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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 94 "                        \+
                Vessiot Tutorial: Levi decomposition of a Lie algebra \+
" }}{PARA 260 "" 0 "" {TEXT 258 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT 
-1 21 "Every  Lie  algebra  " }{TEXT 261 2 "g " }{TEXT -1 47 " can be \+
decomposed into a semi-direct product  " }{TEXT 262 1 "g" }{TEXT -1 3 
" = " }{TEXT 263 1 "r" }{TEXT -1 2 " +" }{TEXT 264 1 "s" }{TEXT -1 10 
" , where  " }{TEXT 265 1 "r" }{TEXT -1 19 " is the radical of " }
{TEXT 266 1 "g" }{TEXT -1 40 ", the maximal solvable  subalgebra and  \+
" }{TEXT 267 1 "s" }{TEXT -1 17 " is semi-simple. " }{TEXT 257 0 "" }
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 117 "The   program  Levi_de
composition  calculates the  Levi_decomposition of an algebra using th
e algorithm presented in " }{TEXT 268 1 " " }{TEXT 269 73 "On the iden
tification of a Lie algebra given by its structure constants I" }
{TEXT 270 86 ",  Rand, Winternitz, and  Zassenhaus, Linear Alg and its
 Applications, 1988, 197--246." }}{PARA 257 "" 0 "" {TEXT -1 67 "In th
is tutorial we  illustrate the various steps in the algorithm." }}
{PARA 257 "" 0 "" {TEXT -1 37 "Levi_decomposition returns the list [" 
}{TEXT 278 1 "r" }{TEXT -1 1 "," }{TEXT 279 1 "s" }{TEXT -1 2 "]." }}
{PARA 257 "" 0 "" {TEXT -1 67 "The  perfect algebra in step 2 is initi
alized as  alg_name_P,  the " }}{PARA 256 "" 0 "" {TEXT 259 22 "Proced
ures Illustrated" }}{PARA 257 "" 0 "" {TEXT -1 95 "Levi_decomposition,
  check_solvable, check_semi_simple, create_semi_direct_product_of_alg
ebras " }}}{PARA 256 "" 0 "" {TEXT 260 8 "Examples" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 53 "restart:with(Vessiot): with(Koszul): with(Ch
evalley):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 256 "" 0 "" {TEXT -1 14 "Special Cases." }}{PARA 257 "" 0 "" 
{TEXT -1 44 "The program  first computes the radical of  " }{TEXT 271 
1 "g" }{TEXT -1 6 ".  If " }{TEXT 272 1 "r" }{TEXT -1 1 "=" }{TEXT 
273 1 "g" }{TEXT -1 34 " then the algebra is solvable and " }{TEXT 
274 2 "s " }{TEXT -1 8 "=0.  If " }{TEXT 275 1 "r" }{TEXT -1 42 "= 0 t
hen the algebra is  semi-simple and  " }{TEXT 276 1 "g" }{TEXT -1 1 "=
" }{TEXT 277 1 "s" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 31 "A SEMI-SIMP
LE EXAMPLE:  sl(3). " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "
" 0 "" {TEXT -1 35 "First  create gl3 and then   sl3R. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gl3:=create_gl(3):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "Lie_alg_init(gl3):" }}{PARA 0 "gl3R > " 0 "" 
{MPLTEXT 1 0 29 "coord_init([x1,x2,x3],[],E3):" }}{PARA 0 "E3 > " 0 "
" {MPLTEXT 1 0 25 "nu:= form([x1,x2,x3],E3):" }}{PARA 0 "E3 > " 0 "" 
