{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 "Times" 1 14 0 0 255 1 0 1 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 "" 1 14 0 0 0 0 0 0 0 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 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 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 "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 "Helveti ca" 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_Tit le1" -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 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 262 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 " " 256 263 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 93 " \+ Vessiot Tutorial: Relative Lie Algebra Cohomology and" }} {PARA 258 "" 0 "" {TEXT -1 47 " the de Rham cohomology of Homogeneous \+ Spaces. " }}{PARA 260 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 " " {TEXT 259 99 "In this tutorial the relative Lie algebra cohomology of some classical Lie algebras is computed. " }}{PARA 257 "" 0 "" {TEXT 260 169 "Let G be a compact Lie group and H a closed subgroup . Then the relative Lie algebra cohomologyH^*(g,h) computes the de Rh am cohomology of the homogeneous space G/H." }}{PARA 257 "" 0 "" {TEXT -1 117 "See the tutorial Classical Matrix Algebras for detail s on the construction of the algebras used in this tutorial." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 258 22 "Proced ures Illustrated" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "L ie _algebra_cohomology" 2 "Lie _algebra_cohomology" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "relative_chains" 2 "relative_chains" "" }{TEXT -1 3 " , " }{HYPERLNK 17 "create_gl" 2 "create_gl" "" }{TEXT -1 3 ", " } {HYPERLNK 17 "create_gl_subalgebra" 2 "create_gl_subalgebra" "" } {TEXT -1 3 ", " }{HYPERLNK 17 "subalgebra_to_Lie_algebra_data" 2 "sub algebra_to_Lie_algebra_data" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "classi fy_Lie_algebra" 2 "classify_Lie_algebra" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "create_intersection_of_subalgebras" 2 "create_intersecti on_of_subalgebras" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "canonical_flat_ metric" 2 "canonical_flat_metric" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "c anonical_symplectic_form" 2 "canonical_sympletic_form" "" }{TEXT -1 20 ", " }{HYPERLNK 17 "canonical_complex_structure" 2 "canonical_complex_structure" "" }{TEXT -1 2 " " }}{EXCHG {PARA 0 " gl6R > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 41 "with(Vessiot):with(Koszul):with(tensors):" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 262 1 " " }{TEXT 266 10 "Example 1." }{TEXT 267 29 " The 3 sphere as s o(4)/so(3)." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 46 "Construct the Lie algebra pair (so(4), so(3))." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 45 "Compute th e relative Lie algebra cohomology." }}{PARA 256 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT 256 46 "Check that (so(4), so(3)) is sym metry pair. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "coord_frame([x1,x2,x3,x4],[],E4):" }}}{EXCHG {PARA 0 "E4>" 0 "" {MPLTEXT 1 0 18 "gl4:=create_gl(4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG% " 0 "" {MPLTEXT 1 0 19 " Lie_alg_init(gl4 ):" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 20 "change_frame_to(E4):" }}} {EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 46 "g:=canonical_flat_metric( 4,0,bas):V:=vect(x4):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 43 "so4_subalg:=create_g l_subalgebra(gl4R,[g]):" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 45 "so3_subalg:=create_gl_subalgebra(gl4R,[g,V]):" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 83 "so4_so3data:=subalgebra_pair_to_Lie_algebra_data_pair (so4_subalg,[so3_subalg],so4):" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 20 "so3:=so4_so3data[2]:" }}}{EXCHG {PARA 0 "so4 > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 24 "C :=relative_chains(so3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG7&7# \"\"\"7#*&%#0~GF'&%'theta1G6#%!GF'7#**F*\"\"\"&F,6#F.F'%#^~GF'&%'theta 2G6#F.F'7#*,&%'theta3G6#F.F'F4F'&%'theta5G6#F.F'F4F'&%'theta6G6#F.F'" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 29 "H:=Lie_algebra_cohomo logy(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7%7#*&%#0~G\"\"\"&%' theta1G6#%!GF)7#**F(\"\"\"&F+6#F-F)%#^~GF)&%'theta2G6#F-F)7#*,&%'theta 3G6#F-F)F3F)&%'theta5G6#F-F)F3F)&%'theta6G6#F-F)" }}}{EXCHG {PARA 0 "s o4 > " 0 "" {MPLTEXT 1 0 34 "M:=reductive_complement(so3,[mu]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7%,&&%#e3G6#%!G\"\"\"*&%#muGF+&% #e4G6#F*F+F+,&*&F-\"\"\"&%#e2G6#F*F+!