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1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 296 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 297 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 300 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 301 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 302 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 303 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 116 " \+ Vessiot Tutorial: Computing normalizers subalgebras in Lie \+ pseudo-groups " }}{PARA 260 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" } {TEXT -1 88 " be a finite dimensional Lie algebra of vector fields on a manifold M and suppose that " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" }{TEXT -1 75 " is an subalgebra of an infinite dimensional Lie algebr a of vector fields " }{XPPEDIT 18 0 "Xi;" "6#%#XiG" }{TEXT -1 27 ". \+ More precisely, take " }{XPPEDIT 18 0 "Xi;" "6#%#XiG" }{TEXT -1 90 " to be an infinitesimal Lie pseudo-group so that coefficients of th e vector fields in " }{XPPEDIT 18 0 "Xi;" "6#%#XiG" }{TEXT -1 133 " a re constrainted by a system of linear PDE. \+ " }} {PARA 257 "" 0 "" {TEXT -1 85 "There are a variety of situations in \+ which one wants to compute the normalizer of " }{XPPEDIT 18 0 "Gamma; " "6#%&GammaG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "Xi;" "6#%#XiG" } {TEXT -1 3 " . " }}{PARA 257 "" 0 "" {TEXT -1 121 "In this tutorial w e derive the additional differential equations that are to be imp osed to compute the normalizer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 18 "Summary of Theory" }}{PARA 257 "" 0 " " {TEXT -1 32 "Let X_i, i=1..p be a basis for " }{XPPEDIT 18 0 "Gamma ;" "6#%&GammaG" }{TEXT -1 34 " . We seek vector fields N in " } {XPPEDIT 18 0 "Xi;" "6#%#XiG" }{TEXT -1 22 " such that [N,X_i] = " } {XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 21 "^j_i X_j, where t he " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 15 " are constan t. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 29 " Of course any vector N in " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" } {TEXT -1 82 " satisfies these equations so we shall exclude these so lutions in what follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Denote the " }{XPPEDIT 18 0 "Gamma;" "6#%&Ga mmaG" }{TEXT -1 40 " invariant vectors in the distribution " } {XPPEDIT 18 0 "Gamma;" "6#%&GammaG" }{TEXT -1 25 " by V_b. Denote \+ the " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" }{TEXT -1 60 " invariant i nvariant vector fields transverse to by W_c. " }{TEXT 262 103 "The \+ following algorithm assumes that there is a complete set of transv erse invariant vector fields." }{TEXT 274 1 " " }{TEXT -1 148 "Under t his assumption it is easy to check that it suffices to solve the n ormalizer equations for vector fields N which lie in the distribution " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 48 "By taking bracket s we find that the matrices " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG " }{TEXT -1 59 " must be infinitesimal automorphisms of the Lie alge bra " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" }{TEXT -1 7 ". The " } {XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 53 " which are inner \+ automorphisms correspond to N in " }{XPPEDIT 18 0 "Gamma;" "6#%&Gamm aG" }{TEXT -1 39 " and