>    r symalg,delete symalg,deltasymalg,directsumsymalg,genbasissymalg,genbasissymalg,generatorssymalg,genhallmon symalg,idealsymalg,isgenerator symalg,simplesymalg,simplify symalg,store<symalg,symbasissymalg,triangular symalg,usew7|zp||  ,, -- -|{{{{{ Gversion2ition of an algebra (d irectsum, factoralg, ideal) or algebras (triangular) in " }{TEXT 263 11 "generators " }{TEXT -1 129 "before they can be used. For others (K acMoody, simple) algebras will be defined on-the-fly from within those functions as needed." }}{PARA 15 "" 0 "" {TEXT -1 73 " To check wheth er a generator is valid in an already defined algebra see " } {HYPERLNK 17 "isgenerator" 2 "symalg, isgenerator" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }} {PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }} {PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Default w eights:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "generators(lie,[x ,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$lieG-%-anglebracketG6%% \"xG%\"yG%\"zG7%7%\6lg, ideal" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "KacMo ody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "wt" 2 "symalg, wt" "" }}}}{MARK "9 1 12" 2 }{VIEWOPTS 1 1 0 1 1 1803 } XT 1 0 24 "genbasis([3,2], G_plus);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$-%#&*G6$-F%6$&%\"eG6#\"\"\"-F%6$F)-F%6$F)&F*6#\"\"# F1-F%6$F-F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Negative weights:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " genbasis([-2,-4], G_minus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%#&* G6$-F%6$-F%6$&%\"fG6#\"\"\"&F,6#\"\"#-F%6$F)F/F/" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "char" 2 "symalg, char" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "genhallmon" 2 "symalg, genhallmon" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "ideal" 2 "symaXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "m := matrix([[2,-3],[-3,2]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "KacMoody(G, m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"GG-%-anglebracketG6%%'G_plusG%'G_zeroG%(G_minusG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "genbasis([3,2], G_plus);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$-%#&*G6$-F%6$&%\"eG6#\"\"\"-F%6$F)-F%6$F)&F*6#\"\"# F1-F%6$F-F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Negative weights:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " genbasis([-2,-4], G_minus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%#&* G6$-F%6$-F%6$&%\"fG6#\"\"\"&F,6#\"\"#-F%6$F)F/F/" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "char" 2 "symalg, char" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "genhallmon" 2 "symalg, genhallmon" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "ideal" 2 "symar expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "Working with names:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "f := (l,n) -> delta[l,n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"lG%\"nG6\"6$%)operatorG%&arrowGF)&%&deltaG6 $9$9%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Now with integers:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " f(3,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "f(5,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }} {PARA 0 "" 0 "" {HYPERLNK 17 "store" 2 "symalg, store" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }}}}{MARK "7 1 2" 3 }{VIEWOPTS 1 1 0 1 1 1803 } load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Wa rning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "War ning, new definition fo 7 symalg,char7symalg,delete7symalg,delta7symalg,genbasis7symalg,generators7symalg,genhallmon7symalg,isgenerator7symalg,&operators7symalg,simplify7symalg,store7 symalg,use7 symalg,wt7symalg,ideal7symalg,directsum7symalg,KacMoody7symalg,factoralg7symalg,simple7symalg,triangular7symalg,symbasis!7symalg,Overview7ibmintelntmaplinputcourimathtimehyperlinkoutputhelpnormallatextexteadingheadwarnbulletitemsimplifapplliealgebrasimplificatruleexpresscallsequencexpralgnamparametrsanynamealgebrasimplifibuiltupothergeneratodefaultinfodescriptfunctusedulesorderconvertssionintosumhallmonomialonlyargumpreswillattemptuseargumentsimplifyaccordspecifirecognizspecialsimplificatfollowtypedirectsymalgdirectsumwofactorfactoralwithtriangulardecompositiekacmoodkacmoodsymalgsimplmwsalgebrasuserdefinedstorcasecharacteristiccharautomaticalfurtherresultbeforreturnthemexamplweneedloadpackagnewdefinitexpandstraightforwardgeneratorxgzgygfumgeneratorsldirsumygfgfresimplifourfirstnowparametricaffinaffinedontprintdeltasimplifhgkgigsgjgtgxgfvirasovoegdeltagfafkgfseealsoalsooperatorfactoralgoverview 