\"\"&%#e5G6#F*F+,&*&F-F3&%#e1G6#F *F+F+&%#e6G6#F*F+" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 28 "ch eck_symmetric_pair(so3,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG " }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 31 "M0:=map(helmsimp,su bs(mu=0,M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M0G7%&%#e3G6#%!G&%# e5G6#F)&%#e6G6#F)" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 29 "ch eck_symmetric_pair(so3,M0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 261 1 " " }{TEXT 268 10 "Example 2." }{TEXT 269 29 " The 4 sphere as so(5) /so(4)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 17 "repeat Example 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 18 "gl5:=create_gl(5):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG% " 0 "" {MPLTEXT 1 0 19 " Lie_al g_init(gl5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~gl5RG" }}}{EXCHG {PARA 0 "gl5R > " 0 "" {MPLTEXT 1 0 36 "coord_frame([x1,x2,x 3,x4,x5],[],E5):" }}}{EXCHG {PARA 0 "E5>" 0 "" {MPLTEXT 1 0 46 "g:=can onical_flat_metric(5,0,bas):V:=vect(x5):" }}}{EXCHG {PARA 0 "E5 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E5 > " 0 "" {MPLTEXT 1 0 43 "s o5_subalg:=create_gl_subalgebra(gl5R,[g]):" }}}{EXCHG {PARA 0 "gl5R > \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl5R > " 0 "" {MPLTEXT 1 0 45 "so4_subalg:=create_gl_subalgebra(gl5R,[g,V]):" }}}{EXCHG {PARA 0 "gl5R > " 0 "" {MPLTEXT 1 0 83 "so5_so4data:=subalgebra_pair_to_Lie_ algebra_data_pair(so5_subalg,[so4_subalg],so5):" }}}{EXCHG {PARA 0 "so 5 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "so5 > " 0 "" {MPLTEXT 1 0 20 "so4:=so5_so4data[2]:" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"&wE#!\"$" }}}{EXCHG {PARA 0 "so5 > " 0 "" {MPLTEXT 1 0 24 "C:=rel ative_chains(so4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG7'7#\"\"\" 7#*&%#0~GF'&%'theta1G6#%!GF'7#**F*\"\"\"&F,6#F.F'%#^~GF'&%'theta2G6#F. F'7#*.F*F1&F,6#F.F'F4F'&F66#F.F'F4F'&%'theta3G6#F.F'7#*0&%'theta4G6#F. F'F4F'&%'theta7G6#F.F'F4F'&%'theta9G6#F.F'F4F'&%(theta10G6#F.F'" }}} {EXCHG {PARA 0 "so5 > " 0 "" {MPLTEXT 1 0 29 "H:=Lie_algebra_cohomolog y(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7&7#*&%#0~G\"\"\"&%'the ta1G6#%!GF)7#**F(\"\"\"&F+6#F-F)%#^~GF)&%'theta2G6#F-F)7#*.F(F0&F+6#F- F)F3F)&F56#F-F)F3F)&%'theta3G6#F-F)7#*0&%'theta4G6#F-F)F3F)&%'theta7G6 #F-F)F3F)&%'theta9G6#F-F)F3F)&%(theta10G6#F-F)" }}}{EXCHG {PARA 0 "so5 > " 0 "" {MPLTEXT 1 0 29 "M:=reductive_complement(so4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"MG7&&%#e4G6#%!G&%#e7G6#F)&%#e9G6#F)&%$e10G6# F)" }}}{EXCHG {PARA 0 "so5 > " 0 "" {MPLTEXT 1 0 28 "check_symmetric_p air(so4,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 270 39 "Example 3. The 5 sphere as su(3)/su(2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 53 "The homogeneous space su(3)/su(2) is not symmetric." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "so5 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_ gl(6)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG% " 0 "" {MPLTEXT 1 0 40 "coord_frame([x1,x2,x3,y1,y2,y3],[u],E6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~E6G" }}}{EXCHG {PARA 0 "E6>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E6>" 0 "" {MPLTEXT 1 0 34 "g:=can onical_flat_metric(6,0,bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG ,.*&&%$dy3G6#%!G\"\"\"&F(6#F*F+F+*&&%$dy1G6#F*F+&F06#F*F+F+*&&%$dx1G6# F*F+&F66#F*F+F+*&&%$dx2G6#F*F+&F<6#F*F+F+*&&%$dy2G6#F*F+&FB6#F*F+F+*&& %$dx3G6#F*F+&FH6#F*F+F+" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "J:=canonical_complex_structure(bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG,.*&&%$dx1G6#%!G\"\"\"&%%D_y1G6#F*F+!\"\"*&&%$dx2G6#F*F+&% %D_y2G6#F*F+F/*&&%$dy3G6#F*F+&%%D_x3G6#F*F+F+*&&%$dy2G6#F*F+&%%D_x2G6# F*F+F+*&&%$dx3G6#F*F+&%%D_y3G6#F*F+F/*&&%$dy1G6#F*F+&%%D_x1G6#F*F+F+" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "E6 > " 0 " " {MPLTEXT 1 0 36 "dz1:=v_zip([1,I],[x1,y1],plus,form):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz2:=v_zip([1,I],[x2,y2],plus,form):" }} {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz3:=v_zip([1,I],[x3,y3],plus,fo rm):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 29 "nu:=dz1&wedge dz2 &wedge dz3:" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuR:= (1/2) &mult ( nu \+ &plus Vessiot_map(conjugate,nu));" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuI:=(I/2) &mult ( nu &minus Vessiot_map(conjugate,nu));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nuRG,**,&%$dx1G6#%!G\"\"\"%#^~GF+&% $dx2G6#F*F+F,F+&%$dx3G6#F*F+F+*,&F(6#F*F+F,F+&%$dy2G6#F*F+F,F+&%$dy3G6 #F*F+!\"\"*,&F.6#F*F+F,F+&%$dy1G6#F*F+F,F+&F:6#F*F+F+*,&F16#F*F+F,F+&F A6#F*F+F,F+&F76#F*F+F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nuIG,**,& %$dx1G6#%!G\"\"\"%#^~GF+&%$dx2G6#F*F+F,F+&%$dy3G6#F*F+!\"\"*,&F(6#F*F+ F,F+&%$dx3G6#F*F+F,F+&%$dy2G6#F*F+F+*,&F.6#F*F+F,F+&F86#F*F+F,F+&%$dy1 G6#F*F+F3*,&FC6#F*F+F,F+&F;6#F*F+F,F+&F16#F*F+F+" }}}{EXCHG {PARA 0 "E 6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 12 "V:=vect(y3):" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 53 "su3_subalg:=create_g l_subalgebra(gl6R,[g,J,nuI,nuR]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %+su3_subalgG7*,*&%$e12G6#%!G\"\"\"&%$e21G6#F*!