hence we need only consider those " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 48 " which are \"outer\" automo rphisms. The command " }{TEXT 258 33 "infinitesimal_outer_automorphi sms" }{TEXT -1 119 " returns a set of matrices which are infinitesima l automorphisms which are not in the span in the inner automorphisms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 17 " Pi ck functions " }{XPPEDIT 18 0 "upsilon;" "6#%(upsilonG" }{TEXT -1 17 "^i_a such that " }{XPPEDIT 18 0 "upsilon;" "6#%(upsilonG" }{TEXT -1 65 "^i_aX_i =0. Then for each point p in M the vector fields Z_ a=" }{XPPEDIT 18 0 "upsilon;" "6#%(upsilonG" }{TEXT -1 106 "^i_a(p)X_i form a basis for the infinitesimal isotropy subalgebra at p. Fr om the equation [N,X_i] =" }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" } {TEXT -1 37 "^j_i X_j we deduce that the vector " }{XPPEDIT 18 0 "up silon;" "6#%(upsilonG" }{TEXT -1 7 "^i_a(p)" }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 135 "^j_i X_j must be in the image of the lin ear istropy representation for Z_a. This places additional constra ints on the matrices " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" } {TEXT -1 142 ". The next step is is to find a basis of outer auto morphism matrices which satisfy these constraints. This is done w ith the command " }{TEXT 259 41 "istotropy_compatible_outer_automorphi sms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 18 "For each matrix " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 71 " found in the previous step find a vector N0 such that [N0,Z _a] =" }{XPPEDIT 18 0 "upsilon;" "6#%(upsilonG" }{TEXT -1 7 "^i_a(p)" }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 42 "^j_i X_j. This i s done with the command " }{TEXT 260 24 "normalizer_distribution0" } {TEXT -1 118 ". Every vector in the normalizer can be expressed in \+ the form N= AN0 +B^bV_b, where the A and B^b are functions. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "The com mand " }{TEXT 261 25 "normalizer_distribution " }{TEXT -1 110 " imple ments all the above steps. It returns a list of pairs of vector fie lds and outer automorphisms [[N0, " }{XPPEDIT 18 0 "Lambda;" "6#%'Lamb daG" }{TEXT -1 7 " ],...]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 98 "We can now substitute into the defining equa tions for N to get a system of PDE for A and B. Use " }{TEXT 263 14 " normalizer_PDE" }{TEXT 256 0 "" }{TEXT -1 219 ". For the time being we shall content ourselves to analysis these equations manually. We ho pe to be able to integrate these PDE with the EDS package in the ne ar future. Under Maple 7, try the Maple PDE package. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 58 "the normalizer \+ is then given by the vector fields in " }{XPPEDIT 18 0 "Gamma;" "6 #%&GammaG" }{TEXT -1 47 ", the invariant vector fields transverse t o " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" }{TEXT -1 76 " and the vecto r fields determined by the two commands determined from " }{TEXT 275 23 "normalizer_distribution" }{TEXT -1 6 " and " }{TEXT 276 16 "n ormalizer_PDE. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "SLxSR \+ > " 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(Koszul):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(Vessiot_library):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "with(eds);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%-check_Cauchy G%-derived_flagG%/normalizer_PDEG%8normalizer_distributionG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(tensors):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT -1 23 "Example 1. Petrov 32.