{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Vessiot_Text" -1 256 "Intrepid" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Vess_Title2" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 0 0 0 2 0 0 0 }1 0 0 0 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess _IO" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 1 0 0 }1 0 0 -1 -1 -1 3 30 0 0 0 0 -1 3 }{PSTYLE "Vess_Title1" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 128 0 64 1 0 0 0 0 0 0 3 0 0 }2 1 0 0 10 10 3 6 3 30 0 0 -1 0 }{PSTYLE "Example" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 90 " \+ Vessiot Tutorial: The nilradical of a Lie algebra" }} {PARA 260 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 107 "This tutorial gives a number of examples of the program for c omputing the nilradical of a Lie algebra " }{TEXT 261 1 "g" }{TEXT -1 20 ". The nilradical " }{TEXT 263 5 "nr(g)" }{TEXT -1 44 " is th e largest nilpotent subalgebra of " }{TEXT 262 5 "g. " }{TEXT -1 82 "It is unique and plays an important role in the classify of solvable algebra." }}{PARA 257 "" 0 "" {TEXT -1 33 "Incontrast to th e the radical of " }{TEXT 264 4 "g, " }{TEXT -1 71 "the computation o f the nilradical is rather complicated. The program " }{TEXT 265 10 "nilradical" }{TEXT -1 36 " implements the algorithm given in" } {TEXT 266 1 " " }{TEXT 267 73 "On the identification of a Lie algebra \+ given by its structure constants I" }{TEXT 268 86 ", Rand, Winternitz , and Zassenhaus, Linear Alg and its Applications, 1988, 197--246." } }{PARA 257 "" 0 "" {TEXT -1 28 "The algorithm is recursive." }}{PARA 257 "" 0 "" {TEXT -1 52 "With _Vessiot_show_intermediate_steps_flag= \+ true, " }{TEXT 269 12 "nilradical " }{TEXT -1 67 "shows the status \+ of each step as described in the above article." }}{PARA 257 "" 0 " " {TEXT -1 277 "The algorithms requires the initalization of many n ew algebras. The name of each new algebra is determined to be the name of the current algebra followed by the step in which the alg ebra is initialized. Each call to nilradical appends an N to the alg ebra name. " }}{PARA 257 "" 0 "" {TEXT -1 148 "If the argument poly \+ is supplied to nilradical, then the program queries the user for infor mation concerning the polynomial constructed in step 6. " }}{PARA 261 "" 0 "" {TEXT 260 22 "Procedures Illustrated" }}{PARA 257 "" 0 "" {TEXT -1 26 "The main routines used by " }{TEXT 270 10 "nilradical" } {TEXT -1 6 " are: " }{HYPERLNK 17 "radical" 2 "radical" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "subalgebra_to_Lie_algebra_data" 2 "subalgebra_t o_Lie_algebra_data" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "create_quotient _algebra" 2 "create_quotient_algebra" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "derived_series" 2 "derived_series" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "upper_central_series" 2 "upper_central_series" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "matrix_to_Lie_algebra_transform" 2 "matrix_to_Lie_alg ebra_transform" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Lie_algebra_transfo rm" 2 "Lie_algebra_transform" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "creat e_inverse_image_subalgebra" 2 "create_inverse_image_subalgebra" "" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT 258 5 "Notes" }}{PARA 257 " " 0 "" {TEXT -1 321 "The branching that occurs in the implementation is basis dependent --- not only on the basis that is used to define \+ the original algebra but also upon the basis used to describe all t he various subalgebras and quotient algebras constructed by the algori thm. To obtain results which are identical on each call to " } {TEXT 271 10 "nilradical" }{TEXT -1 80 ", the various basis that are \+ constructed are put into canonical form using the " }{HYPERLNK 17 "can onical_subspace_basis" 2 "canonical_subspace_basis" "" }{TEXT -1 10 " \+ command. " }}}{PARA 256 "" 0 "" {TEXT 259 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "with(Vessiot):with(Koszul):with(Chevalley):with( Vessiot_library):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 256 "" 0 "" {TEXT -1 25 "Example 1. Special cases " }} {PARA 259 "" 0 "" {TEXT -1 130 "The program treats the cases where t he algebra is semi-simple and where the radical is nilpotent as s pecial cases. In the " }}{PARA 259 "" 0 "" {TEXT -1 244 "first case t he nilradical is 0 and in the second the niradical is the radical. \+ These special cases are checked before entering the genereal algo rithm.The case of the 2 dimensional non-abelian algebra L(2,1) is also treated separately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 29 "L:=Lie_lib(winternitz,[2,1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L):" }}{PARA 0 "w35 \+ > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 0 "w35>" 0 "" {MPLTEXT 1 0 16 "N:=nilradical():" }}{PARA 0 "w21 > " 0 "" {MPLTEXT 1 0 8 "Show(N);" }}{PARA 0 "w21 > " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&%!G%\"|grG&%#e1G6#F(&% #e2G6#F(7&F(%$---G%%----GF27&&F+6#F(F)\"\"!&F+6#F(7&&F.6#F(F),$F7!\"\" F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IBegin~~calculation~of~nilradic al~for~w21G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Step~1G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%9Computing~radical~of~w21G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Radical~~is~2~dimensionalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#,$&%#e1G6#%!G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "L:=Lie_lib(winternitz,[3,5]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L):" }}{PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 0 "w35>" 0 "" {MPLTEXT 1 0 13 "nilradical();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'% !G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(7'F(%$---G%%----GF5F57'&F+6#F(F )\"\"!&F+6#F(,$&F.6#F(!\"#7'&F.6#F(F),$F:!\"\"F9&F16#F(7'&F16#F(F),$F= \"\"#,$FEFDF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IBegin~~calculation~ of~nilradical~for~w35G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Step~1G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Computing~radical~of~w35G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Algebra~is~semi-simpleG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 30 "L:=Lie_lib(winternitz,[5,52]):" }}{PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 0 "w552 > " 0 "" {MPLTEXT 1 0 25 "Lie_br acket_mult_table();" }}{PARA 0 "w552 > " 0 "" {MPLTEXT 1 0 16 "N:=nilr adical():" }}{PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 8 "Show(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~w552G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F (&%#e4G6#F(&%#e5G6#F(7)F(%$---G%%----GF;F;F;F;7)&F+6#F(F)\"\"!,$&F+6#F (\"\"#,$&F.6#F(!\"\"&F76#F(F?7)&F.6#F(F),$FA!\"#F?,$&F16#F(FC&F46#F(,$ &F76#F(FG7)&F16#F(F)FE,$FPFNF?F?&F46#F(7)&F46#F(F),$FHFG,$FRFGF?F?F?7) &F76#F(F)F?FU,$FenFGF?F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%JBegin~~c alculation~of~nilradical~for~w552G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %'Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Computing~radical~of~w5 52G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Radical~~is~abelianG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e5G6#%!G&%#e4G6#F'" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 38 "Example 2. An illustration of Step 6. " }}{PARA 259 "" 0 "" {TEXT -1 46 "1. The algebra is solvalbe so we \+ pass Step1." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 104 "2. The derived algebra is DL= [e1,e2,e4] and D2L =[e1] . Since the derived algebra is not abelian, " }}{PARA 259 "" 0 "" {TEXT -1 89 "we form the factor algebra L/D2L.which is not abelian. \+ The projection map L->L/D2L is" }}{PARA 259 "" 0 "" {TEXT -1 78 "e1- >0, e2->e1 e3->e2, e4->e3, ,e5->e4. The name of this algebra is w540_ N_S2." }}{PARA 259 "" 0 "" {TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 63 "3. The algebra w540_N_S2 has no center so we pass to Step 4 " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 105 " 4. The derived algebra of w540_N_S2 is U=D(L)=[e1,e3], the complement is [e2,e4]. The polynomial f =q. " }}{PARA 259 "" 0 "" {TEXT -1 49 " This has zero constant term so we pass to step 6." }}{PARA 259 "" 0 " " {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 119 "5. B1=[e2,D(L)] = [e 1]. We form the factor algebra w540_N_S2/B1 which is called w540 _N_S2_S6Q. The projection is " }}{PARA 259 "" 0 "" {TEXT -1 29 "e1->0, e2->e1, e3->e2, e4->e3." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 111 "6. We find the nilradical of w540_N_S2_S6Q. Skip down to DONE! where the nilradical is given as [e1,e2], " } }{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 134 "7. \+ The inverse image of [e1,e2] in w540_N_S2_S6Q is M=[e1,e2,e3] in w540_N_S2. We are back in Step 6 for the original algebra." }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 108 "8. T he algebra M is initialized as w540_N_S2_S6M. This algebra is nilpot ent so the nilradical is itself. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 59 "9. This gives the nilradical of w540 _N_S2 as M=[e1,e2,e3]." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 87 "10. The inverse image of M in w540_N is [e1,e2,e 3,e4] which is the nilradical of w540." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "w33>" 0 "" {MPLTEXT 1 0 30 "L:=Lie_lib(winternitz, [5,40]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L):" }} {PARA 0 "w540 > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "nilradical():" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{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)\"\"!FEFEFEFE7)&F/6#%!GF)FEFEFE&F+6#%!G& F/6#%!G7)&F36#%!GF)FEFEFE&F/6#%!GFE7)&F76#%!GF)FE,$FJ!\"\",$FTFfnFE&F7 6#%!G7)&F;6#%!GF)FE,$FMFfnFE,$FhnFfnFE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%JBegin~~calculation~of~nilradical~for~w540G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%'Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Comput ing~radical~of~w540G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1w540~is~solv ableG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MThe~algebra~w540~is~re-init ialized~as~w540_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~ 1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ioThe~~factor~algebra~of~w540_N~by~~~D2L~is~initi alized~as~Lie~algebra~w540_N_S2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'matrixG6#7(7(%!G%\"|grG&%#e1G6#%!G&%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G7( F(%$---G%%----GF" 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 64 "Example 3. A repetition of \+ the previous example in a new basis." }}{PARA 259 "" 0 "" {TEXT -1 50 "We note that the polynomial f changes to q^2. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 82 "The change of basis preserves [e1,e2,e3,e4] so the nilradical remains the same." }} {EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 30 "L:=Lie_lib(winternitz,[5, 40]):" }}}{EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 15 "Lie_alg_int(L): " }}}{EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 77 "A:=matrix([[1,1,-3,1 ,0],[1,1,2,2,0],[-1,-4,1,0,0],[2,4,-1,2,0],[0,0,0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7'7'\"\"\"F*!\"$F*\"\"!7'F*F* \"\"#F.F,7'!\"\"!\"%F*F,F,7'F.\"\"%F0F.F,7'F,F,F,F,F*" }}}{EXCHG {PARA 0 "Ex3>" 0 "" {MPLTEXT 1 0 40 "L:=change_Lie_algebra_basis(A,Ex3 ,w540):" }}}{EXCHG {PARA 0 "w540 > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_ini t(L):" }}}{EXCHG {PARA 0 "Ex3 > " 0 "" {MPLTEXT 1 0 13 "nilradical(): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IBegin~~calculation~of~nilradical ~for~Ex3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Computing~radical~of~Ex3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Ex3~is~solvableG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% KThe~algebra~Ex3~is~re-initialized~as~Ex3_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% -Begin~Step~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%goThe~~factor~algeb ra~of~Ex3_N~by~~~D2L~is~initialized~as~Lie~algebra~Ex3_N_S2G" }}{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 47 "canonical_subspace_basis(% ,frameBaseVectors());" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$7%%%vectG %$Ex3G7\"7#7$7#\"\"\"F,7$F%7#7$7#\"\"#F,7$F%7#7$7#\"\"$F,7$F%7#7$7#\" \"%F," }}}{EXCHG {PARA 0 "Ex3 > " 0 "" {MPLTEXT 1 0 8 "Show(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%#e1G6#%!G&%#e2G6#%!G&%#e3G6#%!G&%# e4G6#%!G" }}}{EXCHG {PARA 0 "Ex3>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 " " 0 "" {TEXT -1 39 "Example 4. An illustratation of Step 7" }}{EXCHG {PARA 0 "w540>" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 259 "" 0 "" {TEXT -1 119 "In this example the ploynomial f constructed in step 5 is a pe rfect square and the algorithm passes through step 7. " }}{PARA 259 " " 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 127 "1. The algebra \+ is solvable, the derived series is abelian and the center is 0. Thus \+ the algorithm passes directly to step 4. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 179 "2. The derived algebra U=[e 1,e2] and the complementary basis is X=[e3]. We find that f =q^2 +2q \+ +1. The constant term is non-zero and f is a perfect square so we go \+ to step 7. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 88 "3. The square free-factor is g=q+1. With ad(e3) as give n below we compute g(ad(e3)). " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 0 "" {TEXT -1 118 "4. Since the derived series U=D(L) is represented by the column vector [0,1,0], the algebra B2 = g(ad(e3)* U is e2. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 108 "5. Form the factor algebra L/B2 -- it is denoted by Ex3_ N_S7Q. Under this quotient e1->e1 e2->0 e3-> e2. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 130 "6. The nilradical of Ex4_N_S7Q is e1. Pull this back to EX4_N by adding in the der ived_series to get the algebra M=[e1,e2]." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 158 "7. The algebra M is i nitalized as Ex4_N_S6M. It is abelian so the a recursive call to ni lradical easily determines the nilradical of M to be [e1,e2]. " }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 70 "8. Th e nilradical of M gives the nilradical of the original algebra." }} {PARA 259 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "Koszul > " 0 "" {MPLTEXT 1 0 89 "C:=structure_equation s_to_Lie_algebra_data([e1,e2,e3],[[e1,e3]=e1 +e2, [e2,e3]=e2], Ex4):" }}}{EXCHG {PARA 0 "Koszul > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(C):" }}{PARA 0 "Ex3>" 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#%!G7'F(%$---G%%----GF8F87'&F+6#%!GF)\"\"!F=,&&F+6#%! G\"\"\"&F/6#%!GFB7'&F/6#%!GF)F=F=&F/6#%!G7'&F36#%!GF),&F?!\"\"FCFR,$FJ FRF=" }}}{EXCHG {PARA 0 "Ex4>" 0 "" {MPLTEXT 1 0 7 "ad(e3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!\"\"\"\"!F)7%F(F(F)7%F)F)F) " }}}{EXCHG {PARA 0 "Ex3 > " 0 "" {MPLTEXT 1 0 16 "N:=nilradical():" } }{PARA 0 "Ex4 > " 0 "" {MPLTEXT 1 0 8 "Show(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IBegin~~calculation~of~nilradical~for~Ex4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%9Computing~radical~of~Ex4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Ex4~ is~solvableG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThe~algebra~Ex4~is~r e-initialized~as~Ex4_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~S tep~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%FDerived~Algebra~~of~~Ex4_N~is~AbelianG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~2G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%H Computing~Upper~central~series~of~Ex4_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Center~of~Ex4_N~is~0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End ~of~~Step~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~4G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%=Derived~Algebra~~U~~of~Ex4_NG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e1G6#%!G&%#e2G6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%9Complementary~basis~to~UG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e3G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~ of~~Step~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~5G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%#j=G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3The~vector~x_j~is~G&%#e3G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7The~polynomial~f_j~is~G,(*$)%#_qG\"\"#\"\"\"\"\"\"F*F *F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 85 "Example 5 A demonstration of how a change of basis changes the algorithm branching. " }}{PARA 259 "" 0 "" {TEXT -1 89 "We take the algebra from Example 3 and use the Lie algebra classification command to" }}{PARA 259 "" 0 "" {TEXT -1 34 " identify this algebra as L(3,3). " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 160 "1. When we compute the nilradical this time, the polynomial f = q+1 which is square-free, non-zero constant ter m. Accordingly, we pass directly to step 8. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 179 "2. In step 8, the c entralizer M of e1 is computed to be [e1,e2]. Since this coincides \+ with the derived algebra and the derived algebra contains the nilradi cal we are done. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "w 21>" 0 "" {MPLTEXT 1 0 26 "classify_Lie_algebra(Ex3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7%7%!\"\"\"\"\"\"\"!7%F*F+F+7%F+F+F* 7$%+winternitzG7$\"\"$F1" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 46 "E xample 6. The 5 dimensional algebra in RWZ." }}{PARA 259 "" 0 "" {TEXT -1 37 "This is the second example in RWZ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "w552>" 0 "" {MPLTEXT 1 0 30 "L:=Lie_li b(winternitz,[5,26]):" }}{PARA 0 "w526 > " 0 "" {MPLTEXT 1 0 16 "Lie_a lg_init(L):" }}{PARA 0 "w526 > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mul t_table();" }}{PARA 0 "w552>" 0 "" {MPLTEXT 1 0 16 "N:=nilradical():" }}{PARA 0 "w552>" 0 "" {MPLTEXT 1 0 8 "Show(N);" }}{PARA 0 "w552>" 0 " " {MPLTEXT 1 0 0 "" }}{PARA 0 "Ex3 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#F(&%#e2G6# F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(7)F(%$---G%%----GF;F;F;F;7)&F+6#F(F)\" \"!F?F?F?&F+6#F(7)&F.6#F(F)F?F?&F+6#F(F?F?7)&F16#F(F)F?,$FE!\"\"F?F?,& &F16#F(\"\"\"&F46#F(FO7)&F46#F(F)F?F?F?F?&F46#F(7)&F76#F(F),$F@FKF?,&F MFKFPFK,$FUFKF?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%JBegin~~calculatio n~of~nilradical~for~w526G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Step~1G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Computing~radical~of~w526G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%1w526~is~solvableG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%MThe~algebra~w526~is~re-initialized~as~w526_NG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~1G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%-Begin~Step~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G Derived~Algebra~~of~~w526_N~is~AbelianG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~St ep~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IComputing~Upper~central~ser ies~of~w526_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6Center~of~w526_N~i s~0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Derived~Algebra~~U~~of~w526_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e1G6#%!G&%#e3G6#F'&%#e4G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Complementary~basis~to~UG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e2G6#%!G&%#e5G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~St ep~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#j=G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3The~vector~x_j~is~G&%#e2G6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%7The~polynomial~f_j~is~G%#_qG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% -Begin~Step~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4The~~algebra~B1~is :G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e1G6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%aoThe~~quotient~of~w526_N~by~B1~is~initialized~as~Li e~algebra~w526_N_S6QG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7 (7(%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(7(F(%$---G%%----G F8F8F87(&F+6#F(F)\"\"!FFI,$FCFIF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% 0End~of~~Step~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%GDerived~Algebra~~U~~of~w526_N_S6Q_ N_S3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e1G6#%!G&%#e2G6#F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%9Complementary~basis~to~UG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e3G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~St ep~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#j=G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3The~vector~x_j~is~G&%#e3G6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%7The~polynomial~f_j~is~G,(*$)%#_qG\"\"#\"\"\"\"\"\"F *F*F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~5G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 200 "As a final example we look \+ at the first example from Rand, Winternitz, Zassenhaus. The procedu re used by our program is different because of the construct of a different intermediate basis. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 0 "" {TEXT -1 56 "The Lie algebra is defined in Table 1 on page 240. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 75 "We compare the output from nilradical with the discus sion in the article. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 " " 0 "" {TEXT -1 87 "1. The radical of RWZ1 is computed and it is dete rmined that the algebra is solvable. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 95 "2. The second derived algebra D2L \+ is computed and the factor algebra RWZ1_N_S2 is computed. " }} {PARA 259 "" 0 "" {TEXT -1 60 "The displayed multiplication table c oincides with (4.50) " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 " " 0 "" {TEXT -1 35 "3. The center of RWZ1_N_S2 is 0. " }}{PARA 259 " " 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 81 "4. The derived a lgebra of RWZ1_N_S2 is called U. It coincides with (4.51a). " }} {PARA 259 "" 0 "" {TEXT -1 46 "The same complement as in (4.51b) is c hosen. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 77 "5. Step 5 begins with e5, the polynomial (4.53) coincides wi th f = _q-2. " }}{PARA 259 "" 0 "" {TEXT -1 70 "Since f is square fre e and the constant term is zero, go to step 8. " }}{PARA 259 "" 0 " " {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 75 "6. The Step5-Step 8 \+ loop is performed 3 more times with j=2, j=3, j=4. " }}{PARA 259 " " 0 "" {TEXT -1 40 "The polynomials f are given by (4.54). " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 85 "6. In step \+ 8 the centralizer M is computed and agrees with that given by (4.5 5). " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 68 "7. The subalgbra M is initialized as the Lie algebra RWZ1_N_S2_S8M . " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 90 "8. This algebra is solvable, D2M =0 but now there is a 1 dimensio nal center. This is " }}{PARA 259 "" 0 "" {TEXT -1 54 "factored out t o give the algebra RWZ1_N_S2_S8M_N_S3. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 75 "9. The Step 5-Step8 loop is p erformed twice since f1=_q -1 and f2=_q+1. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 108 "10. In contrast to th e description in RWZ, we find that for the last element (j=3) e5 the polynomial " }}{PARA 259 "" 0 "" {TEXT -1 62 "f has a zero const ant term. Accordingly we pass to step 6. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 31 "11. The algebra B1 is [ e3,e3]. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 98 "12. The factor algebra of RWZ1_N_S2_S8M_N_S3 by B1 is init ialized as RWZ1_N_S2_S8M_N_S3_S6Q. " }}{PARA 259 "" 0 "" {TEXT -1 69 " This is a 4 dim algebra and now we begin anew to find its radical. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 109 " 13. The hypercenter of this algebra is 2 dimensional. We quotient \+ by it in step3 to form the algebra " }}{PARA 259 "" 0 "" {TEXT -1 109 "RWZ1_N_S2_S8M_N_S3_S6Q_S3. This is a 2 dimensional solvable alge bra. We pass through the Step5-Step8 loop " }}{PARA 259 "" 0 "" {TEXT -1 72 "once to find that the nil radical of RWZ1_N_S2_S8M_N_S3 _S6Q_S3 is e1. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 " " {TEXT -1 88 "14. The hypercenter is added back in to get the nilra dical of RWZ1_N_S2_S8M_N_S3_S6Q. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" } }{PARA 259 "" 0 "" {TEXT -1 127 "15. The algebra B1 is added back in t o get the subalgebra M. This algebra is initialized and RWZ1_N_S2_S 8M_N_S3_S6M and its " }}{PARA 259 "" 0 "" {TEXT -1 295 "nilradical is now computed to be [e1,e2,e3]. Look for the second DONE! after \+ the initialization of this algebra. This completes step 6 for the con struction of the nilradical of RWZ1_N_S2_S8M. The program passes to \+ step 9 and finds the nilradical for this algebra to be [e1,e2,e3,e4 ]" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 134 "16. This puts us back in step 8 for the original algebra, working wi th RWZ1_N_S2 from which the final nilradical is constructed. " }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 9 "17. Wh ew!" }}{PARA 259 "" 0 "" {TEXT 256 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "table1:=[ [e1,e5]=e2, [e1,e6]=e3, [e1,e7]=e4, [e1,eb] =-2*e1, [e1,ec]=-e1,[e1,ed]=-e1, [e1,ee]=-e1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "[e2,e8]=e3,[e2,e9]=e4,[e2,eb]=-e2,[e2,ec]=-2*e2,[e2,e d]=-e2,[e2,ee]=-e2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "[e3,ea]=e4, [e3,eb]=-e3,[e3,ec]=-e3,[e3,ed]=-2*e3,[e3,ee]=-e3, \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "[e4,eb]=-e4,[e4,ec]=-e4,[e4,ed]=-e4,[e4,ee]=-2*e4," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "[e5,e8]=e6,[e5,e9]=e7,[e5,eb]=e5,[ e5,ec]=-e5," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "[e6,ea]=e7,[e6,eb]=e 6,[e6,ed]=-e6," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "[e7,eb]=e7,[e7,ee ]=-e7," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "[e8,ea]=e9,[e8,ec]=e8,[e8 ,ed]=-e8," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "[e9,ec]=e9,[e9,ee]=-e9 ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "[ea,ed]=ea,[ea,ee]=-ea]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "V:=[e1,e2,e3,e4,e5,e6,e7,e8,e9,ea,eb,ec,ed,ee];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"VG70%#e1G%#e2G%#e3G%#e4G%#e5G%#e6G%#e7G%#e8G%#e9G %#eaG%#ebG%#ecG%#edG%#eeG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "table1;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7H/7$%#e1G%#e5G%#e2G/7$F&% #e6G%#e3G/7$F&%#e7G%#e4G/7$F&%#ebG,$F&!\"#/7$F&%#ecG,$F&!\"\"/7$F&%#ed GF9/7$F&%#eeGF9/7$F(%#e8GF,/7$F(%#e9GF0/7$F(F3,$F(F:/7$F(F8,$F(F5/7$F( F=FI/7$F(F@FI/7$F,%#eaGF0/7$F,F3,$F,F:/7$F,F8FV/7$F,F=,$F,F5/7$F,F@FV/ 7$F0F3,$F0F:/7$F0F8Fjn/7$F0F=Fjn/7$F0F@,$F0F5/7$F'FCF+/7$F'FFF//7$F'F3 F'/7$F'F8,$F'F:/7$F+FSF//7$F+F3F+/7$F+F=,$F+F:/7$F/F3F//7$F/F@,$F/F:/7 $FCFSFF/7$FCF8FC/7$FCF=,$FCF:/7$FFF8FF/7$FFF@,$FFF:/7$FSF=FS/7$FSF@,$F SF:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L:=structure_equatio ns_to_Lie_algebra_data(V,table1,RWZ1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Lie_alg_init(L,V,[epsilon]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~RWZ1G" }}}{EXCHG {PARA 0 "RWZ1 > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7272%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F (&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(&%#e7G6#F(&%#e8G6#F(&%#e9G6#F(&%#eaG6#F (&%#ebG6#F(&%#ecG6#F(&%#edG6#F(&%#eeG6#F(72F(%$---G%%----GFVFVFVFVFVFV FVFVFVFVFVFVFV72&F+6#F(F)\"\"!FZFZFZ&F.6#F(&F16#F(&F46#F(FZFZFZ,$&F+6# F(!\"#,$&F+6#F(!\"\",$&F+6#F(Fbo,$&F+6#F(Fbo72&F.6#F(F)FZFZFZFZFZFZFZ& F16#F(&F46#F(FZ,$&F.6#F(Fbo,$&F.6#F(F^o,$&F.6#F(Fbo,$&F.6#F(Fbo72&F16# F(F)FZFZFZFZFZFZFZFZFZ&F46#F(,$&F16#F(Fbo,$&F16#F(Fbo,$&F16#F(F^o,$&F1 6#F(Fbo72&F46#F(F)FZFZFZFZFZFZFZFZFZFZ,$&F46#F(Fbo,$&F46#F(Fbo,$&F46#F (Fbo,$&F46#F(F^o72&F76#F(F),$FenFboFZFZFZFZFZFZ&F:6#F(&F=6#F(FZ&F76#F( ,$&F76#F(FboFZFZ72&F:6#F(F),$FgnFboFZFZFZFZFZFZFZFZ&F=6#F(&F:6#F(FZ,$& F:6#F(FboFZ72&F=6#F(F),$FinFboFZFZFZFZFZFZFZFZFZ&F=6#F(FZFZ,$&F=6#F(Fb o72&F@6#F(F)FZ,$F\\pFboFZFZ,$F`sFboFZFZFZFZ&FC6#F(FZ&F@6#F(,$&F@6#F(Fb oFZ72&FC6#F(F)FZ,$F^pFboFZFZ,$FbsFboFZFZFZFZFZFZ&FC6#F(FZ,$&FC6#F(Fbo7 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"6#%\\oDerived~Algebra~~of~~RWZ1_N_S2_S8M_N_S3_S6M_N_S3_S8 M_N~is~AbelianG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~2G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%^oComputing~Upper~central~series~of~RWZ1_N_S2_S8M_N_ S3_S6M_N_S3_S8M_NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"7#&%#e1G6# %!GF#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%)infinityG7#&%#e1G6#%!G\"\" \"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%[tThe~~factor~algebra~~of~~RWZ1 _N_S2_S8M_N_S3_S6M_N_S3_S8M_N~by~the~hypercenter~is~initialized~as~Lie ~algebra~RWZ1_N_S2_S8M_N_S3_S6M_N_S3_S8M_N_S3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&%!G%\"|grG&%#e1G6#F(&%#e2G6#F(7&F(%$--- G%%----GF27&&F+6#F(F)\"\"!,$&F+6#F(\"\"&7&&F.6#F(F),$F8!\"&F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%0End~of~~Step~~3G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%-Begin~Step~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%f nDerived~Algebra~~U~~of~RWZ1_N_S2_S8M_N_S3_S6M_N_S3_S8M_N_S3G" }} {PARA 11 "" 1 "" {XPPMATH 20 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11 "" 1 "" {XPPMATH 20 "6%%3The~~nilradical~ofG%BRWZ1_N_S2_S8M_N_S3_S6 M_N_S3_S8M_NG%$~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e1G6#%!G&% #e2G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&DONE!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2The~nilradical~ofG%@RWZ1_N_S2_S8M_N_S3_S6M_N_S3_S8 MG%$~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e1G6#%!G&%#e2G6#F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%+End~Step~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%3The~~nilradical~ofG%