\"\"&%$e45G6#F*F+&%$e54 G6#F*F/,*&%$e13G6#F*F+&%$e31G6#F*F/&%$e46G6#F*F+&%$e64G6#F*F/,*&%$e14G 6#F*F+&%$e36G6#F*F/&%$e41G6#F*F/&%$e63G6#F*F+,*&%$e15G6#F*F+&%$e24G6#F *F+&%$e42G6#F*F/&%$e51G6#F*F/,*&%$e16G6#F*F+&%$e34G6#F*F+&%$e43G6#F*F/ &%$e61G6#F*F/,*&%$e23G6#F*F+&%$e32G6#F*F/&%$e56G6#F*F+&%$e65G6#F*F/,*& %$e25G6#F*F+&FH6#F*F/&%$e52G6#F*F/&FN6#F*F+,*&%$e26G6#F*F+&%$e35G6#F*F +&%$e53G6#F*F/&%$e62G6#F*F/" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 55 "su2_subalg:=create_gl_subalgebra(gl6R,[g,J,nuI,nuR,V] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+su2_subalgG7%,*&%$e12G6#%!G\" \"\"&%$e21G6#F*!\"\"&%$e45G6#F*F+&%$e54G6#F*F/,*&%$e14G6#F*F+&%$e25G6# F*F/&%$e41G6#F*F/&%$e52G6#F*F+,*&%$e15G6#F*F+&%$e24G6#F*F+&%$e42G6#F*F /&%$e51G6#F*F/" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 84 "su3_ su2_data:=subalgebra_pair_to_Lie_algebra_data_pair(su3_subalg,[su2_sub alg],su3):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 21 "su2:=su3_ su2_data[2]:" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 24 "C:=r elative_chains(su2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"CG7(7#\"\" \"7#,&&%'theta3G6#%!GF'&%'theta7G6#F-F'7%,&*(&%'theta2G6#F-F'%#^~GF'&% 'theta6G6#F-F'!\"\"*(&%'theta5G6#F-F'F7F'&%'theta8G6#F-F'F',&*(&F56#F- F'F7F'&FA6#F-F'F'*(&F>6#F-F'F7F'&F96#F-F'F',&*(&F56#F-F'F7F'&F>6#F-F'F '*(&F96#F-F'F7F'&FA6#F-F'F'7%,**,&F56#F-F'F7F'&F+6#F-F'F7F'&FA6#F-F'F; *,&F56#F-F'F7F'&F/6#F-F'F7F'&FA6#F-F'F;*,&F+6#F-F'F7F'&F>6#F-F'F7F'&F9 6#F-F'F'*,&F>6#F-F'F7F'&F96#F-F'F7F'&F/6#F-F'F',**,&F56#F-F'F7F'&F+6#F -F'F7F'&F>6#F-F'F'*,&F56#F-F'F7F'&F>6#F-F'F7F'&F/6#F-F'F;*,&F+6#F-F'F7 F'&F96#F-F'F7F'&FA6#F-F'F;*,&F96#F-F'F7F'&F/6#F-F'F7F'&FA6#F-F'F',**,& F56#F-F'F7F'&F+6#F-F'F7F'&F96#F-F'F'*,&F56#F-F'F7F'&F96#F-F'F7F'&F/6#F -F'F;*,&F+6#F-F'F7F'&F>6#F-F'F7F'&FA6#F-F'F'*,&F>6#F-F'F7F'&F/6#F-F'F7 F'&FA6#F-F'F;7#*0&F56#F-F'F7F'&F>6#F-F'F7F'&F96#F-F'F7F'&FA6#F-F'7#,&* 4&F56#F-F'F7F'&F+6#F-F'F7F'&F>6#F-F'F7F'&F96#F-F'F7F'&FA6#F-F'F'*4&F56 #F-F'F7F'&F>6#F-F'F7F'&F96#F-F'F7F'&F/6#F-F'F7F'&FA6#F-F'F'" }}} {EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 29 "H:=Lie_algebra_cohomology(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7'7#*&%#0~G\"\"\"&%'theta1G6#%!GF)7#**F(\"\"\"&F+ 6#F-F)%#^~GF)&%'theta2G6#F-F)7#*.F(F0&F+6#F-F)F3F)&F56#F-F)F3F)&%'thet a3G6#F-F)7#*2F(F0&F+6#F-F)F3F)&F56#F-F)F3F)&F>6#F-F)F3F)&%'theta4G6#F- F)7#,&*4&F56#F-F)F3F)&F>6#F-F)F3F)&%'theta5G6#F-F)F3F)&%'theta6G6#F-F) F3F)&%'theta8G6#F-F)F)*4&F56#F-F)F3F)&FS6#F-F)F3F)&FV6#F-F)F3F)&%'thet a7G6#F-F)F3F)&FY6#F-F)F)" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 29 "M:=reductive_complement(su2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"MG7'&%#e2G6#%!G,&*&&#\"\"\"\"\"#6#F)F.&%#e3G6#F)F.F.*&F,\"\"\"&%# e7G6#F)F.F.&%#e5G6#F)&%#e6G6#F)&%#e8G6#F)" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 28 "check_ symmetric_pair(su2,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} }{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 271 40 " Example 4. The 7 sphere as sp(2)/sp(1)." }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 62 "The \+ homogeneous space sp(2)/sp(1) is not a symmetric space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 46 " coord_frame([x1,y1,u1,v1,x2,y2,u2,v2],[u],E8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~E8G" }}}{EXCHG {PARA 0 "E8>" 0 "" {MPLTEXT 1 0 36 "J:=canonical_complex_structure(bas):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 67 "K1 :=evalV( -dx1 &t D_u1 - dy1 & t D_v1 +du1 &t D_x1 + dv1 &t D_y1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#K1G,**&&%$dx1G6#%!G\"\"\"&%%D_u1G6#F*F+!\"\"*&&%$dy1G6#F*F+&%%D_v1 G6#F*F+F/*&&%$du1G6#F*F+&%%D_x1G6#F*F+F+*&&%$dv1G6#F*F+&%%D_y1G6#F*F+F +" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 66 "K2:=evalV( -dx2 &t \+ D_u2 - dy2 &t D_v2 +du2 &t D_x2 + dv2 &t D_y2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K2G,**&&%$dx2G6#%!G\"\"\"&%%D_u2G6#F*F+!\"\"*&&%$dy2 G6#F*F+&%%D_v2G6#F*F+F/*&&%$du2G6#F*F+&%%D_x2G6#F*F+F+*&&%$dv2G6#F*F+& %%D_y2G6#F*F+F+" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 17 "K:= K1 &minus K2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"KG,2*&&%$dx1G6#%!G\"\"\"&%%D_u1G6#F*F+!\"\"* &&%$dy1G6#F*F+&%%D_v1G6#F*F+F/*&&%$du1G6#F*F+&%%D_x1G6#F*F+F+*&&%$dv1G 6#F*F+&%%D_y1G6#F*F+F+*&&%$dx2G6#F*F+&%%D_u2G6#F*F+F+*&&%$dy2G6#F*F+&% %D_v2G6#F*F+F+*&&%$du2G6#F*F+&%%D_x2G6#F*F+F/*&&%$dv2G6#F*F+&%%D_y2G6# F*F+F/" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 34 "g:=canonical_f lat_metric(8,0,bas):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 12 " V:=vect(v2):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(8) ):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG% " 0 "" {MPLTEXT 1 0 47 "sp2_subalg:=create_gl_subalgebra(gl8R,[J,K,g]):" }}}{EXCHG {PARA 0 "g l8R > " 0 "" {MPLTEXT 1 0 49 "sp1_subalg:=create_gl_subalgebra(gl8R,[J ,K,g,V]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " gl8R > " 0 "" {MPLTEXT 1 0 84 "sp2_sp1_data:=subalgebra_pair_to_Lie_al gebra_data_pair(sp2_subalg,[sp1_subalg],sp2):" }}}{EXCHG {PARA 0 "sp2 \+ > " 0 "" {MPLTEXT 1 0 21 "sp1:=sp2_sp1_data[2]:" }}}{EXCHG {PARA 0 "sp 2 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 24 "C:=relative_chains(sp1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% \"CG7*7#\"\"\"7%&%(theta10G6#%!G&%'theta8G6#F,&%'theta9G6#F,7(*(&F.6#F ,F'%#^~GF'&F*6#F,F',&*(&%'theta1G6#F,F'F7F'&%'theta7G6#F,F'F'*(&%'thet a3G6#F,F'F7F'&%'theta5G6#F,F'F',&*(&F=6#F,F'F7F'&FD6#F,F'F'*(&FG6#F,F' F7F'&F@6#F,F'F'*(&F.6#F,F'F7F'&F16#F,F',&*(&F=6#F,F'F7F'&FG6#F,F'F'*(& FD6#F,F'F7F'&F@6#F,F'!