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "co ord_init([x1,x2,x3,x4],[],petrov);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %3frame~name:~petrovG" }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 44 "Gamma:=Lie_lib(petrov,[32,3],[x1,x2,x3,x4]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gq*This~group~action~is~identical~to~Petrov~32.19,~via ~the~transformation~x1=y2~~~~~x2=y3~~~~~x3=-y1~~~~~x4=y4G" }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 12 "Show(Gamma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%%D_x2G6#%!G&%%D_x3G6#%!G,&&%%D_x1G6#%!G!\"\"*&%# x3G\"\"\"&F%6#%!GF4F4,&*&%#x1GF4&F.6#%!GF4F1*&F3\"\"\"&F)6#%!GF4F4" }} }{EXCHG {PARA 262 "" 0 "" {TEXT 266 9 "Step 1: " }}{PARA 263 "" 0 "" {TEXT 267 35 "Find the invariant vector fields. " }}{PARA 264 "" 0 " " {TEXT 268 75 "We are often able to do this using the isotropy_inva riant_tensor command." }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 76 "invVectors:=isotropy_invariant_tensors(Gamma,frameBaseVectors(),[a ,b,c,d]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+invVectorsG7$7$7%%%ve ctG%'petrovG7\"7#7$7#\"\"%\"\"\"7$F'7#7$7#\"\"#F/" }}}{EXCHG {PARA 0 " petrov>" 0 "" {MPLTEXT 1 0 8 "Show(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%%D_x4G6#%!G&%%D_x2G6#%!G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 68 "The vector D_x4 is the invariant vector tranverse to the orb its of" }{TEXT 269 2 " " }{XPPEDIT 270 0 "Gamma;" "6#%&GammaG" } {TEXT 271 3 ", " }{TEXT 265 63 "the vector D_x2 is an invariant vect or tangent to the orbits." }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT 272 7 "Step 2:" }}{PARA 266 "" 0 "" {TEXT 273 109 "Create the associated abstract Lie algebra and find a repre sentative basis for the outer automorphisms. " }}}{EXCHG {PARA 0 "p etrov>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 39 "alg_data:=vect_to_Lie_alg(Gamma,p_alg);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%)alg_dataG7$7%%(Lie_algG%&p_algG7#\"\"%7%7$7% \"\"#\"\"$\"\"\"F07$7%F.F*F.F07$7%F/F*F/!\"\"" }}}{EXCHG {PARA 0 "petr ov>" 0 "" {MPLTEXT 1 0 23 "Lie_alg_init(alg_data):" }}}{EXCHG {PARA 0 "p_alg>" 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(%!G%\"|grG&%#e1G6#%!G&%#e2G6#%!G &%#e3G6#%!G&%#e4G6#%!G7(F(%$---G%%----GF" 0 "" {MPLTEXT 1 0 41 "Out:=infinitesimal_outer_automorphisms();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$OutG7$-%'matrixG6#7&7&\"\"#\"\"!F,F,7&F,\"\"\"F,F,7& F,F,F.F,7&F,F,F,F,-F'6#7&7&F,F,F,F.F0F0F0" }}}{EXCHG {PARA 0 "p_alg>" 0 "" {MPLTEXT 1 0 53 "N0:=normalizer_distribution(Gamma,Out,[x1,x2,x3, x4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N0G7#7$7$7%%%vectG%'petrov G7\"7$7$7#\"\"\",$%#x1G!