{MPLTEXT 1 0 44 "sl3Rsubalg:=create_gl_subalgebra(gl3R,[nu]):" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~vector~basis,~wh
ere~GF#%^ois~the~matrix~with~1~in~the~i~th~row,~j~th~column,~zeros~els
ewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilonG6#%#ijG%3Is~the
~dual~basis.G" }}}{EXCHG {PARA 0 "sl3R > " 0 "" {MPLTEXT 1 0 17 "Show(
sl3Rsubalg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*,&&%$e11G6#%!G\"\"\"
&%$e33G6#F(!\"\"&%$e12G6#F(&%$e13G6#F(&%$e21G6#F(,&&%$e22G6#F(F)&F+6#F
(F-&%$e23G6#F(&%$e31G6#F(&%$e32G6#F(" }}}{EXCHG {PARA 12 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "gl3R > " 0 "" {MPLTEXT 1 0 58 "sl3Rda
ta:=subalgebra_to_Lie_algebra_data(sl3Rsubalg,sl3R):" }}}{EXCHG {PARA 
0 "gl3R > " 0 "" {MPLTEXT 1 0 23 "Lie_alg_init(sl3Rdata);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~sl3RG" }}}{EXCHG {PARA 0 "sl3R >
 " 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(7,F(%$---G%%----G
FDFDFDFDFDFDFD7,&F+6#F(F)\"\"!&F.6#F(,$&F16#F(\"\"#,$&F46#F(!\"\"FH&F:
6#F(,$&F=6#F(!\"#,$&F@6#F(FR7,&F.6#F(F),$FIFRFHFH,&&F+6#F(\"\"\"&F76#F
(FR&F.6#F(&F16#F(,$&F@6#F(FRFH7,&F16#F(F),$FLFXFHFH,$&F:6#F(FR,$&F16#F
(FRFH&F+6#F(&F.6#F(7,&F46#F(F)FP,&F[oFRF^oF]oF\\pFH,$&F46#F(FRFHFH,$&F
=6#F(FR7,&F76#F(F)FH,$F`oFRF_pFjpFH,$&F:6#F(FN,$&F=6#F(FR,$&F@6#F(FX7,
&F:6#F(F),$FSFR,$FboFRFHFH,$FdqFXFH&F46#F(&F76#F(7,&F=6#F(F),$FVFNFeo,
$FapFRFHFgq,$FbrFRFHFH7,&F@6#F(F)FZFH,$FcpFRF]q,$FjqFN,$FdrFRFHFH" }}}
{EXCHG {PARA 0 "sl3R > " 0 "" {MPLTEXT 1 0 25 "LD:=Levi_decomposition(
):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~1G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%:Computing~Radical~of~sl3RG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%7Algebra~is~semi-simpleG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%.End~of~Step~1G" }}}{EXCHG {PARA 0 "sl3R > " 0 "" {MPLTEXT 1 0 
12 "Show(LD[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG 
{PARA 0 "sl3R > " 0 "" {MPLTEXT 1 0 12 "Show(LD[2]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#7*&%#e1G6#%!G&%#e2G6#F'&%#e3G6#F'&%#e4G6#F'&%#e5G6#F
'&%#e6G6#F'&%#e7G6#F'&%#e8G6#F'" }}}{EXCHG {PARA 0 "sl3R>" 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 64 " A SOLVABLE
 EXAMPLE:  THE  ALGEBRA OF UPPER TRIANGLAR MATRICES. " }}}{EXCHG 
{PARA 0 "sl3R>" 0 "" {MPLTEXT 1 0 23 "t4:=create_upper_gl(4):" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~vector~basis,~wh
ere~GF#%jnis~the~matrix~with~1~in~the~i~th~row,~j~th~column,~0~elsewhe
re.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilonG6#%#ijG%3Is~the~dua
l~basis.G" }}}{EXCHG {PARA 0 "sl3R>" 0 "" {MPLTEXT 1 0 17 "Lie_alg_ini
t(t4):" }}}{EXCHG {PARA 0 "upper_gl4 > " 0 "" {MPLTEXT 1 0 25 "Lie_bra
cket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7.7.