\"\"*(&F16#F,F'F7F'&F*6#F,F'7,,&*,&F=6#F,F'F7F'& FD6#F,F'F7F'&F*6#F,F'F'*,&FG6#F,F'F7F'&F@6#F,F'F7F'&F*6#F,F'F',&*,&F=6 #F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F'*,&FD6#F,F'F7F'&FG6#F,F'F7F'&F.6#F,F' F',&*,&F=6#F,F'F7F'&F@6#F,F'F7F'&F16#F,F'F'*,&FD6#F,F'F7F'&FG6#F,F'F7F '&F16#F,F'F',&*,&F=6#F,F'F7F'&F@6#F,F'F7F'&F*6#F,F'F'*,&FD6#F,F'F7F'&F G6#F,F'F7F'&F*6#F,F'F',&*,&F=6#F,F'F7F'&FG6#F,F'F7F'&F.6#F,F'F^o*,&FD6 #F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F',&*,&F=6#F,F'F7F'&FG6#F,F'F7F'&F16#F, F'F^o*,&FD6#F,F'F7F'&F@6#F,F'F7F'&F16#F,F'F'*,&F.6#F,F'F7F'&F16#F,F'F7 F'&F*6#F,F',&*,&F=6#F,F'F7F'&FD6#F,F'F7F'&F16#F,F'F'*,&FG6#F,F'F7F'&F@ 6#F,F'F7F'&F16#F,F'F',&*,&F=6#F,F'F7F'&FD6#F,F'F7F'&F.6#F,F'F'*,&FG6#F ,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F',&*,&F=6#F,F'F7F'&FG6#F,F'F7F'&F*6#F,F' F^o*,&FD6#F,F'F7F'&F@6#F,F'F7F'&F*6#F,F'F'7,,&*0&F=6#F,F'F7F'&FG6#F,F' F7F'&F.6#F,F'F7F'&F*6#F,F'F'*0&FD6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'& F*6#F,F'F^o,&*0&F=6#F,F'F7F'&FD6#F,F'F7F'&F.6#F,F'F7F'&F*6#F,F'F'*0&FG 6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F*6#F,F'F'*0&F=6#F,F'F7F'&FD6#F,F 'F7F'&FG6#F,F'F7F'&F@6#F,F',&*0&F=6#F,F'F7F'&FG6#F,F'F7F'&F.6#F,F'F7F' &F16#F,F'F'*0&FD6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F^o,&*0&F =6#F,F'F7F'&FD6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F'*0&FG6#F,F'F7F'&F@6#F, F'F7F'&F.6#F,F'F7F'&F16#F,F'F',&*0&F=6#F,F'F7F'&FD6#F,F'F7F'&F16#F,F'F 7F'&F*6#F,F'F'*0&FG6#F,F'F7F'&F@6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F',&*0 &F=6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F*6#F,F'F'*0&FD6#F,F'F7F'&FG6# F,F'F7F'&F.6#F,F'F7F'&F*6#F,F'F',&*0&F=6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F 'F7F'&F16#F,F'F'*0&FD6#F,F'F7F'&FG6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F',& *0&F=6#F,F'F7F'&F@6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F'*0&FD6#F,F'F7F'&FG 6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F',&*0&F=6#F,F'F7F'&FG6#F,F'F7F'&F16#F ,F'F7F'&F*6#F,F'F'*0&FD6#F,F'F7F'&F@6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F^ o7(,&*4&F=6#F,F'F7F'&FD6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F' *4&FG6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F',&*4& F=6#F,F'F7F'&FG6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F^o*4&FD6# F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F',&*4&F=6#F,F 'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F'*4&FD6#F,F'F7F' &FG6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'F'*4&F=6#F,F'F7F'&FD6# F,F'F7F'&FG6#F,F'F7F'&F@6#F,F'F7F'&F*6#F,F'*4&F=6#F,F'F7F'&FD6#F,F'F7F '&FG6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'*4&F=6#F,F'F7F'&FD6#F,F'F7F'&FG6#F ,F'F7F'&F@6#F,F'F7F'&F16#F,F'7%*8&F=6#F,F'F7F'&FD6#F,F'F7F'&FG6#F,F'F7 F'&F@6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F'*8&F=6#F,F'F7F'&FD6#F,F'F7F'&FG6# F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F*6#F,F'*8&F=6#F,F'F7F'&FD6#F,F'F7F '&FG6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'7#*<&F=6#F,F'F7F'&FD6 #F,F'F7F'&FG6#F,F'F7F'&F@6#F,F'F7F'&F.6#F,F'F7F'&F16#F,F'F7F'&F*6#F,F' " }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 29 "H:=Lie_algebra_coho mology(C);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"HG7)7#*&%#0~G\"\"\"& %'theta1G6#%!GF)7#**F(\"\"\"&F+6#F-F)%#^~GF)&%'theta2G6#F-F)7#*.F(F0&F +6#F-F)F3F)&F56#F-F)F3F)&%'theta3G6#F-F)7#*2F(F0&F+6#F-F)F3F)&F56#F-F) F3F)&F>6#F-F)F3F)&%'theta4G6#F-F)7#*6F(F0&F+6#F-F)F3F)&F56#F-F)F3F)&F> 6#F-F)F3F)&FI6#F-F)F3F)&%'theta5G6#F-F)7#*:F(F0&F+6#F-F)F3F)&F56#F-F)F 3F)&F>6#F-F)F3F)&FI6#F-F)F3F)&FV6#F-F)F3F)&%'theta6G6#F-F)7#*<&F+6#F-F )F3F)&F>6#F-F)F3F)&FV6#F-F)F3F)&%'theta7G6#F-F)F3F)&%'theta8G6#F-F)F3F )&%'theta9G6#F-F)F3F)&%(theta10G6#F-F)" }}}{EXCHG {PARA 0 "sp2 > " 0 " " {MPLTEXT 1 0 29 "M:=reductive_complement(sp1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7)&%#e1G6#%!G&%#e3G6#F)&%#e5G6#F)&%#e7G6#F)&%#e8G 6#F)&%#e9G6#F)&%$e10G6#F)" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 28 "check_symmetric_pair(sp1,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %&falseG" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 272 36 "Example 5. The 6 sphere as g2/su(3)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 49 "restart: with(Vessiot):with(Koszul):with(tensors):" }}}{EXCHG {PARA 0 "so8 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~gl7RG" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 43 "coord_frame([x1,x2,x3,x4,x5,x6,x7],[u],E7):" }}} {EXCHG {PARA 0 "E7>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl7R > \+ " 0 "" {MPLTEXT 1 0 149 "phi:= evalV( dx1 &w dx2 &w dx3 + dx1 &w dx4 \+ &w dx5 -dx1 &w dx6 &w dx7 + dx2 &w dx4 &w dx6+ dx2 &w dx5 &w dx7 +dx3 \+ &w dx4 &w dx7 -dx3 &w dx5 &w dx6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%$phiG,0*,&%$dx1G6#%!G\"\"\"%#^~GF+&%$dx2G6#%!GF+%#^~GF+&%$dx3G6#%!GF +F+*,&F(6#%!GF+%#^~GF+&%$dx4G6#%!GF+%#^~GF+&%$dx5G6#%!GF+F+*,&F(6#%!GF +%#^~GF+&%$dx6G6#%!GF+%#^~GF+&%$dx7G6#%!GF+!