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT symalg,&operators7symalg,KacMoody7symalg,Overview!7 symalg,char 7 symalg,delete7 symalg,delta7symalg,directsum7symalg,factoralg7symalg,genbasis7symalg,generators7symalg,genhallmon7 symalg,ideal7symalg,isgenerator7 symalg,simple7symalg,simplify7 symalg,store7symalg,symbasis7symalg,triangular7 symalg,use7 symalg,wt7mber(RI,TrueGens) and (LI[] > 0 and not(LI[] > nops(RI))) ) then RETURN(add(RI[j] * M[LI[],j],j=1..nops(RI)) * op(0,GENERATORS[1])[RI[]]);  17 "factoralg" 2 "symalg, factoralg" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "ideal" 2 "symalg, ideal" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " simple" 2 "symalg, simple" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "store" 2 "symalg, store" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg , Overview" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "triangular" 2 "symalg, \+ triangular" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "use" 2 "symalg, use" " " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{MARK "11 1 15" 2 }{VIEWOPTS 1 1 0 1 1 1803 } PLTEXT 1 0 22 "i sgenerator(e[0,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "isgenerator(e[1,1,1,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 0 {PARA 260 "" 0 "s eealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {HYPERLNK 17 "directsum" 2 "symalg, directsum" "" }{TEXT -1 2 ", " } {HYPERLNK allowargumentbuildcommandcutoffwtdeltagealnamenvironmexpandfiR genhallmon]headibmSintel2lessmmapmptng\option)packplac3reforrulesets simplificatstorkteswtypicalxvariablQwit0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 6 "delta " } {TEXT 256 31 "- simulates the delta function" }}{PARA 257 "" 0 "" {TEXT -1 17 &operators,symalg7KacMoody,symalg7Overview,symalg!7 char,symalg 7 delete,symalg7 delta,symalg7directsum,symalg7factoralg,symalg7genbasis,symalg7generators,symalg7genhallmon,symalg7 ideal,symalg7isgenerator,symalg7 simple,symalg7simplify,symalg7 store,symalg7symbasis,symalg7triangular,symalg7 use,symalg7 wt,symalg7mber(RI,TrueGens) and (LI[] > 0 and not(LI[] > nops(RI))) ) then RETURN(add(RI[j] * M[LI[],j],j=1..nops(RI)) * op(0,GENERATORS[1])[RI[]]); " 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 17 " d elta[i, j]" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 257 8 " i, j" }{TEXT 23 4 " - " } {TEXT -1 17 "names or integers" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 258 5 "D elta" }{TEXT -1 146 " returns an unevaluated expression when either of the two parameters is of type name, otherwise if i = j then returns 1 , and 0 in all other cases." }}{PARA 15 "" 0 "" {TEXT -1 116 " Please \+ note the usage of the angled brackets ( [] ), these are very important for the function to perform properly." }}}{SECT 0 {PARA 259 "" 0 "exa mples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need t o load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Wa rning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "War ning, new definition fosymalg,&operators7symalg,KacMoody7symalg,Overview!7 symalg,char 7 symalg,delete7 symalg,delta7symalg,directsum7symalg,factoralg7symalg,genbasis7symalg,generators7symalg,genhallmon7 symalg,ideal7symalg,isgenerator7 symalg,simple7symalg,simplify7 symalg,store7symalg,symbasis7symalg,triangular7 symalg,use7 symalg,wt73 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 7{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 }{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 "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0  0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PST7_o^  777777-777#7Y7g7$7%7)7077277A 7<<<<<<.<<<$ " 0 "" {MPLTEXT 1 0 13 "wi th(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition fo r simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "TypicaTH 20 "6#,&&%\"hG6#\"\"!! \"#%\"kG!