%!G%\"|grG&%$e11G6#F(&%$e12G6#F(&%$e13G6#F(&%$e14G6#F(&%$e22G6#F(&%$e2
3G6#F(&%$e24G6#F(&%$e33G6#F(&%$e34G6#F(&%$e44G6#F(7.F(%$---G%%----GFJF
JFJFJFJFJFJFJFJ7.&F+6#F(F)\"\"!&F.6#F(&F16#F(&F46#F(FNFNFNFNFNFN7.&F.6
#F(F),$FO!\"\"FNFNFN&F.6#F(&F16#F(&F46#F(FNFNFN7.&F16#F(F),$FQFYFNFNFN
FNFNFN&F16#F(&F46#F(FN7.&F46#F(F),$FSFYFNFNFNFNFNFNFNFN&F46#F(7.&F76#F
(F)FN,$FZFYFNFNFN&F:6#F(&F=6#F(FNFNFN7.&F:6#F(F)FN,$FfnFYFNFN,$F\\pFYF
NFN&F:6#F(&F=6#F(FN7.&F=6#F(F)FN,$FhnFYFNFN,$F^pFYFNFNFNFN&F=6#F(7.&F@
6#F(F)FNFN,$F^oFYFNFN,$FepFYFNFN&FC6#F(FN7.&FC6#F(F)FNFN,$F`oFYFNFN,$F
gpFYFN,$FeqFYFN&FC6#F(7.&FF6#F(F)FNFNFN,$FfoFYFNFN,$F^qFYFN,$F]rFYFN" 
}}}{EXCHG {PARA 0 "upper_gl4>" 0 "" {MPLTEXT 1 0 25 "LD:=Levi_decompos
ition():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~1G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%?Computing~Radical~of~upper_gl4G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%4Algebra~is~solvableG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.End~of~Step~1G" }}}{EXCHG {PARA 0 "upper_gl4 > " 0 "
" {MPLTEXT 1 0 12 "Show(LD[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,&
%$e22G6#%!G&%$e14G6#F'&%$e11G6#F'&%$e23G6#F'&%$e13G6#F'&%$e12G6#F'&%$e
24G6#F'&%$e34G6#F'&%$e44G6#F'&%$e33G6#F'" }}}{EXCHG {PARA 0 "upper_gl4
>" 0 "" {MPLTEXT 1 0 12 "Show(LD[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7\"" }}}{EXCHG {PARA 0 "upper_gl4>" 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 256 "" 0 "" {TEXT -1 27 "The Euclidean algebra e(3)." }}
{PARA 257 "" 0 "" {TEXT -1 135 "We  use the vect_to_alg command to  co
nstruct  e(3). The  translations  generate the radical and the  rotati
ons the semi-simple part.  " }}{PARA 257 "" 0 "" {TEXT -1 63 "The alge
bra e3 is  a perfect algebra: [e3,e3] =e3  so e3_P =e3." }}{PARA 257 "
" 0 "" {TEXT -1 68 "The  radical is  abelian so  the algoritm passes d
irectly to Step 4." }}{EXCHG {PARA 0 "upper_gl4>" 0 "" {MPLTEXT 1 0 
23 "coord_init([x,y,z],[]):" }}}{EXCHG {PARA 0 "euclid>" 0 "" 
{MPLTEXT 1 0 8 "Tx:=D_x:" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 
0 8 "Ty:=D_y:" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 8 "Tz:=D_
z:" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 42 "  Rxy:=(x &mult \+
D_y) &minus (y &mult D_x):" }}}{EXCHG {PARA 0 "euclid > " 0 "" 
{MPLTEXT 1 0 40 "Rzx:=(z &mult D_x) &minus (x &mult D_z):" }}}{EXCHG 
{PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 40 "Ryz:=(y &mult D_z) &minus (z
 &mult D_y):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 53 "e3Data:=vect_to_Lie_a
lg([Tx,Ty,Tz,Rxy,Rzx,Ryz],euc3):" }}}{EXCHG {PARA 0 "euclid > " 0 "" 
{MPLTEXT 1 0 21 "Lie_alg_init(e3Data):" }}}{EXCHG {PARA 0 "euc3 > " 0 
"" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7*7*%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F
(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(7*F(%$---G%%----GF>F>F>F>F>7*&F+6#F(F)
\"\"!FBFB&F.6#F(,$&F16#F(!\"\"FB7*&F.6#F(F)FBFBFB,$&F+6#F(FHFB&F16#F(7
*&F16#F(F)FBFBFBFB&F+6#F(,$&F.6#F(FH7*&F46#F(F),$FCFHFMFBFB&F:6#F(,$&F
76#F(FH7*&F76#F(F)FFFB,$FTFH,$FgnFHFB&F46#F(7*&F:6#F(F)FB,$FOFHFWFjn,$
FaoFHFB" }}}{EXCHG {PARA 0 "euc3>" 0 "" {MPLTEXT 1 0 25 "LD:=Levi_deco
mposition():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~1G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Computing~Radical~of~euc3G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%/Begin~~Step~2~G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9Computing~Derived~SeriesG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"!7(&%#e1G6#%!G&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6