\"\"*,&F.6#%!GF+%#^~GF+&F< 6#%!GF+%#^~GF+&FJ6#%!GF+F+*,&F.6#%!GF+%#^~GF+&FA6#%!GF+%#^~GF+&FO6#%!G F+F+*,&F36#%!GF+%#^~GF+&F<6#%!GF+%#^~GF+&FO6#%!GF+F+*,&F36#%!GF+%#^~GF +&FA6#%!GF+%#^~GF+&FJ6#%!GF+FR" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 44 "g2_subalg:=create_gl_subalgebra(gl7R,[phi]):" }}} {EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 57 "su3_subalg:=create_gl_s ubalgebra(gl7R,[phi,vect(x1,E7)]):" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 81 "g2 _su3_data:=subalgebra_pair_to_Lie_algebra_data_pair(g2_subalg,[su3_sub alg],g2):" }}}{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 20 "su3:=g2_su3 _data[2]:" }}}{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 29 "Chains:=rel ative_chains(su3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ChainsG7)7#\" \"\"7#*&%#0~GF'&%'theta1G6#%!GF'7#,(*(&F,6#%!GF'%#^~GF'&%'theta2G6#%!G F'F'*(&%'theta3G6#%!GF'%#^~GF'&%'theta4G6#%!GF'F'*(&%'theta5G6#%!GF'%# ^~GF'&%'theta6G6#%!GF'!\"\"7$,**,&F,6#%!GF'%#^~GF'&F<6#%!GF'%#^~GF'&FK 6#%!GF'FN*,&F,6#%!GF'%#^~GF'&FA6#%!GF'%#^~GF'&FF6#%!GF'F'*,&F76#%!GF'% #^~GF'&F<6#%!GF'%#^~GF'&FF6#%!GF'F'*,&F76#%!GF'%#^~GF'&FA6#%!GF'%#^~GF '&FK6#%!GF'F',**,&F,6#%!GF'%#^~GF'&F<6#%!GF'%#^~GF'&FF6#%!GF'FN*,&F,6# %!GF'%#^~GF'&FA6#%!GF'%#^~GF'&FK6#%!GF'FN*,&F76#%!GF'%#^~GF'&F<6#%!GF' %#^~GF'&FK6#%!GF'FN*,&F76#%!GF'%#^~GF'&FA6#%!GF'%#^~GF'&FF6#%!GF'F'7#, (*0&F,6#%!GF'%#^~GF'&F76#%!GF'%#^~GF'&F<6#%!GF'%#^~GF'&FA6#%!GF'F'*0&F ,6#%!GF'%#^~GF'&F76#%!GF'%#^~GF'&FF6#%!GF'%#^~GF'&FK6#%!GF'FN*0&F<6#%! GF'%#^~GF'&FA6#%!GF'%#^~GF'&FF6#%!GF'%#^~GF'&FK6#%!GF'FN7#*6F*\"\"\"&F ,6#%!GF'%#^~GF'&F76#%!GF'%#^~GF'&F<6#%!GF'%#^~GF'&FA6#%!GF'%#^~GF'&FF6 #%!GF'7#*8&F,6#%!GF'%#^~GF'&F76#%!GF'%#^~GF'&F<6#%!GF'%#^~GF'&FA6#%!GF '%#^~GF'&FF6#%!GF'%#^~GF'&FK6#%!GF'" }}}{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 34 "H:=Lie_algebra_cohomology(Chains);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%\"HG7(7#*&%#0~G\"\"\"&%'theta1G6#%!GF)7#**F(\"\"\"& F+6#%!GF)%#^~GF)&%'theta2G6#%!GF)7#*.F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%# ^~GF)&%'theta3G6#%!GF)7#*2F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%#^~GF)&FD6#% !GF)%#^~GF)&%'theta4G6#%!GF)7#*6F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%#^~GF) &FD6#%!GF)%#^~GF)&FV6#%!GF)%#^~GF)&%'theta5G6#%!GF)7#*8&F+6#%!GF)%#^~G F)&F66#%!GF)%#^~GF)&FD6#%!GF)%#^~GF)&FV6#%!GF)%#^~GF)&Ffo6#%!GF)%#^~GF )&%'theta6G6#%!GF)" }}}{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 261 "" 0 "" {TEXT -1 38 " Example 6. The 7 sphere as \+ so(7)/g2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 256 171 "In this example we find that in order to compute the relativ e chains , it is essential to change to a basis in so(7) adapted to g2 to avoid serious memory problems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "so7 > " 0 "" {MPLTEXT 1 0 49 "restart:with(Vess iot):with(Koszul):with(tensors):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(7)):" }}}{EXCHG {PARA 0 "gl7R \+ > " 0 "" {MPLTEXT 1 0 43 "coord_frame([x1,x2,x3,x4,x5,x6,x7],[u],E7): " }}}{EXCHG {PARA 0 "E7>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E7 >" 0 "" {MPLTEXT 1 0 149 "phi:= evalV( dx1 &w dx2 &w dx3 + dx1 &w dx4 &w dx5 -dx1 &w dx6 &w dx7 + dx2 &w dx4 &w dx6+ dx2 &w dx5 &w dx7 +dx3 &w dx4 &w dx7 -dx3 &w dx5 &w dx6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%$phiG,0*,&%$dx1G6#%!G\"\"\"%#^~GF+&%$dx2G6#%!GF+%#^~GF+&%$dx3G6#%!G F+F+*,&F(6#%!GF+%#^~GF+&%$dx4G6#%!GF+%#^~GF+&%$dx5G6#%!GF+F+*,&F(6#%!G F+%#^~GF+&%$dx6G6#%!GF+%#^~GF+&%$dx7G6#%!GF+!\"\"*,&F.6#%!GF+%#^~GF+&F <6#%!GF+%#^~GF+&FJ6#%!GF+F+*,&F.6#%!GF+%#^~GF+&FA6#%!GF+%#^~GF+&FO6#%! GF+F+*,&F36#%!GF+%#^~GF+&F<6#%!GF+%#^~GF+&FO6#%!GF+F+*,&F36#%!GF+%#^~G F+&FA6#%!GF+%#^~GF+&FJ6#%!GF+FR" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 34 "g:=c anonical_flat_metric(7,0,bas):" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 43 "so7_ subalg:=create_gl_subalgebra(gl7R,[g]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+so7_subalgG77,&&%$e12G6#%!G\"\"\"&%$e21G6#%!G!\"\",&&%$e13G6# %!GF+&%$e31G6#%!GF0,&&%$e14G6#%!GF+&%$e41G6#%!GF0,&&%$e15G6#%!GF+&%$e5 1G6#%!GF0,&&%$e16G6#%!GF+&%$e61G6#%!GF0,&&%$e17G6#%!GF+&%$e71G6#%!GF0, &&%$e23G6#%!GF+&%$e32G6#%!GF0,&&%$e24G6#%!GF+&%$e42G6#%!GF0,&&%$e25G6# %!GF+&%$e52G6#%!GF0,&&%$e26G6#%!GF+&%$e62G6#%!GF0,&&%$e27G6#%!GF+&%$e7 2G6#%!GF0,&&%$e34G6#%!GF+&%$e43G6#%!GF0,&&%$e35G6#%!GF+&%$e53G6#%!GF0, &&%$e36G6#%!GF+&%$e63G6#%!GF0,&&%$e37G6#%!GF+&%$e73G6#%!GF0,&&%$e45G6# %!GF+&%$e54G6#%!GF0,&&%$e46G6#%!GF+&%$e64G6#%!GF0,&&%$e47G6#%!GF+&%$e7 4G6#%!GF0,&&%$e56G6#%!GF+&%$e65G6#%!GF0,&&%$e57G6#%!GF+&%$e75G6#%!GF0, &&%$e67G6#%!GF+&%$e76G6#%!GF0" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 46 "g2_subalg:=create_gl_subalgebra(gl7R,[phi,g]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%*g2_subalgG70,*&%$e12G6#%!G\"\"\"&%$ e21G6#%!G!