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simplify((e [2]&*f[1])&*(f[3]&*e[-6]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"kG !\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " - W2:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "generators(W2 , [x[i,j],y[i,j]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "store([x[i, s]&*x[j,t]=(j-i)*x[i+j,s+t],y[s,i]&*y[t,j]=(j-i)*y[s+t,i+j],x[i,s]&*y[ j,t]=j*y[i+j,s+t]-s*x[i+j,s+t], W2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"xG6$%\"iG%\"sG&F(6$%\"jG%\"tG*&,&F.\"\"\"F*!\"\"F2&F(6 $,&F*F2F.F2,&F+F2F/F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&% \"xG6$%\"jG%\"tG&F(6$%\"iG%\"sG,$*&,&F*\"\"\"F.!\"\"F3&F(6$,&F.F3F*F3, &F/F3F+F3F3F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"yG6$%\"s G%\"iG&F(6$%\"tG%\"jG*&,&F/\"\"\"F+!\"\"F2&F(6$,&F*F2F.F2,&F+F2F/F2F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"yG6$%\"tG%\"jG&F(6$% \"sG%\"iG,$*&,&F+\"\"\"F/!\"\"F3&F(6$,&F.F3F*F3,&F/F3F+F3F3F4" }} {PARA 1e as index for generators if they will have special simp lification " }{HYPERLNK 17 "rules defined" 2 "symalg, store" "" } {TEXT -1 76 " that depend on these index values. Parametric generators are recognized by " }{HYPERLNK 17 "simplify" 1 "symalgsimplify.mws" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "wt" 1 "symalgwt.mws" "" }{TEXT -1 74 ", etc. and used accordingly. As an example see the implementation of the " }{HYPERLNK 17 "Virasovo algebra" 2 "symalg, simplify" "" } {TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 61 " Note the existance of special algebra constructors, such as " }{HYPERLNK 17 "directsum" 2 " symalg, directsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "sy malg, factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "ideal" 2 "symalg, \+ ideal" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMood y" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simple" 2 "symalg, simple" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "triangular" 2 "symalg, triangular" "" } {TEXT -1 112 ". Some of these require the definguments , it expects the algebra name and the list of generators for the algeb ra. The order of the generators in the list decides the " }{HYPERLNK 17 "lexicographical ordering" 2 "symalg, &operators" "" }{TEXT -1 57 " of the generators (first one being the smallest, etc). " }}{PARA 15 "" 0 "" {TEXT -1 30 "When no weights are passed to " }{TEXT 264 10 "ge nerators" }{TEXT -1 162 " (optional third argument) default weights ar e built for the generators. If there are user defined weights, these \+ have to be thrown into a list and passed in to " }{TEXT 262 10 "genera tors" }{TEXT -1 213 " as the third argument. The list should contain o ne weight per generator (a weight is a list: [i,...,n] where i,...,n \+ are integers or names if the weight is parametric) and each weight mus t be of the same length." }}{PARA 15 "" 0 "" {TEXT -1 63 " It is possi ble to define parametric generators and parametric " }{HYPERLNK 17 "we ights" 2 "symalg, wt" "" }{TEXT -1 111 ". It is not recommended to us e the same nam" }{TEXT -1 88 "a number of integer lists , each integer list corresponding to one of the generator names" }} {SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 " " 0 "" {TEXT -1 4 " In " }{TEXT 260 10 "generators" }{TEXT -1 133 " it is possible to define a new algebra that can be later used in the map le session. Once an algebra is defined all functions in the " } {HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT -1 183 " package \+ are able to work with it. It is advisable to use such names for the n ame of the algebra and for the generators that will not be assigned va lues elsewhere during the session." }}{PARA 15 "" 0 "" {TEXT -1 53 " W hen the first algebra is defined, it is set as the " }{HYPERLNK 17 "de fault algebra" 2 "symalg, use" "" }{TEXT -1 105 ", and all symalg func tions will use that algebra if no algebra is specified for them among \+ the arguments." }}{PARA 15 "" 0 "" {TEXT -1 18 " In the case when " } {TEXT 261 10 "generators" }{TEXT -1 156 " is called with two