F(F(F'F& } {RTABLE M7R0 I5RTABLE_SAVE/15956956X,%)anythingG6"6"[gl!"%!!!#[o")")""!F'F'F'F'F'F'F'F'*&^#" ""F*%#t2GF*F'F'F'F'F'F'F'F'*&^#!""F*F+F*F'F'F'F'F'F'F'F'*&F)F*%#t1GF*F'F'F'F'F' F'F'F'*&F-F*F0F*F'F'F'F'F'F'F'F',&F/F*F(F*F'F'F'F'F'F'F'F',&F1F*F,F*F'F'F'F'F'F 'F'F'F'F& } {RTABLE M7R0 I4RTABLE_SAVE/3239700X,%)anythingG6"6"[gl!"%!!!#[o")")"""""!F(F(F(F(F(F(F(%#t2G F(F(F(F(F(F(F(F(*$F)!""F(F(F(F(F(F(F(F(%#t1GF(F(F(F(F(F(F(F(*$F,F+F(F(F(F(F(F(F (F(*&F,F'F)F'F(F(F(F(F(F(F(F(*&F,F+F)F+F(F(F(F(F(F(F(F(F'F& } {RTABLE M7R0 I5RTABLE_SAVE/11642420X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'^#!"#F'F 'F'F'F'F'F(^#""#F'F'F'F'F'F'F*F'F'F'F'F'^#"""^#""$F'F'F'F'F'F'^#!"$^#!""F'F'F'F 'F,F0F'F'F'F'F'F'F.F2F'F'F'F'F2F,F'F'F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/12634812X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'F' F'"""""$F'F'F'F'F'F'F)F(F'!"#F'F'F'F'F'F'F*F*F'F'F'F'F'F'F*F'!""!"$F'F'F'F'F'F' F,F+F'F'F'F'F'F'F'F'F(F(F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/12991492X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'F' F'^#!""^#!"$F'F'F'F'F'F'^#""$^#"""F'^#!"#F'F'F'F'F'F'F0^#""#F'F'F'F'F'F'F2F'F(F ,F'F'F'F'F'F'F*F.F'F'F'F'F'F'F'F'F(F.F'F'F'F& } {RTABLE M7R0 I4RTABLE_SAVE/2848920X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'""#F'F'F' F'F'F'F(F(F'F'F'F'F'F'F(F'F'F'F'F'"""""$F'F'F'F'F'F'F*F)F'F'F'F'!""!"$F'F'F'F'F 'F'F,F+F'F'F'F'F+F+F'F'F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/14418436X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'F' !"#F'F'F'F'F'F'F(F'F'F'F'F'F'""#F'F'F'F'F'F'F)F'F'F'F'F'F'!"%F'F'F'F'F'F'F*F*F' F'F'F'F'F'F*F'F'F'F'F'F)F)F'F& } {RTABLE M7R0 I5RTABLE_SAVE/11480048X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'^# !"$^#"""F'F'F'F'F'F'^#!""^#""$F'F'F'F'F(F,F'F'F'F'F'F'F*F.F'F'F'F'F'^#""#F'F'F' F'F'F'F0^#!"#F'F'F'F'F'F'F2F'F'F'F'F'F*F,F'F& } {RTABLE M7R0 I4RTABLE_SAVE/3000424X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'!"$ """F'F'F'F'F'F'F)F(F'F'F'F'""$!""F'F'F'F'F'F'F+F*F'F'F'F'F'""#F'F'F'F'F'F'F,F,F 'F'F'F'F'F'F,F'F'F'F'F'F+F+F'F& } {RTABLE M7R0 I4RTABLE_SAVE/2576436X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'""#F'F'F' F'F'F'F(F(F'F'F'F'F'F'F(F'F'F'F'F'"""!"$F'F'F'F'F'F'F*F)F'F'F'F'!""""$F'F'F'F'F 'F'F,F+F'F'F'F'F+F+F'F'F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/12890980X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'F' F'^#!"#F'F'F'F'F'F'F'F'^#""#F'^#!"%F'F'F'F'F'F'F,^#""%F'F'F'F'F'F'F.F'F(F'F'F'F 'F'F'F'F'F*F'F'F'F'F'F'F'F'F(F*F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/11420060X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'^# ""$^#"""F'F'F'F'F'F'^#!""^#!"$F'F'F'F'F(F,F'F'F'F'F'F'F*F.F'F'F'F'F'^#""#F'F'F' F'F'F'F0^#!"#F'F'F'F'F'F'F2F'F'F'F'F'F*F,F'F& } {RTABLE M7R0 I5RTABLE_SAVE/13201680X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'F'F'F'F' F'!""""$F'F'F'F'F'F'F)F(F'""#F'F'F'F'F'F'F*F*F'F'F'F'F'F'F*F'"""!"$F'F'F'F'F'F' F,F+F'F'F'F'F'F'F'F'F(F(F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/12214644X*%)anythingG6"6"[gl!#%!!!"[o"[o""!F'F'F'F'F'F'F'^#""%F'F 'F'F'F'F'F(^#!"%F'F'F'F'F'F'F*F'F'F'F'F'^#!"#F'F'F'F'F'F'F'F'^#""#F'F'F'F'F,F'F 'F'F'F'F'F'F'F.F'F'F'F'F.F,F'F'F'F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/12024176X,%)anythingG6"6"[gl!"%!!!#[o")")""!,&*&^#""#"""%$t2|irGF ,F,F-""%,&*&^#!"#F,F-!""F,*$F-F3F.*&^#!"'F,%$t1|irGF,*&^#""'F,F8F3,&*&F8F,F-F,F 2*(^#F.F,F8F,F-F,F,,&*&F8F3F-F3F2*(^#!"%F,F8F3F-F3F,F'F'F'F',$*&F-F3F8F,""$,&FA F,*(F*F,F8F3F-F3F,*&^#!"$F,F8F,,&*&F-F2F8F3F7*(FKF,F8F3F-F2F,,&*&^#F,F,F-F3F,F4 F2F'F'F',&F=F,*(F1F,F8F,F-F,F,,$*&F-F,F8F3FG,&*&F8F,F-F+F7*(^#FGF,F8F,F-F+F,*&F ZF,F8F3,&*&^#F3F,F-F,F,F-F2F',$FVFL,&FAF3*(F1F,F8F3F-F3F,F'F'Ffn*(FZF,F8F2F-F3* &FKF,F8F3F',&F=F3*(F*F,F8F,F-F,F,,$FFFLF'F'*(FKF,F8F+F-F,FP*&FZF,F8F,F'F]o,&FNF ;*(FZF,F-F2F8F3F,,&*&FhnF,F-F3F,F4F+*(FKF,F8F2F-F3F'F'FHF',&FXF;*(FKF,F-F+F8F,F ,Fbo*(FZF,F8F+F-F,,&*&FRF,F-F,F,F-F+F'F'FSF'F(F/F5F9F