\"\"&%$e56G6#%!GF+&%$e65G6#%!GF0,*&%$e13G6#%!GF+&%$e31G6#%!G F0&%$e57G6#%!GF+&%$e75G6#%!GF0,*&%$e14G6#%!GF+&%$e36G6#%!GF0&%$e41G6#% !GF0&%$e63G6#%!GF+,*&%$e15G6#%!GF+&%$e37G6#%!GF0&%$e51G6#%!GF0&%$e73G6 #%!GF+,*&%$e16G6#%!GF+&%$e34G6#%!GF+&%$e43G6#%!GF0&%$e61G6#%!GF0,*&%$e 17G6#%!GF+&%$e35G6#%!GF+&%$e53G6#%!GF0&%$e71G6#%!GF0,*&%$e23G6#%!GF+&% $e32G6#%!GF0&%$e67G6#%!GF+&%$e76G6#%!GF0,*&%$e24G6#%!GF+&F]q6#%!GF+&%$ e42G6#%!GF0&Faq6#%!GF0,*&%$e25G6#%!GF+&F\\p6#%!GF0&F`p6#%!GF+&%$e52G6# %!GF0,*&%$e26G6#%!GF+&F[o6#%!GF0&%$e62G6#%!GF0&Fco6#%!GF+,*&%$e27G6#%! GF+&FP6#%!GF+&FX6#%!GF0&%$e72G6#%!GF0,*&%$e45G6#%!GF+&%$e54G6#%!GF0&Fb r6#%!GF+&Ffr6#%!GF0,*&%$e46G6#%!GF+&FC6#%!GF0&%$e64G6#%!GF0&FG6#%!GF+, *&%$e47G6#%!GF+&F26#%!GF+&F66#%!GF0&%$e74G6#%!GF0" }}}{EXCHG {PARA 0 " gl7R > " 0 "" {MPLTEXT 1 0 82 "so7_g2_data:=subalgebra_pair_to_Lie_alg ebra_data_pair(so7_subalg,[g2_subalg],so7):" }}}{EXCHG {PARA 0 "so7 > \+ " 0 "" {MPLTEXT 1 0 19 "g2:=so7_g2_data[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#g2G70,&&%#e1G6#%!G\"\"\"&%$e19G6#%!GF+,&&%#e2G6#%!GF +&%$e20G6#%!GF+,&&%#e3G6#%!GF+&%$e14G6#%!G!\"\",&&%#e4G6#%!GF+&%$e15G6 #%!GFB,&&%#e5G6#%!GF+&%$e12G6#%!GF+,&&%#e6G6#%!GF+&%$e13G6#%!GF+,&&%#e 7G6#%!GF+&%$e21G6#%!GF+,&&%#e8G6#%!GF+&Fen6#%!GF+,&&%#e9G6#%!GF+&FR6#% !GFB,&&%$e10G6#%!GF+&FI6#%!GFB,&&%$e11G6#%!GF+&F?6#%!GF+,&&%$e16G6#%!G F+&F^o6#%!GF+,&&%$e17G6#%!GF+&F66#%!GFB,&&%$e18G6#%!GF+&F-6#%!GF+" }}} {EXCHG {PARA 0 "so7 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "so7 \+ > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "so7 > " 0 "" {MPLTEXT 1 0 66 "new_so7_g2_data:=subalgebra_to_adapted_Lie_algebra_basis(g2,nso7 ):" }}}{EXCHG {PARA 0 "nso7 > " 0 "" {MPLTEXT 1 0 24 "ng2:=new_so7_g2_ data[2]:" }}}{EXCHG {PARA 0 "nso7 > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "nso7 > " 0 "" {MPLTEXT 1 0 29 "Chains:=relative_chains (ng2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ChainsG7*7#\"\"\"7#*&%#0~ GF'&%'theta1G6#%!GF'7#**F*\"\"\"&F,6#%!GF'%#^~GF'&%'theta2G6#%!GF'7#,0 *,&%(theta15G6#%!GF'%#^~GF'&%(theta16G6#%!GF'%#^~GF'&%(theta21G6#%!GF' F'*,&F>6#%!GF'%#^~GF'&%(theta17G6#%!GF'%#^~GF'&%(theta20G6#%!GF'!\"\"* ,&F>6#%!GF'%#^~GF'&%(theta18G6#%!GF'%#^~GF'&%(theta19G6#%!GF'F'*,&FC6# %!GF'%#^~GF'&FQ6#%!GF'%#^~GF'&F_o6#%!GF'F'*,&FC6#%!GF'%#^~GF'&Fjn6#%!G F'%#^~GF'&FV6#%!GF'F'*,&FQ6#%!GF'%#^~GF'&Fjn6#%!GF'%#^~GF'&FH6#%!GF'F' *,&F_o6#%!GF'%#^~GF'&FV6#%!GF'%#^~GF'&FH6#%!GF'FY7#,0*0&F>6#%!GF'%#^~G F'&FC6#%!GF'%#^~GF'&FQ6#%!GF'%#^~GF'&Fjn6#%!GF'F'*0&F>6#%!GF'%#^~GF'&F C6#%!GF'%#^~GF'&F_o6#%!GF'%#^~GF'&FV6#%!GF'FY*0&F>6#%!GF'%#^~GF'&FQ6#% !GF'%#^~GF'&F_o6#%!GF'%#^~GF'&FH6#%!GF'FY*0&F>6#%!GF'%#^~GF'&Fjn6#%!GF '%#^~GF'&FV6#%!GF'%#^~GF'&FH6#%!GF'FY*0&FC6#%!GF'%#^~GF'&FQ6#%!GF'%#^~ GF'&FV6#%!GF'%#^~GF'&FH6#%!GF'FY*0&FC6#%!GF'%#^~GF'&Fjn6#%!GF'%#^~GF'& F_o6#%!GF'%#^~GF'&FH6#%!GF'F'*0&FQ6#%!GF'%#^~GF'&Fjn6#%!GF'%#^~GF'&F_o 6#%!GF'%#^~GF'&FV6#%!GF'FY7#*6F*F1&F,6#%!GF'%#^~GF'&F76#%!GF'%#^~GF'&% 'theta3G6#%!GF'%#^~GF'&%'theta4G6#%!GF'%#^~GF'&%'theta5G6#%!GF'7#*:F*F 1&F,6#%!GF'%#^~GF'&F76#%!GF'%#^~GF'&F_z6#%!GF'%#^~GF'&Fdz6#%!GF'%#^~GF '&Fiz6#%!GF'%#^~GF'&%'theta6G6#%!GF'7#*<&F>6#%!GF'%#^~GF'&FC6#%!GF'%#^ ~GF'&FQ6#%!GF'%#^~GF'&Fjn6#%!GF'%#^~GF'&F_o6#%!GF'%#^~GF'&FV6#%!GF'%#^ ~GF'&FH6#%!GF'" }}}{EXCHG {PARA 0 "nso7 > " 0 "" {MPLTEXT 1 0 34 "H:=L ie_algebra_cohomology(Chains);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\" HG7)7#*&%#0~G\"\"\"&%'theta1G6#%!GF)7#**F(\"\"\"&F+6#%!GF)%#^~GF)&%'th eta2G6#%!GF)7#*.F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%#^~GF)&%'theta3G6#%!GF )7#*2F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%#^~GF)&FD6#%!GF)%#^~GF)&%'theta4G 6#%!GF)7#*6F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%#^~GF)&FD6#%!GF)%#^~GF)&FV6 #%!GF)%#^~GF)&%'theta5G6#%!GF)7#*:F(F0&F+6#%!GF)%#^~GF)&F66#%!GF)%#^~G F)&FD6#%!GF)%#^~GF)&FV6#%!GF)%#^~GF)&Ffo6#%!GF)%#^~GF)&%'theta6G6#%!GF )7#*<&%(theta15G6#%!GF)%#^~GF)&%(theta16G6#%!GF)%#^~GF)&%(theta17G6#%! GF)%#^~GF)&%(theta18G6#%!GF)%#^~GF)&%(theta19G6#%!GF)%#^~GF)&%(theta20 G6#%!GF)%#^~GF)&%(theta21G6#%!GF)" }}}{EXCHG {PARA 0 "nso7 > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 55 "Example 7. Complex projective space CP^4 as su(3)/u(2)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 49 "restart:with(Vessiot):with(Koszul):with(tensors):" }} }{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl (6)):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 39 "coord_frame([ x1,x2,x3,y1,y2,y3],[],E6):" }}}{EXCHG {PARA 0 "E6>" 0 "" {MPLTEXT 1 0 34 "g:=canonical_flat_metric(6,0,bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,.*&&%$dx2G6#%!G\"\"\"&F(6#%!GF+F+*&&%$dx3G6#%!GF+&F16#%!G F+F+*&&%$dx1G6#%!GF+&F96#%!GF+F+*&&%$dy3G6#%!GF+&FA6#%!GF+F+*&&%$dy2G6 #%!GF+&FI6#%!GF+F+*&&%$dy1G6#%!GF+&FQ6#%!GF+F+" }}}{EXCHG {PARA 0 "E6 \+ > " 0 "" {MPLTEXT 1 0 36 "J:=canonical_complex_structure(bas);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG,.