#F(&%#e6G6#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%)infinityG7(&%#
e2G6#%!G,$&%#e3G6#F(!\"\",$&%#e1G6#F(F-&%#e6G6#F(,$&%#e5G6#F(F-&%#e4G6
#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Begin~~Step~2~G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Computing~Derived~SeriesG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%[pThe~terminal~algebra~in~the~~derived~seri
es~is~initialized~as~Lie~algebra~euc3_PG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7*7*%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#
F(&%#e5G6#F(&%#e6G6#F(7*F(%$---G%%----GF>F>F>F>F>7*&F+6#F(F)\"\"!FBFB,
$&F.6#F(!\"\"FB&F16#F(7*&F.6#F(F)FBFBFB&F+6#F(,$&F16#F(FFFB7*&F16#F(F)
FBFBFBFB&F.6#F(,$&F+6#F(FF7*&F46#F(F)FD,$FLFFFBFB&F:6#F(,$&F76#F(FF7*&
F76#F(F)FBFO,$FTFF,$FgnFFFB&F46#F(7*&F:6#F(F),$FGFFFBFWFjn,$FaoFFFB" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<Computing~Radical~of~euc3_PG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%,End~~Step~2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ERa
dical~of~alg_name2~is~~now~AbelianG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%-Begin~Step~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%QComputing~Semi-S
imple~Part~of~Levi~DecompositionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
&Done!G" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 12 "Show(LD[1])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e1G6#%!G&%#e2G6#F'&%#e3G6#F'
" }}}{EXCHG {PARA 0 "euc3>" 0 "" {MPLTEXT 1 0 12 "Show(LD[2]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e6G6#%!G,$&%#e5G6#F'!\"\"&%#e4G6#
F'" }}}{EXCHG {PARA 0 "euc3>" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 256 "" 0 "" {TEXT -1 28 "The  Euclidean algebra e(4)." }}{PARA 
257 "" 0 "" {TEXT -1 132 "This  time we  generate the Euclidean algebr
a  e(4) as the algebra of isometries of  the  standard metric on R^4 u
sing the  package " }{TEXT 280 10 "isometries" }}{PARA 257 "" 0 "" 
{TEXT -1 18 "The 4 translations" }}{EXCHG {PARA 0 "euclid>" 0 "" 
{MPLTEXT 1 0 31 "with(tensors):with(isometries):" }}}{EXCHG {PARA 0 "e
uclid>" 0 "" {MPLTEXT 1 0 29 "coord_init([x1,x2,x3,x4],[]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{EXCHG {PARA 0 "eu
clid>" 0 "" {MPLTEXT 1 0 30 "g:=canonical_flat_metric(4,0):" }}}
{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 8 "show(g);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,**&&%$dx1G6#%!G\"\"\"&F&6#F(F)F)*&&%$dx2G6#F(F)
&F.6#F(F)F)*&&%$dx3G6#F(F)&F46#F(F)F)*&&%$dx4G6#F(F)&F:6#F(F)F)" }}}
{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 37 "C:=offel2(g): R:=curvat
ure_tensor(C):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 34 "KD
:=Killing_data(g,C,R,[0,0,0,0]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@
Computing~~curvature~derivativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$%?Maximal~~size~of~symmetry~alg:G\"#5" }}}{EXCHG {PARA 0 "euclid
 > " 0 "" {MPLTEXT 1 0 52 "L:=Killing_data_to_Lie_algebra(KD,R,[0,0,0,
0],euc4):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 17 "Lie_alg
_init(L):\n" }}}{EXCHG {PARA 0 "euc4 > " 0 "" {MPLTEXT 1 0 25 "Lie_bra
cket_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(&%$e10G6#F(7.F(%$---G%%----GFJFJFJFJFJFJ
FJFJFJ7.&F+6#F(F)\"\"!FNFNFN*&&#!\"\"\"\"#6#F(\"\"\"&F.6#F(FU*&FP\"\"
\"&F16#F(FU*&FPFY&F46#F(FUFNFNFN7.&F.6#F(F)FNFNFNFN*&&#FUFSFTFU&F+6#F(
FUFNFN*&FPFY&F16#F(FU*&FPFY&F46#F(FUFN7.&F16#F(F)FNFNFNFNFN*&F]oFY&F+6
#F(FUFN*&F]oFY&F.6#F(FUFN*&FPFY&F46#F(FU7.&F46#F(F)FNFNFNFNFNFN*&F]oFY
&F+6#F(FUFN*&F]oFY&F.6#F(FU*&F]oFY&F16#F(FU7.&F76#F(F),$FOFR,$F\\oFRFN
FNFN*&F]oFY&F@6#F(FU*&F]oFY&FC6#F(FU*&FPFY&F:6#F(FU*&FPFY&F=6#F(FUFN7.