\"\"7$7#\"\"$,$%#x3GF2-%'matrixG6#7&7&\"\"#\" \"!F>F>7&F>F/F>F>7&F>F>F/F>7&F>F>F>F>" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 277 44 "The normalizer_PDE command has arguments: " }}{PARA 269 "" 0 "" {TEXT 278 53 "normalizer_PDE(N,InvV,Gamma, ind_var, dep_va rs,flag) " }}{PARA 270 "" 0 "" {TEXT 279 6 "where " }}{PARA 271 "" 0 " " {TEXT 280 97 "-N: a vector matrix pair which is the out put of the normalizer_distribution program " }}{PARA 272 "" 0 "" {TEXT 281 50 "-InvV: the invariant vector fields in Gamma" }} {PARA 273 "" 0 "" {TEXT 282 44 "-Gamma: the Lie algebra of vector fi elds. " }}{PARA 274 "" 0 "" {TEXT 283 74 "-ind_var: a list of indep endent variables to be used in the dep_vars. " }}{PARA 275 "" 0 "" {TEXT 284 145 "-dep_vars: a list of dependent variables, the first is the coefficient of the vector in N, the rest are the coefficients of \+ the vectors in InvV." }}{PARA 276 "" 0 "" {TEXT 285 161 "-flag: \+ an optional argument -- if set to _ExpSolve then the defining pde 's are solved for all derivatives and also for the coeff of thevect or in N." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " petrov > " 0 "" {MPLTEXT 1 0 66 "normalizer_PDE(N0[1],[vect(x2)],Gamma ,[x1,x2,x3],[a,b],_ExpSolve);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)/-% %diffG6$-%\"aG6%%#x1G%#x2G%#x3GF-\"\"!/-F&6$F(F,F./-F&6$-%\"bGF*F+F./- F&6$F5F-F./F(\"\"\"/-F&6$F5F,!\"#/-F&6$F(F+F." }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT 286 32 "The solution is \+ b = -2*x2, a=1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "pe trov > " 0 "" {MPLTEXT 1 0 46 "N1:=v_zip([1,-2*x2],[N0[1][1],vect(x2)] ,plus):" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 9 "show(N1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#x1G\"\"\"&%%D_x1G6#%!GF&!\" \"*&%#x2GF&&%%D_x2G6#%!GF&!\"#*&%#x3GF&&%%D_x3G6#%!GF&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 78 "We check the answer by computing the s tructure equations for the normalizer." }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 41 "L:=vect_to_Lie_alg([N1, op(Gamma)],norm):" }}} {EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~normG" }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#%!G&%#e2G6#%! G&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%!G7)F(%$---G%%----GF@F@F@F@7)&F+6#%!GF )\"\"!,$&F/6#%!G\"\"#&F36#%!G&F76#%!GFE7)&F/6#%!GF),$FG!\"#FEFEFEFE7)& F36#%!GF),$FK!\"\"FEFE&F/6#%!G&F36#%!G7)&F76#%!GF),$FNFfnFE,$FgnFfnFE, $&F76#%!GFfn7)&F;6#%!GF)FEFE,$FjnFfnFdoFE" }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 256 "" 0 "" {TEXT -1 23 "Example 2. Petrov 32.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 36 "coord_ini t([x1,x2,x3,x4],[],petrov);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame ~name:~petrovG" }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 45 "Gamm a:=Lie_lib(petrov,[32,11],[x1,x2,x3,x4]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\r*This~group~action~is~identical~to~Petrov~32.27~(e= -1),~via~the~transformation~x1=y1~~~~~x2=y2~~~~x3=y3~~~~~x4=y4G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'%Aand~rearanging~the~vector~fieldsG/&% \"XG6#\"\"\"&%\"YG6#\"\"#/&F&F+&F*6#\"\"$/&F&F0,$&F*F'!\"\"/&F&6#\"\"% ,$&F*F9F6" }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 12 "Show(Gamm a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%%D_x2G6#%!G&%%D_x3G6#%!G,$& %%D_x1G6#%!G!