*&&%$dx2G6#%!G\"\"\"&%%D_y2G6# %!GF+!\"\"*&&%$dy1G6#%!GF+&%%D_x1G6#%!GF+F+*&&%$dy2G6#%!GF+&%%D_x2G6#% !GF+F+*&&%$dx3G6#%!GF+&%%D_y3G6#%!GF+F0*&&%$dx1G6#%!GF+&%%D_y1G6#%!GF+ F0*&&%$dy3G6#%!GF+&%%D_x3G6#%!GF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "dz1:=v_zip([1,I],[x1,y1],plus,form):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz2:=v_zip([1,I],[x2,y2],plus,form):" }} {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz3:=v_zip([1,I],[x3,y3],plus,fo rm):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 29 "nu:=dz1&wedge dz2 &wedge dz3:" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuR:= (1/2) &mu lt ( nu &plus Vessiot_map(conjugate,nu));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nuRG,**,&%$dx1G6#%!G\"\"\"%#^~GF+&%$dx2G6#%!GF+%#^~G F+&%$dx3G6#%!GF+F+*,&F(6#%!GF+%#^~GF+&%$dy2G6#%!GF+%#^~GF+&%$dy3G6#%!G F+!\"\"*,&F.6#%!GF+%#^~GF+&%$dy1G6#%!GF+%#^~GF+&FA6#%!GF+F+*,&F36#%!GF +%#^~GF+&FK6#%!GF+%#^~GF+&F<6#%!GF+FD" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuI:=(I/2) &mult ( nu &minus Vessiot_map(conjugate,n u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nuIG,**,&%$dx1G6#%!G\"\"\"% #^~GF+&%$dx2G6#%!GF+%#^~GF+&%$dy3G6#%!GF+!\"\"*,&F(6#%!GF+%#^~GF+&%$dx 3G6#%!GF+%#^~GF+&%$dy2G6#%!GF+F+*,&F.6#%!GF+%#^~GF+&F=6#%!GF+%#^~GF+&% $dy1G6#%!GF+F6*,&FO6#%!GF+%#^~GF+&FB6#%!GF+%#^~GF+&F36#%!GF+F+" }}} {EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E6 > \+ " 0 "" {MPLTEXT 1 0 12 "V:=vect(y3):" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 53 "su3_subalg:=create_gl_subalgebra(gl6R,[g,J,nuR,nuI]); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+su3_subalgG7*,*&%$e12G6#%!G\"\" \"&%$e21G6#%!G!\"\"&%$e45G6#%!GF+&%$e54G6#%!GF0,*&%$e13G6#%!GF+&%$e31G 6#%!GF0&%$e46G6#%!GF+&%$e64G6#%!GF0,*&%$e14G6#%!GF+&%$e36G6#%!GF0&%$e4 1G6#%!GF0&%$e63G6#%!GF+,*&%$e15G6#%!GF+&%$e24G6#%!GF+&%$e42G6#%!GF0&%$ e51G6#%!GF0,*&%$e16G6#%!GF+&%$e34G6#%!GF+&%$e43G6#%!GF0&%$e61G6#%!GF0, *&%$e23G6#%!GF+&%$e32G6#%!GF0&%$e56G6#%!GF+&%$e65G6#%!GF0,*&%$e25G6#%! GF+&FP6#%!GF0&%$e52G6#%!GF0&FX6#%!GF+,*&%$e26G6#%!GF+&%$e35G6#%!GF+&%$ e53G6#%!GF0&%$e62G6#%!GF0" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 46 "u2_subalg:=create_gl_subalgebra(gl6R,[g,J,V]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*u2_subalgG7&,*&%$e12G6#%!G\"\"\"&%$e21G6#%!G!\" \"&%$e45G6#%!GF+&%$e54G6#%!GF0,&&%$e14G6#%!GF+&%$e41G6#%!GF0,*&%$e15G6 #%!GF+&%$e24G6#%!GF+&%$e42G6#%!GF0&%$e51G6#%!GF0,&&%$e25G6#%!GF+&%$e52 G6#%!GF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 92 "Note that u2, as constructed here, is not a subalgebr a of su3. The correct subalgebra is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 73 "new_u2_subalg:=[su3_ subalg[1],su3_subalg[3],su3_subalg[4],su3_subalg[7]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.new_u2_subalgG7&,*&%$e12G6#%!G\"\"\"&%$e21G6#%! G!\"\"&%$e45G6#%!GF+&%$e54G6#%!GF0,*&%$e14G6#%!GF+&%$e36G6#%!GF0&%$e41 G6#%!GF0&%$e63G6#%!GF+,*&%$e15G6#%!GF+&%$e24G6#%!GF+&%$e42G6#%!GF0&%$e 51G6#%!GF0,*&%$e25G6#%!GF+&F?6#%!GF0&%$e52G6#%!GF0&FG6#%!GF+" }}} {EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 86 "su3_u2_data:=subalgebra _pair_to_Lie_algebra_data_pair(su3_subalg,[new_u2_subalg],su3):" }}} {EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 40 "Chains:=relative_chains( su3_u2_data[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ChainsG7'7#\"\" \"7#*&%#0~GF'&%'theta1G6#%!GF'7#,&*(&%'theta2G6#%!GF'%#^~GF'&%'theta5G 6#%!GF'F'*(&%'theta6G6#%!GF'%#^~GF'&%'theta8G6#%!GF'F'7#*.F*\"\"\"&F,6 #%!GF'%#^~GF'&F36#%!GF'%#^~GF'&%'theta3G6#%!GF'7#*0&F36#%!GF'%#^~GF'&F 86#%!GF'%#^~GF'&F=6#%!GF'%#^~GF'&FB6#%!GF'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 84 " Example 8 The orie nted Grassmannian of 3 planes in R^6 as so(6)/so(3)times so(3)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 82 "This exa mple is taken from Wolf's book Spaces of Constant Curvature, p. 253 -- " }}{PARA 0 "" 0 "" {TEXT 256 64 "actually Wolf looks at the \+ case SO(8)/So(4) times So(4). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 49 "restart:with(Vessiot):wi th(Koszul):with(tensors):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(6)):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 37 "coord_frame([x1,x2,x3,x4,x5,x6],[u]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{EXCHG {PARA 0 "euclid> " 0 "" {MPLTEXT 1 0 34 "g:=canonical_flat_metric(6,0,bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,.