&F:6#F(F),$FXFRFN,$FjoFRFN,$FdqFRFN*&F]oFY&FF6#F(FU*&F]oFY&F76#F(FUFN*
&FPFY&F=6#F(FU7.&F=6#F(F),$FfnFRFNFN,$FfpFR,$FgqFR,$FfrFRFNFN*&F]oFY&F
76#F(FU*&F]oFY&F:6#F(FU7.&F@6#F(F)FN,$FaoFR,$F]pFRFN,$FjqFR,$FirFRFNFN
*&F]oFY&FF6#F(FU*&FPFY&FC6#F(FU7.&FC6#F(F)FN,$FdoFRFN,$FipFR,$F]rFRFN,
$FfsFR,$FctFRFN*&F]oFY&F@6#F(FU7.&FF6#F(F)FNFN,$F`pFR,$F\\qFRFN,$F\\sF
R,$FisFR,$FftFR,$FauFRFN" }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 
25 "LD:=Levi_decomposition():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Beg
in~Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Computing~Radical~of~e
uc4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Begin~~Step~2~G" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%9Computing~Derived~SeriesG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"\"!7,&%#e1G6#%!G&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5
G6#F(&%#e6G6#F(&%#e7G6#F(&%#e8G6#F(&%#e9G6#F(&%$e10G6#F(\"#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%%)infinityG7,*&&#!\"\"\"\"#6#%!G\"\"\"&%#e2G
6#F+F,*&F&\"\"\"&%#e3G6#F+F,*&F&F1&%#e4G6#F+F,*&&#F,F)F*F,&%#e1G6#F+F,
*&F:F1&%#e8G6#F+F,*&F:F1&%#e9G6#F+F,*&F&F1&%#e6G6#F+F,*&F&F1&%#e7G6#F+
F,*&F:F1&%$e10G6#F+F,*&F:F1&%#e5G6#F+F,\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%/Begin~~Step~2~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9
Computing~Derived~SeriesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[pThe~te
rminal~algebra~in~the~~derived~series~is~initialized~as~Lie~algebra~eu
c4_PG" }}{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(&%$e10G6#F(7.F(%$---G%%----GFJFJFJFJFJFJFJFJFJ7.&F+6#
F(F)\"\"!FNFNFN*&&#!\"\"\"\"%6#F(\"\"\"&F.6#F(FU*&FP\"\"\"&F16#F(FUFNF
NFN*&FPFY&F46#F(FU7.&F.6#F(F)FNFNFNFN*&&#FUFSFTFU&F+6#F(FUFN*&F]oFY&F4
6#F(FUFN*&FPFY&F16#F(FUFN7.&F16#F(F)FNFNFNFNFN*&F]oFY&F+6#F(FUFN*&F]oF
Y&F46#F(FU*&F]oFY&F.6#F(FUFN7.&F46#F(F)FNFNFNFNFNFN*&FPFY&F.6#F(FU*&FP
FY&F16#F(FUFN*&F]oFY&F+6#F(FU7.&F76#F(F),$FOFR,$F\\oFRFNFNFN*&F]oFY&FC
6#F(FU*&F]oFY&FF6#F(FUFN*&FPFY&F:6#F(FU*&FPFY&F=6#F(FU7.&F:6#F(F),$FXF
RFN,$FjoFRFN,$FdqFRFNFN*&F]oFY&FF6#F(FU*&F]oFY&F76#F(FU*&FPFY&F@6#F(FU
7.&F=6#F(F)FN,$FaoFRFN,$FfpFR,$FgqFRFNFN*&F]oFY&FC6#F(FU*&FPFY&F@6#F(F
U*&F]oFY&F76#F(FU7.&F@6#F(F)FNFN,$F]pFR,$FipFRFN,$FfrFR,$FesFRFN*&F]oF
Y&F=6#F(FU*&F]oFY&F:6#F(FU7.&FC6#F(F)FN,$FdoFR,$F`pFRFN,$FjqFR,$FirFR,
$FhsFR,$FetFRFNFN7.&FF6#F(F),$FfnFRFNFN,$F\\qFR,$F]rFR,$F\\sFR,$F[tFR,
$FhtFRFNFN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<Computing~Radical~of~e