\"\",&*&%#x3G\"\"\"&F%6#%!GF5F5*&%#x2GF5&F)6#%!GF5F5" }}} {EXCHG {PARA 287 "" 0 "" {TEXT 301 9 "Step 1: " }}{PARA 288 "" 0 "" {TEXT 302 35 "Find the invariant vector fields. " }}{PARA 289 "" 0 " " {TEXT 303 75 "We are often able to do this using the isotropy_inva riant_tensor command." }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 76 "invVectors:=isotropy_invariant_tensors(Gamma,frameBaseVectors(),[a ,b,c,d]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+invVectorsG7$7$7%%%ve ctG%'petrovG7\"7#7$7#\"\"\"F.7$F'7#7$7#\"\"%F." }}}{EXCHG {PARA 0 "pet rov>" 0 "" {MPLTEXT 1 0 8 "Show(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$&%%D_x1G6#%!G&%%D_x4G6#%!G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 68 "The vector D_x4 is the invariant vector tranverse to the orbits of " }{TEXT 304 2 " " }{XPPEDIT 18 0 "Gamma;" "6#%&GammaG" }{TEXT 305 3 ", " }{TEXT 300 63 "the vector D_x2 is an invariant vector tangent \+ to the orbits." }}}{EXCHG {PARA 292 "" 0 "" {TEXT -1 0 "" }}{PARA 290 "" 0 "" {TEXT 306 7 "Step 2:" }}{PARA 291 "" 0 "" {TEXT 307 109 "Creat e the associated abstract Lie algebra and find a representative bas is for the outer automorphisms. " }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 39 "al g_data:=vect_to_Lie_alg(Gamma,p_alg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)alg_dataG7$7%%(Lie_algG%&p_algG7#\"\"%7$7$7%\"\"\"F*\"\"#F.7$7%F /F*F.F." }}}{EXCHG {PARA 0 "petrov>" 0 "" {MPLTEXT 1 0 23 "Lie_alg_ini t(alg_data):" }}}{EXCHG {PARA 0 "p_alg>" 0 "" {MPLTEXT 1 0 25 "Lie_bra cket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7( %!G%\"|grG&%#e1G6#%!G&%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G7(F(%$---G%%---- GF " 0 "" {MPLTEXT 1 0 41 "Out:=infinitesimal_outer_automorphisms();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$OutG7%-%'matrixG6#7&7&\"\"\"\"\"!F, F,7&F,F+F,F,7&F,F,F,F,F.-F'6#7&F.F.7&F,F,F,F+F.-F'6#7&F.F.7&F,F,F+F,F. " }}}{EXCHG {PARA 0 "p_alg>" 0 "" {MPLTEXT 1 0 53 "N0:=normalizer_dist ribution(Gamma,Out,[x1,x2,x3,x4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#N0G7$7$7$7%%%vectG%'petrovG7\"7$7$7#\"\"#,$%#x2G!\"\"7$7#\"\"$,$%#x 3GF2-%'matrixG6#7&7&\"\"\"\"\"!F>F>7&F>F=F>F>7&F>F>F>F>F@7$7$F(7#7$7#F =F>-F96#7&F@F@7&F>F>F=F>F@" }}}{EXCHG {PARA 293 "" 0 "" {TEXT 308 44 " The normalizer_PDE command has arguments: " }}{PARA 294 "" 0 "" {TEXT 309 53 "normalizer_PDE(N,InvV,Gamma, ind_var, dep_vars,flag) " } }{PARA 295 "" 0 "" {TEXT 310 6 "where " }}{PARA 296 "" 0 "" {TEXT 311 97 "-N: a vector matrix pair which is the output of the n ormalizer_distribution program " }}{PARA 297 "" 0 "" {TEXT 312 50 "-In vV: the invariant vector fields in Gamma" }}{PARA 298 "" 0 "" {TEXT 313 44 "-Gamma: the Lie algebra of vector fields. " }}{PARA 299 "" 0 "" {TEXT 314 74 "-ind_var: a list of independent variables to be used in the dep_vars. " }}{PARA 300 "" 0 "" {TEXT 315 145 "-de p_vars: a list of dependent variables, the first is the coefficient o f the vector in N, the rest are the coefficients of the vectors in Inv V." }}{PARA 301 "" 0 "" {TEXT 316 161 "-flag: an optional arg ument -- if set to _ExpSolve then the defining pde's are solved for \+ all derivatives and also for the coeff of thevector in N." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 66 "normalizer_PDE(N0[1],[vect(x1)],Gamma,[x1,x2,x3],[a,b ],_ExpSolve);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)/-%%diffG6$-%\"bG6% %#x1G%#x2G%#x3GF,\"\"!/-F&6$F(F+F./-F&6$-%\"aGF*F,F./-F&6$F5F+F./-F&6$ F(F-F./F5\"\"\"/-F&6$F5F-F." }}}{EXCHG {PARA 303 "" 0 "" {TEXT -1 0 " " }}{PARA 302 "" 0 "" {TEXT 317 27 "The solution is b =0, a=1" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 42 "N1:=v_zip([1,0],[N0[1][1],vect(x1)],plus):" }}} {EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 9 "show(N1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#x2G\"\"\"&%%D_x2G6#%!GF&!