*&&%$dx3G6#%!G\"\"\"&F(6#%!GF+F+*&&% $dx4G6#%!GF+&F16#%!GF+F+*&&%$dx1G6#%!GF+&F96#%!GF+F+*&&%$dx6G6#%!GF+&F A6#%!GF+F+*&&%$dx2G6#%!GF+&FI6#%!GF+F+*&&%$dx5G6#%!GF+&FQ6#%!GF+F+" }} }{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 68 "h:=array_to_tens(lin alg[diag](1,1,1,0,0,0),[[cov_bas,cov_bas],[] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,(*&&%$dx3G6#%!G\"\"\"&F(6#%!GF+F+*&&%$dx1G6#%!GF +&F16#%!GF+F+*&&%$dx2G6#%!GF+&F96#%!GF+F+" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 43 "so6_subalg:=create_gl_subalgebra(gl6R,[g]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 49 "so3xso3_subalg:=crea te_gl_subalgebra(gl6R,[g,h]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 92 "so6_so3xso3_data:=subalgebra_pair_to_Lie_algebra_data _pair(so6_subalg,[so3xso3_subalg],so6):" }}}{EXCHG {PARA 0 "so6 > " 0 "" {MPLTEXT 1 0 29 "so3xso3:=so6_so3xso3_data[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(so3xso3G7(&%#e1G6#%!G&%#e2G6#%!G&%#e6G6#%!G&%$e13G6# %!G&%$e14G6#%!G&%$e15G6#%!G" }}}{EXCHG {PARA 0 "so6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "so6 > " 0 "" {MPLTEXT 1 0 76 "new_so6_so3x so3_data:=subalgebra_to_adapted_Lie_algebra_basis(so3xso3,nso6):" }}} {EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 30 "nso3:=new_so6_so3xso3_d ata[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nso3G7(&%#e1G6#%!G&%#e2G 6#%!G&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%!G&%#e6G6#%!G" }}}{EXCHG {PARA 0 " nso6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 30 "Chains:=relative_chains(nso3):" }}}{EXCHG {PARA 0 "ns o6 > " 0 "" {MPLTEXT 1 0 17 "map(nops,Chains);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"\"F$F$F$F$F$F$F$F$F$" }}}{EXCHG {PARA 0 "nso6 > \+ " 0 "" {MPLTEXT 1 0 13 "Chains[1..4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7#\"\"\"7#*&%#0~GF%&%'theta1G6#%!GF%7#**F(\"\"\"&F*6#%!GF%%#^~GF% &%'theta2G6#%!GF%7#*.F(F/&F*6#%!GF%%#^~GF%&F56#%!GF%%#^~GF%&%'theta3G6 #%!GF%" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 10 "Chains[5];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7#,4*0&%'theta7G6#%!G\"\"\"%#^~GF*&% 'theta8G6#%!GF*%#^~GF*&%(theta10G6#%!GF*%#^~GF*&%(theta11G6#%!GF*F**0& F'6#%!GF*%#^~GF*&F-6#%!GF*%#^~GF*&%(theta13G6#%!GF*%#^~GF*&%(theta14G6 #%!GF*F**0&F'6#%!GF*%#^~GF*&%'theta9G6#%!GF*%#^~GF*&F26#%!GF*%#^~GF*&% (theta12G6#%!GF*F**0&F'6#%!GF*%#^~GF*&FR6#%!GF*%#^~GF*&FD6#%!GF*%#^~GF *&%(theta15G6#%!GF*F**0&F-6#%!GF*%#^~GF*&FR6#%!GF*%#^~GF*&F76#%!GF*%#^ ~GF*&Fen6#%!GF*F**0&F-6#%!GF*%#^~GF*&FR6#%!GF*%#^~GF*&FI6#%!GF*%#^~GF* &Ffo6#%!GF*F**0&F26#%!GF*%#^~GF*&F76#%!GF*%#^~GF*&FD6#%!GF*%#^~GF*&FI6 #%!GF*F**0&F26#%!GF*%#^~GF*&Fen6#%!GF*%#^~GF*&FD6#%!GF*%#^~GF*&Ffo6#%! GF*F**0&F76#%!GF*%#^~GF*&Fen6#%!GF*%#^~GF*&FI6#%!GF*%#^~GF*&Ffo6#%!GF* F*" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 10 "Chains[6];" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7#,4*4&%'theta7G6#%!G\"\"\"%#^~GF*&%'t heta8G6#%!GF*%#^~GF*&%'theta9G6#%!GF*%#^~GF*&%(theta10G6#%!GF*%#^~GF*& %(theta13G6#%!GF*F**4&F'6#%!GF*%#^~GF*&F-6#%!GF*%#^~GF*&F26#%!GF*%#^~G F*&%(theta11G6#%!GF*%#^~GF*&%(theta14G6#%!GF*F**4&F'6#%!GF*%#^~GF*&F-6 #%!GF*%#^~GF*&F26#%!GF*%#^~GF*&%(theta12G6#%!GF*%#^~GF*&%(theta15G6#%! GF*F**4&F'6#%!GF*%#^~GF*&F76#%!GF*%#^~GF*&FM6#%!GF*%#^~GF*&F]o6#%!GF*% #^~GF*&F<6#%!GF*F**4&F'6#%!GF*%#^~GF*&F76#%!GF*%#^~GF*&F<6#%!GF*%#^~GF *&FR6#%!GF*%#^~GF*&Fbo6#%!GF*F**4&F-6#%!GF*%#^~GF*&F76#%!GF*%#^~GF*&FM 6#%!GF*%#^~GF*&F]o6#%!GF*%#^~GF*&FR6#%!GF*F**4&F-6#%!GF*%#^~GF*&FM6#%! GF*%#^~GF*&F<6#%!GF*%#^~GF*&FR6#%!GF*%#^~GF*&Fbo6#%!GF*F**4&F26#%!GF*% #^~GF*&F76#%!GF*%#^~GF*&FM6#%!GF*%#^~GF*&F]o6#%!GF*%#^~GF*&Fbo6#%!GF*F **4&F26#%!GF*%#^~GF*&F]o6#%!GF*%#^~GF*&F<6#%!GF*%#^~GF*&FR6#%!GF*%#^~G F*&Fbo6#%!GF*F*" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 14 "Cha ins[7..10];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&7#*:%#0~G\"\"\"&%'the ta1G6#%!GF'%#^~GF'&%'theta2G6#%!GF'%#^~GF'&%'theta3G6#%!GF'%#^~GF'&%'t heta4G6#%!GF'%#^~GF'&%'theta5G6#%!GF'%#^~GF'&%'theta6G6#%!GF'7#*>F&\" \"\"&F)6#%!GF'%#^~GF'&F.6#%!GF'%#^~GF'&F36#%!GF'%#^~GF'&F86#%!GF'%#^~G F'&F=6#%!GF'%#^~GF'&FB6#%!GF'%#^~GF'&%'theta7G6#%!GF'7#*BF&FG&F)6#%!GF '%#^~GF'&F.6#%!GF'%#^~GF'&F36#%!GF'%#^~GF'&F86#%!GF'%#^~GF'&F=6#%!GF'% #^~GF'&FB6#%!GF'%#^~GF'&F[o6#%!GF'%#^~GF'&%'theta8G6#%!GF'7#*D&F[o6#%! GF'%#^~GF'&F]q6#%!GF'%#^~GF'&%'theta9G6#%!GF'%#^~GF'&%(theta10G6#%!GF' %#^~GF'&%(theta11G6#%!GF'%#^~GF'&%(theta12G6#%!GF'%#^~GF'&%(theta13G6# %!GF'%#^~GF'&%(theta14G6#%!GF'%#^~GF'&%(theta15G6#%!GF'" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 34 "H:=Lie_algebra_cohomology(Chains):" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 12 "map(nops,H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"\"F$F$F$F$F$F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 70 "Each relative chain det ermines a unique cohomology representative. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 54 "seq(co eff_set(H[i][1] &minus Chains[i+1][1]), i=1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+<#\"\"!F#F#F#F#F#F#F#F#" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 30 "M:=reductive_complement(nso3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"MG7+&%#e7G6#%!G&%#e8G6#%!G&%#e9G6#%!G&%$e10G6#%!G&%$e11G6#%!G&%$e12G 6#%!G&%$e13G6#%!G&%$e14G6#%!G&%$e15G6#%!G" }}}{EXCHG {PARA 0 "nso6 > \+ " 0 "" {MPLTEXT 1 0 29 "check_symmetric_pair(nso3,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "nso6 > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT 274 19 "updated 01/26/03:IA" }}}{MARK "0 0" 58 }{VIEWOPTS 1 1 0 3 4 1802 }