uc4_PG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,End~~Step~2G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%ERadical~of~alg_name2~is~~now~AbelianG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%-Begin~Step~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%QCo
mputing~Semi-Simple~Part~of~Levi~DecompositionG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%&Done!G" }}}{EXCHG {PARA 0 "euc4 > " 0 "" {MPLTEXT 1 
0 12 "Show(LD[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%#e1G6#%!G&%#
e4G6#F'&%#e2G6#F'&%#e3G6#F'" }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 
1 0 12 "Show(LD[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(*&&#\"\"\"\"
\"#6#%!GF'&%#e8G6#F*F'*&F%\"\"\"&%#e9G6#F*F'*&&#!\"\"F(F)F'&%#e6G6#F*F
'*&F4F/&%#e7G6#F*F'*&F%F/&%$e10G6#F*F'*&F%F/&%#e5G6#F*F'" }}}{EXCHG 
{PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 54 "C:=canonical_subspace_basis(LD[2
],frameBaseVectors()):" }}}{EXCHG {PARA 0 "euc4 > " 0 "" {MPLTEXT 1 0 
8 "Show(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%#e5G6#%!G&%#e6G6#F'
&%#e7G6#F'&%#e8G6#F'&%#e9G6#F'&%$e10G6#F'" }}}{EXCHG {PARA 0 "euc4>" 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 48 "A se
mi-direct product of so(3) and L(4,2,[1,1])." }}{PARA 257 "" 0 "" 
{TEXT -1 91 "The following  example comes from the  paper  by L. Basar
ab-Horwath, V. Lahno, R. Zhdanov, " }{TEXT 281 96 "The structure of Li
e algebras and the classification problem for partial differential equ
ations." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc4>" 0 
"" {MPLTEXT 1 0 22 "with(Vessiot_library);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7#%(Lie_libG" }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 
1 0 30 "L1:=Lie_lib(winternitz,[3,6]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#L1G7$7%%(Lie_algG%$w36G7#\"\"$7%7$7%\"\"\"\"\"#F*F.7$7%F.F*F/
!\"\"7$7%F/F*F.F." }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 36 "L2:
=Lie_lib(winternitz,[4,2],[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#L2G7$7%%(Lie_algG%$w42G7#\"\"%7%7$7%\"\"\"F*F.F.7$7%\"\"#F*F1F.7$7%
\"\"$F*F4F." }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 13 "?create_s
emi;" }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 256 "" 0 "" {TEXT -1 26 "The optical group opt(2,1)" }}{PARA 
257 "" 0 "" {TEXT -1 97 "This is the example used to illustrate the pr
ogram in Rand, Winternitz, Zassenhaus, page 224-226." }}{PARA 257 "" 