ar7ibmintelntmaplinputcourimathtimehyperlinkoutputhelpnormaltextheadwarnbulletitemdirectsumuseddefinalgebrcallsequencdirsumalgnamalgparameternamealgebrapredfinedinfodescriptiwithpossiblalreadexistingfirstargumdirectumsecondexpectebralisttheycanappearwithoutorderonlyentrassumotherusedefaultsymalghesimplifunctrecognizapplspecialsimplificatruleermscontaingeneratorbothzeroalltermimplfiaccordparticularincluduserstorexamplweneedloadpackagsymnewdefinitsimplifyexpantypicalusaglielieganglebracketgxgygslfghgdsdsganglebracketgliegexprexprgygfspecspecgpgqgrglieandspeclieandspecgpgfseealsoalsofactoralgidealisgeneratorkacmoodsimploverviewwt0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT rk \+ with the default algebra, which may be changed using the " }{HYPERLNK 17 "use" 2 "symalg, use" "" }{TEXT -1 9 " command." }}{PARA 15 "" 0 " " {TEXT -1 25 " The expressions are not " }{HYPERLNK 17 "simplified" 2 "symalg, simplify" "" }{TEXT -1 31 " during the comparison process. " }}{PARA 15 "" 0 "" {TEXT -1 157 " When either of the expressions is \+ a sum, then the leading term (lexicographically smallest term) is pick ed from the sum and the comparison is based on that." }}{PARA 15 "" 0 "" {TEXT -1 106 " Only comparisons of type name are supported (ie x ca nnot be compared to 0, or any other integer or real)." }}{PARA 15 "" 0 "" {TEXT -1 150 " Note that for the right results brackets must be p laced around the expressions when comparing them, unless the expressio ns are assigned to variables." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 63 "We need to load \+ the symalg package, and define some generators." }{MPLTEXT 1 0 0 "" }} {EXCHG {PARA 0 "> " 0 ;7{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 "Courier" 1 10 0 0 0 0 0 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0uveryfirstargumpassanynumbcanoncewereappliieexpressionsimplfisimplifyagainwillproducdifferanswhancaseexamplweneedloadpackagnewitiondefinitionexpandsomeslhgegfgstoreglieliegfailgseealsoalsooverviewT -1 22 "Parametric generators:" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simple(S, D[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"SG-%-anglebracketG6$&%\"eG6&%#k1G%#k2G%# k3G%#k4G&%\"hG6#%#i1G7$7&F+F,F-F.7&\"\"!F6F6F6" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "use(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" SG-%-anglebracketG6$&%\"eG6&%#k1G%#k2G%#k3G%#k4G&%\"hG6#%#i1G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "isgenerator(h[k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "isgenerator(e[k1,k2,k3,k4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "i sgenerator(e[0,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "isgenerator(e[1,1,1,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 0 {PARA 260 "" 0 "s eealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {HYPERLNK 17 "directsum" 2 "symalg, directsum" "" }{TEXT -1 2 ", " } {HYPERLNK 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Straight forward exa mple:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "generators(lie,[x,y ,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "isgenerator([x,z]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Indexed generators with integer in dices:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "m := matrix([[2,-3 ],[-3,2]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "KacMoody(G, m);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"GG-%-anglebracketG6%%'G_plusG%'G_z eroG%(G_minusG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "use();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"GG-%-anglebracketG6(&%\"eG6#\"\"\" &F)6#\"\"#&%\"hGF*&F0F-&%\"fGF*&F3F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "isgenerator(h[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "isgenerator(h[- 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEX& 0 "" 0 "" {TEXT -1 16 "Special weights:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "generators(lie2, [p,q,r,w], [[1,0,0],[0,1,0],[0,0,1], [1,1,1]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "genhallmon([1,1,1], l ie2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%#&*G6$-F%6$%\"pG%\"rG%\"q G-F%6$-F%6$F)F+F*%\"wG" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "&<" 2 "symalg, &operat ors" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "genbasis" 2 "symalg, genbasis " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "wt" 2 "symalg, wt" "" }}}}{MARK "9 1 6 " 3 }{VIEWOPTS 1 1 0 1 1 1803 } {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "generators(lie, [x,y,z]):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "genhallmon([1,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%#&*G6$-F%6$-F%6$%\"xG%\"zG%\"yGF--F%6$-F%6$-F%6$F+F -F,F--F%6$-F%6$F2F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA(optional7777777order777777ormal7ors7ossibl7oth7other# 77777777otherwis7our7777outputO8 7777777777777777777ov7ove7over7overload!7overrid 77overviewO 7777777777777777777pa7pack7777!7gene7777gener 77genera 77generat7777generato 77 generatorO 777777777 7777777 77!7 genhallmon7777!7or7generators" 2 "symalg, generators" "" }}{PARA 15 "" 0 " " {TEXT -1 113 " When an algebra is specified in the list of rules, th e name of the algebra must be the last element in the list." }}{PARA 15 "" 0 "" {TEXT -1 88 " If an algebra is not specified with the set o f rules, then the default algebra is used." }}{PARA 15 "" 0 "" {TEXT -1 102 " When using the DONTPRINT option to turn off the echo, it has \+ to be the very first argument passed to " }{TEXT 260 5 "store" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 39 " For deleting rules for an al gebra see " }{HYPERLNK 17 "delete" 2 "symalg, delete" "" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {TEXT -1 92 " There are examples for the storage and usage of parametric generators on the help page for " }{HYPERLNK 17 "symalg, simplify" 2 "symalg, simplify" "" }{TEXT -1 1 "." }}} {SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13O74ibmintelntmaplinputcourimathtimehyperlinkoutputhelpnormaltextcourierheadwarnmaplebulletitemlexicographicallessbooleanoperatorequallexicographicalgreatcallsequencexpropvarparameteranylieexpressvariablwithxpressassignthemcomparisonoperatorsinfodescripteneratorbuildupmustdeclarbeforehandgeneratorsymalgfunctiononlyworkdefaultalgebramaychangusingusecommandsimplifisimplifduringproceseithsumleadtermsmallestpickedbasedtypenamesupportiecannotcomparotherintegrealnoterightresultbracketlacedaroundunlesexpressionsexamplweneedloadpackagdefinsomenewdefinitexpandlieganglebracketgxgygzgorderabovtruegstraightforwardusagbracktsithlhsrhslhsgrhsgfalsegconstantfinetoosumsygfseealsoalsosymaloverview0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 7]{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 } {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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 01 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 2 "&<" }{TEXT 258 53 " - lexicographical 'less than' boolean operator" }} {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 3 "&<=" }{TEXT 30 1 " " } {TEXT -1 0 "" }{TEXT 259 60 " - lexicographical 'less than or equal t o' boolean operator" }}{PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 3 "&<>" }{TEXT 260 49 " - lexicographical 'not equal' boolean operator " }}{PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 6 "&= " }{TEXT 261 43 "- lexicographical 'equal' boolean operator" }}{PARA 3 "" 0 " " {MPLTEXT 0 21 0 "" }{TEXT -1 6 "&> " }{TEXT 262 50 "- lexicograp hical 'greater than' boolean operator" }}{PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 4 "&>= " }{TEXT 263 63 " B-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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -/interact7into 77 introduct!7ion7irectsum7irst7 isgenerator7777!7isn7istic 7itemS 7777777777777777777!7ith777ition7jg777kac 77 7kacmo7kacmood3.77777777 777!7kg 777kgf7kind7kmalg7kmalgna7kmalgnam7kn7laced7lar7last  77later777latex 77le77777lead7length7les7less7typical# 7777777 7 plealgnam7pleas77!7ples7plif7plus 777plusg7777po7posit7777possi7possibl 7777777pr 77pred 77predefin7pres7777press7previous 777prg7print 77prior7777procedur7proces 77produc7product 7!7prompt7proper7provid7qg77777ra 77rameter7ras7rath7rator7777re 77real7recogn7recogniz 77 recommend7refor7acmoodmatrixgganglebracketgplusgzerogminusgplusegminufgseealsogeneratorgenhallmonidealkacmoodyoverviewARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "A straight forward exampl e:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gen erators(lie,[x,y,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a := (3*(x &*z)&*y)&*(y&*z)+z&*y:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify( a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%#&*G6$-F%6$-F%6$-F%6$%\"xG% \"zG%\"yGF/F.\"\"$-F%6$-F%6$F)F.F/!