\"\"*&%#x3GF&&%% D_x3G6#%!GF&F+" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 66 "no rmalizer_PDE(N0[2],[vect(x1)],Gamma,[x1,x2,x3],[a,b],_ExpSolve);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<)/-%%diffG6$-%\"bG6%%#x1G%#x2G%#x3GF, \"\"!/-F&6$F(F-F./-F&6$-%\"aGF*F-F3/-F&6$F5F,F8/-F&6$F5F+F;/-F&6$F(F+! \"\"/F5F5" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 44 "N2:=v_z ip([0,-x1],[N0[1][1],vect(x1)],plus):" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 25 "Show([N1, N2,op(Gamma)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(, &*&%#x2G\"\"\"&%%D_x2G6#%!GF'!\"\"*&%#x3GF'&%%D_x3G6#%!GF'F,,$*&%#x1GF '&%%D_x1G6#%!GF'F,&F)6#%!G&F06#%!G,$&F76#%!GF,,&*&F.\"\"\"&F)6#%!GF'F' *&F&FF&F06#%!GF'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 78 "We check t he answer by computing the structure equations for the normalizer." }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 44 "L:=vect_to_Lie_alg ([N1, N2,op(Gamma)],norm):" }}}{EXCHG {PARA 0 "petrov > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% 2Lie~algebra:~normG" }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 25 "L ie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG 6#7*7*%!G%\"|grG&%#e1G6#%!G&%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%! G&%#e6G6#%!G7*F(%$---G%%----GFDFDFDFDFD7*&F+6#%!GF)\"\"!FI&F36#%!G&F76 #%!GFIFI7*&F/6#%!GF)FIFIFIFI&F;6#%!GFI7*&F36#%!GF),$FJ!\"\"FIFIFIFI&F7 6#%!G7*&F76#%!GF),$FMFfnFIFIFIFI&F36#%!G7*&F;6#%!GF)FI,$FTFfnFIFIFIFI7 *&F?6#%!GF)FIFI,$FgnFfn,$F_oFfnFIFI" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 320 69 "We check that the original algebra is an ideal in the \+ algebra norm." }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 27 "check_i deal([e3,e4,e5,e6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 256 "" 0 "" {TEXT -1 30 "Example 3. Pommaret, page 346" }}{PARA 279 "" 0 "" {TEXT 288 74 "The normalizer of the 3 dimension Lie algebra Gamma on R^3 is computed ." }}{PARA 280 "" 0 "" {TEXT 289 113 "Since the action is free, there will be 3 invariant vector fields and one element of the normalize r for each " }}{PARA 281 "" 0 "" {TEXT 290 103 "outer automorphism (4) . Since the center of Gamma is 1 we get the dimension of the norma lizer to be" }}{PARA 282 "" 0 "" {TEXT 291 15 "3+ 3 + 4 -1 =9." }} {PARA 283 "" 0 "" {TEXT 292 0 "" }}{PARA 284 "" 0 "" {TEXT 293 96 "The point of this example is that one can compute the normalizer of N t o get a 10 dim algebra." }}{PARA 286 "" 0 "" {TEXT 298 12 "Interesting !" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "norm> " 0 "" {MPLTEXT 1 0 35 "coord_init([x1,x2,x3],[],pommaret):" }}} {EXCHG {PARA 0 "pommaret>" 0 "" {MPLTEXT 1 0 48 "Gamma:=[D_x1, D_x2 &p lus (x1 &mult D_x3), D_x3]:" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 12 "Show(Gamma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%% D_x1G6#%!G,&&%%D_x2G6#F'\"\"\"*&%#x1GF,&%%D_x3G6#F'F,F,&F06#F'" }}} {EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 26 "L:=vect_to_Lie_alg( Gamma):" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 16 "Lie_alg _init(L):" }}}{EXCHG {PARA 0 "Koszul>" 0 "" {MPLTEXT 1 0 25 "Lie_brack et_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'%! G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(7'F(%$---G%%----GF5F57'&F+6#F(F) \"\"!&F16#F(F97'&F.6#F(F),$F:!