0 "" {TEXT -1 151 "The  derived series  terminates at a 6 dimensional \+
subalgebra which is  initialized as opt21_P -- the basis is  K1->e1,K2
->e2,K3->e3,M->e4,Q->e5,N->e6." }}{PARA 257 "" 0 "" {TEXT -1 47 "This \+
 algebra s  re-initialized as opt21_S4_0. " }}{PARA 257 "" 0 "" {TEXT 
-1 87 "Its radical is  e4,e5,e6 which has derived algebra e6. Thus the
 radical is not abelian." }}{PARA 257 "" 0 "" {TEXT -1 56 "Factor out \+
e6. The result is the algebra opt21_S4_0_Q.  " }}{PARA 257 "" 0 "" 
{TEXT -1 46 "Find the Levi decomposition of  opt21_S4_0_Q. " }}{PARA 
257 "" 0 "" {TEXT -1 49 "Construct the  subalgebra  L_1  by equation 3
.21." }}{PARA 257 "" 0 "" {TEXT -1 70 "Loop through Step 3 again.   Th
is time the  radical of L_1 is abelian." }}{PARA 257 "" 0 "" {TEXT -1 
105 " In Step 4 construct the Levi-decomposition for the original alge
bra from the Levi decomposition of L_1. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc4>" 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "euc4>" 0 "" {MPLTEXT 1 0 242 "opt21Data:=structure_equ
ations_to_Lie_algebra_data([W,K1,K2,L3,M,Q,N],[[W,M]=1/2*M, [W,N]=N, [
W,Q]=1/2*Q,[K1,K2]=-L3,[K1,L3]=-K2, [K1,M]=-1/2*M,[K1,Q]=1/2*Q ,[K2,L3
]=K1, [K2,M]=1/2*Q, [K2,Q]=1/2*M, [L3,M]=-1/2*Q,[L3,Q]=1/2*M,[M,Q]=-N]
,opt21):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Lie_alg_init(op
t21Data,[W,K1,K2,L3,M,Q,N],[alpha]):" }}}{EXCHG {PARA 0 "opt21 > " 0 "
" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7+7+%!G%\"|grG&%\"WG6#%!G&%#K1G6#%!G&%#K2G6
#%!G&%#L3G6#%!G&%\"MG6#%!G&%\"QG6#%!G&%\"NG6#%!G7+F(%$---G%%----GFHFHF
HFHFHFH7+&F+6#%!GF)\"\"!FMFMFM*&&#\"\"\"\"\"#6#F(FQ&F;6#%!GFQ*&FO\"\"
\"&F?6#%!GFQ&FC6#%!G7+&F/6#%!GF)FMFM,$&F76#%!G!\"\",$&F36#%!GFao*&&#Fa
oFRFSFQ&F;6#%!GFQ*&FOFX&F?6#%!GFQFM7+&F36#%!GF)FMF^oFM&F/6#%!G*&FOFX&F
?6#%!GFQ*&FOFX&F;6#%!GFQFM7+&F76#%!GF)FMFco,$FdpFaoFM*&FgoFX&F?6#%!GFQ
*&FOFX&F;6#%!GFQFM7+&F;6#%!GF),$FNFao,$FfoFao,$FgpFao,$FdqFaoFM,$&FC6#
%!GFaoFM7+&F?6#%!GF),$FWFao,$F\\pFao,$F[qFao,$FhqFaoFerFMFM7+&FC6#%!GF
),$FfnFaoFMFMFMFMFMFM" }}}{EXCHG {PARA 0 "opt21 > " 0 "" {MPLTEXT 1 0 
25 "LD:=Levi_decomposition();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Beg
in~Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;Computing~Radical~of~o
pt21G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Begin~~Step~2~G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%9Computing~Derived~SeriesG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"\"!7)&%\"WG6#%!G&%#K1G6#%!G&%#K2G6#%!G&%#L3G6#%!