\"\"F9F97'&F16#F(F)F9F9F9" }}}{EXCHG {PARA 0 "Koszul>" 0 "" {MPLTEXT 1 0 41 "Out:=infinitesimal_outer_autom orphisms();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$OutG7&-%'matrixG6#7% 7%\"\"!\"\"\"F+7%F+F+F+F--F'6#7%F-7%F,F+F+F--F'6#7%7%!\"\"F+F+F*F--F'6 #7%F1F-7%F+F+F," }}}{EXCHG {PARA 0 "Koszul>" 0 "" {MPLTEXT 1 0 48 "Inv V:=[D_x1 &plus((x2) &mult D_x3), D_x2, D_x3]:" }}}{EXCHG {PARA 285 "" 0 "" {TEXT 294 44 "Check that these are invariant vector fields" }}} {EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 62 "matrix(3,3, (i,j)-> coeff_set(Lie_bracket(Gamma[i],InvV[j])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%<#\"\"!F(F(F'F'" }}}{EXCHG {PARA 0 "pom maret > " 0 "" {MPLTEXT 1 0 12 "Show(Gamma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%%D_x1G6#%!G,&&%%D_x2G6#F'\"\"\"*&%#x1GF,&%%D_x3G6#F 'F,F,&F06#F'" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 15 "sh ow(ND[1][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%%D_x1G 6#%!GF%" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 50 "ND:=nor malizer_distribution(Gamma,Out,[x1,x2,x3]):" }}}{EXCHG {PARA 0 "pommar et > " 0 "" {MPLTEXT 1 0 3 "ND;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&7 $7$7%%%vectG%)pommaretG7\"7#7$7#\"\"\"\"\"!-%'matrixG6#7%7%F.F.F.7%F-F .F.F37$F%-F06#7%F4F37%F.F.F-7$F%-F06#7%7%!\"\"F.F.7%F.F-F.F37$F%-F06#7 %F@F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 47 "Solve the PDE for each vector/outer pair in ND." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 70 "normalizer_PDE(ND[1],InvV[1..3],Gamma,[x1,x2,x3],[a,b ,c,d],_ExpSolve);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 39 "N1:=v_zip([-x1,0,x1*x2 -x3],InvV,plus); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N1G7$7%%%vectG%)pommaretG7\"7$7 $7#\"\"\",$%#x1G!\"\"7$7#\"\"$,$%#x3GF0" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 70 "normalizer_PDE(ND[2],InvV[1..3],Gamma,[x1,x2, x3],[a,b,c,d],_ExpSolve);" }}{PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#F+F./-F&6$ FGF+FP/-F&6$FGF,FS/-F&6$F>F-F." }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 33 "N2:=v_zip([0,-x2,-x3],InvV,plus);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#N2G7$7%%%vectG%)pommaretG7\"7$7$7#\"\"#,$%#x2G!\" \"7$7#\"\"$,$%#x3GF0" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 70 "normalizer_PDE(ND[3],InvV[1..3],Gamma,[x1,x2,x3],[a,b,c,d],_ExpS olve);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 " " {MPLTEXT 1 0 38 "N3:=v_zip([-x2,0,1/2*x2^2],InvV,plus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N3G7$7%%%vectG%)pommaretG7\"7$7$7#\"\"\",$%# x2G!\"\"7$7#\"\"$,$*$)F/\"\"#\"\"\"#F0F7" }}}{EXCHG {PARA 0 "pommaret \+ > " 0 "" {MPLTEXT 1 0 70 "normalizer_PDE(ND[4],InvV[1..3],Gamma,[x1,x2 ,x3],[a,b,c,d],_ExpSolve);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 39 "N4:=v_zip([0,-x1,-1/2*x1^2 ],InvV,plus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N4G7$7%%%vectG%)po mmaretG7\"7$7$7#\"\"#,$%#x1G!\"\"7$7#\"\"$,$*$)F/F-\"\"\"#F0F-" }}} {EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 54 "normalizer:=[N1,N2, N3,N4, op(Gamma), InvV[1],InvV[2]]:" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 17 "Show(normalizer);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+,&*&%#x1G\"\"\"&%%D_x1G6#%!GF'!\"\"*&%#x3GF'&%%D_x3G6#F+F'F,,& *&%#x2GF'&%%D_x2G6#F+F'F,*&F.