G&%\"MG6#%!G&%\"QG6#%!G&%\"NG6#%!G\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"\"\"7(*&&#F#\"\"#6#%!GF#&%\"MG6#%!GF#*&F&\"\"\"&%\"QG6#%!GF#&%
\"NG6#%!G,$&%#L3G6#%!G!\"\",$&%#K2G6#%!GF>&%#K1G6#%!G\"\"'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%%)infinityG7(*&&#!\"\"\"\"%6#%!G\"\"\"&%\"NG
6#%!GF,*&F&\"\"\"&%\"QG6#%!GF,*&&#F,F)F*F,&%\"MG6#%!GF,,$&%#K1G6#%!GF(
,$&%#K2G6#%!GF(,$&%#L3G6#%!GF(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%/Begin~~Step~2~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gnThe~basis~for
~the~terminal~algebra~in~the~derived~series~is:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7(&%#K1G6#%!G&%#K2G6#%!G&%#L3G6#%!G&%\"MG6#%!G&%\"QG6#%
!G&%\"NG6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\pThe~terminal~alge
bra~in~the~~derived~series~is~initialized~as~Lie~algebra~opt21_PG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7*7*%!G%\"|grG&%#e1G6#%!G&
%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%!G&%#e6G6#%!G7*F(%$---G%%----
GFDFDFDFDFD7*&F+6#%!GF)\"\"!,$&F36#%!G!\"\",$&F/6#%!GFN*&&#FN\"\"#6#F(
\"\"\"&F76#%!GFX*&&#FXFVFWFX&F;6#%!GFXFI7*&F/6#%!GF)FKFI&F+6#%!G*&Fgn
\"\"\"&F;6#%!GFX*&FgnFdo&F76#%!GFXFI7*&F36#%!GF)FP,$F`oFNFI*&FTFdo&F;6
#%!GFX*&FgnFdo&F76#%!GFXFI7*&F76#%!GF),$FSFN,$FcoFN,$FapFNFI,$&F?6#%!G
FNFI7*&F;6#%!GF),$FfnFN,$FhoFN,$FepFNFaqFIFI7*&F?6#%!GF)FIFIFIFIFIFI" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Computing~Radical~of~opt21_PG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%,End~~Step~2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RTh
e~~radical~of~the~subalgebra~L_0~is~not~AbelianG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%VInitializing~subalgebra~L_0~as~Lie~algebra~opt21_S4_0
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>The~radical~of~opt21_S4_0~is:G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e5G6#%!G&%#e4G6#%!G&%#e6G6#%!
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe~derived~algebra~of~~radical
~of~opt21_S4_0~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e6G6#%!G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%LThe~~complementary~basis~to~the~ra
dical~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'&%#e1G6#%!G&%#e2G6#%!G
&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
fnThe~Factor~Algebra~of~opt21_S4_0~by~Radical~is~opt21_S4_0_QG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#%!G&
%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%!G7)F(%$---G%%----GF@F@F@F@7)
&F+6#%!GF)\"\"!,$&F36#%!G!\"\",$&F/6#%!GFJ*&&#FJ\"\"#6#F(\"\"\"&F76#%!
GFT*&&#FTFRFSFT&F;6#%!GFT7)&F/6#%!GF)FGFE&F+6#%!G*&FY\"\"\"&F;6#%!GFT*
&FYF`o&F76#%!GFT7)&F36#%!GF)FL,$F\\oFJFE*&FPF`o&F;6#%!GFT*&FYF`o&F76#%
!GFT7)&F76#%!GF),$FOFJ,$F_oFJ,$F]pFJFEFE7)&F;6#%!GF),$FXFJ,$FdoFJ,$Fap
FJFEFE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCompute~Levi~decomposition
~of~Factor~AlgebraG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%AThe~~radical~
of~opt21_S4_0_Q~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e5G6#%!G&
%#e4G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OThe~semi-simple~subalg
ebra~of~opt21_S4_0_Q~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e1G6
#%!G&%#e2G6#%!G&%#e3G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~su
balgebra~L_1isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%\"NG6#%!G&%#K1G
6#%!G&%#K2G6#%!G&%#L3G6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin
~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ERadical~of~alg_name2~is~
~now~AbelianG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~4G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%QComputing~Semi-Simple~Part~of~Levi~D
ecompositionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Done!G" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%#LDG7$7&7$7%%%vectG%&opt21G7\"7#7$7#\"\"&\"\"
\"7$F(7#7$7#F0F07$F(7#7$7#\"\"(F07$F(7#7$7#\"\"'F07%7$F(7#7$7#\"\"#F07
$F(7#7$7#\"\"$F07$F(7#7$7#\"\"%F0" }}}{EXCHG {PARA 0 "opt21 > " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{MARK "8 9 0" 12 }{VIEWOPTS 1 1 0 3 4 1802 }