\"\"\"&F06#F+F'F,,&*&F4F9&F)6#F+F'F,*&&,$ *$)F4\"\"#F9#F,FE6#F+F'&F06#F+F'F',&*&F&F9&F66#F+F'F,*&&,$*$)F&FEF9FFF GF'&F06#F+F'F'&F)6#F+,&&F66#F+F'*&F&F9&F06#F+F'F'&F06#F+,&&F)6#F+F'*&F 4F9&F06#F+F'F'&F66#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 296 82 "Check that the vectors we have found for m a Lie algebra in which Gamma is normal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 37 "L1:=vect _to_Lie_alg(normalizer,norm);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#L1 G7$7%%(Lie_algG%%normG7#\"\"*747$7%\"\"\"\"\"$F/F.7$7%F.\"\"%F2!\"\"7$ 7%F.\"\"&F6F.7$7%F.\"\"(F9F.7$7%F.\"\")F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 17 "Lie_alg_init(L1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~normG" }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7-7-%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F (&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(&%#e7G6#F(&%#e8G6#F(&%#e9G6#F(7-F(%$--- G%%----GFGFGFGFGFGFGFGFG7-&F+6#F(F)\"\"!FK&F16#F(,$&F46#F(!\"\"&F76#F( FK&F=6#F(&F@6#F(FK7-&F.6#F(F)FKFK,$&F16#F(FQ&F46#F(FK&F:6#F(&F=6#F(FK& FC6#F(7-&F16#F(F),$FLFQFfnFK,&&F+6#F(\"\"\"&F.6#F(FQFK&F76#F(FKFK&F@6# F(7-&F46#F(F)FO,$FhnFQ,&FeoFQFhoFgoFK&F:6#F(FKFK&FC6#F(FK7-&F76#F(F),$ FRFQFKFK,$FcpFQFK&F=6#F(FKFKFK7-&F:6#F(F)FK,$FjnFQ,$FjoFQFK,$F\\qFQFKF KFKFK7-&F=6#F(F),$FTFQ,$F\\oFQFKFKFKFKFKFKFK7-&F@6#F(F),$FVFQFKFK,$Fep FQFKFKFKFK,$&F=6#F(FQ7-&FC6#F(F)FK,$F^oFQ,$F\\pFQFKFKFKFKF_rFK" }}} {EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 24 "check_ideal([e5,e6,e7]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 297 58 "Now we go again and find the normalizer of this alge bra." }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 41 "Out:=infinitesimal_outer_a utomorphisms();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$OutG7#-%'matrixG 6#7+7+\"\"!F+F+F+F+F+F+F+F+F*F*F*7+F+F+F+F+F+F+F+\"\"\"F+7+F+F+F+F+F+F +F+F+F-F*7+F+F+F+F+F-F+F+F+F+7+F+F+F+F+F+F-F+F+F+" }}}{EXCHG {PARA 0 " pommaret > " 0 "" {MPLTEXT 1 0 26 "change_frame_to(pommaret);" }}} {EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "norm>" 0 "" {MPLTEXT 1 0 67 "isotropy_invariant_tensors(normalizer,fr ameBaseVectors(),[a,b,c]); " }}{PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "pomma ret > " 0 "" {MPLTEXT 1 0 56 "NND:=normalizer_distribution(normalizer, Out,[x1,x2,x3]):" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 62 "normalizer_PD E(NND[1],[],normalizer,[x1,x2,x3],[a],_ExpSolve);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/-%%diffG6$-%\"aG6%%#x1G%#x2G%#x3GF+\"\"!/-F&6$F(F-F. /-F&6$F(F,F./F(\"\"\"" }}}{EXCHG {PARA 0 "norm2>" 0 "" {MPLTEXT 1 0 16 "show(NND[1][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#x1G\"\" \"&%%D_x1G6#%!GF&!\"\"*&%#x2GF&&%%D_x2G6#F*F&F+*(F-\"\"\"F%F2&%%D_x3G6 #F*F&F+" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "norm2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pomm aret > " 0 "" {MPLTEXT 1 0 54 "L3:=vect_to_Lie_alg([NND[1][1],op(norma lizer)],norm2):" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 17 "Lie_alg_init(L3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Lie~algebra:~n orm2G" }}}{EXCHG {PARA 0 "norm2>" 0 "" {MPLTEXT 1 0 43 "check_ideal([e 2,e3,e4,e5,e6,e7,e8,e9,e10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}}{EXCHG {PARA 0 "norm2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pommaret > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{MARK "35 31 0 0" 64 }{VIEWOPTS 1 1 0 3 4 1802 }