\"$-F%6$F/F.!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Example with a directs um algebra:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "generators(sl 2,[e,f,h]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "directsum(DirSum, [l ie,sl2]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(e&*(x&*h)+z&* y-f, DirSum);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#&*G6$%\"yG%\"zG! \"\"%\"fGF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Resimplify our first expression, now with a characteristic defi ned:" }}{EXCHYPERLNK 17 "Kac Moody" 2 "symalg, KacMoody" "" } {TEXT 271 9 " algebra)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 12 " " }{TEXT 23 2 "- " }{HYPERLNK 17 "" 1 "symalgsimple.mws" "" }{HYPERLNK 17 "simple" 1 "symalgsimple.mws" "" }{TEXT 273 9 " algeb ras" }}{PARA 0 "" 0 "" {TEXT 267 12 " " }{TEXT 23 2 "- " } {TEXT 268 0 "" }{TEXT 269 14 "algebras with " }{HYPERLNK 17 "user defi ned simplification" 2 "symalg, store" "" }{TEXT 274 6 " rules" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 28 "In the case when there is a " }{HYPERLNK 17 "characteristic" 2 "symalg, char" "" }{TEXT -1 10 " defi ned, " }{TEXT 275 8 "simplify" }{TEXT -1 73 " will automatically furth er simplify its results before returning them. " }}}{SECT 0 {PARA 260 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package." }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{P5pared!):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Lhs := (y&*z) - x; Rhs := x + (x&*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LhsG,&-%#&*G6$%\"yG%\"zG\"\"\"%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RhsG,&%\"xG\"\"\"-%#&*G6$F&%\"yGF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Lhs &<> Rhs;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symal g, Overview" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "use" 2 "symalg, use" " " }}}}{MARK "0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 } PPMATH 20 "6#>%$RhsG,$-%#&*G6$-F'6$%\"xG%\"zG%\"yG\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Lhs &>= Rhs;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "An example with sums (only the lexicographicall y smallest terms are com8ueG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Example w ith assigning the expressions to variables:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "Lhs := x&*(z&*y); Rhs := (x&*z)&*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LhsG-%#&*G6$%\"xG-F&6$%\"zG%\"yG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$RhsG-%#&*G6$-F&6$%\"xG%\"zG%\"yG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Lhs &<> Rhs;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 23 "Constants are fine too:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "Lhs := -Lhs; Rhs := Rhs * 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LhsG,$-%#&*G6$%\"xG-F'6$%\"zG%\"yG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RhsG,$-%#&*G6$-F'6$%\"xG%\"zG%\"yG\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Lhs &>= Rhs;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "An example with sums (only the lexicographicall y smallest terms are com8"" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "generators(lie,[x,y,z]);" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$lieG-%-anglebracketG6%%\"xG%\"yG%\"zG7%7%\"\"\"\"\"! F.7%F.F-F.7%F.F.F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 55 "Lexicographical order for the generators defined above: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x &< y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y &< z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x &< (x&*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "A straight forward example (note the usage of bracke ts):" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "( (x&*z)&*y) &< (y&*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr:7ibmintelntmaplinputcourimathtimehyperlinkoutputhelpnormalormaltextheadwarnbulletitemkacmooduseddefinkacmoodalgebracallsequenckmalgnamcmatrixgennamweightparameternamecartanmatrixdescribrelationshipbetweengeneratorlistuseeightpositgeneratorinfodescriptwithpossiblspecialetsimplificatrulesetupalongwitfollowautomaticallyfunctgivendefaultspecififirstargumplusfactorweightsminunegatzeroliefreekmalgidealonceanyabovcaneedednotenootherpriorefinitwillsymalgsesssecondadvisablpassbutalsoacceptoptionalthirdfourththeiroverridendefaultnaturaliekmalgnamemalgnamoverridthesexpectalltypeformgensnewonlyneedsamegeneratorwnexamplweloadpackagdefinitsimplifexpandtypicalusaggganglebracketgplusgzerogminusgoureghgffgfevalmatrixgmynammynamesgynamesgxgyghgpgqgseealsodirectsumfactoralggenbasisimploverviewtriangular -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 0 0 }{CSTYLE "" -1 282 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "Courier" 1 10 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 289 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 " Courier" 1 10 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "Courier" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 299ge :" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for \+ simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for e xpand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "An example of the typical usage:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "generators(lie, [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$lieG-%-anglebracketG6$%\"xG%\"yG7$7$\"\"\"\"\"!7$F-F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ideal(L, [x&*(x&*(x&*(x&*y)) ),(((x&*y)&*y)&*y)&*y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"LG-%-a nglebracketG6$-%#&*G6$%\"xG-F)6$F+-F)6$F+-F)6$F+%\"yG-F)6$-F)6$-F)6$F0 F2F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factoralg(F, [lie , L], [2,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG-%-anglebracket G6$%$lieG%\"LG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "### non-h omogenous expression" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e1 := ((((x &*y)&*y)&*y)&*y)+(x&*y)&*(y&*(x&ules " }{TEXT 23 8 " - " }{TEXT -1 47 "si mplification rules for the triangular algebra" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 6 " Wi th " }{TEXT 330 10 "triangular" }{TEXT -1 102 " it is possible to defi ne an algebra with triangular decomposition (to define a Kac-Moody alg ebra see " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 3 "). " }}{PARA 15 "" 0 "" {TEXT -1 24 " As the second argument " } {TEXT 331 10 "triangular" }{TEXT -1 67 " expects a list algebras prede fined algebras (typically defined in " }{HYPERLNK 17 "generators" 2 "s ymalg, generators" "" }{TEXT -1 246 "). Among the three algebras ther e has to be a plus algebra (one with positive weights), a zero algebra (with zero weights) and a minus algebra (with negative weights). The re is no particular order in which they have to be apperaing in the li st." }}{PARA 15 "" 0 "" {TEXT -1 44 " When there is no third argument \+ present to " }{TEXT 332 10 "triangular" }{TEXT- lexicographical 'greater than or equal to' boolean operator" }{TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ (expr1) op (expr2)" }}{PARA 0 "" 0 "" {TEXT -1 25 " var1 o p var2" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT 256 16 " expr1, expr2" }{TEXT 23 5 " - " }{TEXT -1 18 "any lie expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 17 " \+ var1, var2 " }{TEXT 23 4 " - " }{TEXT -1 47 "variables with lie e xpressions assigned to them" }}{PARA 0 "" 0 "" {TEXT 257 19 " op \+ " }{TEXT 23 2 "- " }{TEXT -1 31 "one of the comparison opera tors" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 77 " The g enerators that build up the expressions must be declared beforehand in " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 1 ". " }}{PARA 15 "" 0 "" {TEXT -1 93 " The comparison functions only wo#40 ". If only one argume nt is present, then " }{TEXT 259 8 "simplify" }{TEXT -1 66 " will atte mpt to use the default algebra for the simplification. " }}{PARA 15 " " 0 "" {TEXT -1 30 "When there are two arguments, " }{TEXT 258 8 "simp lify" }{TEXT -1 92 " will simplify the lie expression according to the specified algebra's simplification rules." }}{PARA 15 "" 0 "" {TEXT 261 8 "simplify" }{TEXT -1 84 " recognizes and applies a special simpl ification to the following types of algebras:" }}{PARA 0 "" 0 "" {TEXT 262 12 " " }{TEXT 23 2 "- " }{TEXT -1 2 "a " } {HYPERLNK 17 "direct sum" 2 "symalg, directsum" "" }{TEXT -1 28 " of t wo lie algebras " }}{PARA 0 "" 0 "" {TEXT 263 12 " \+ " }{TEXT 23 2 "- " }{HYPERLNK 17 "factor algebras" 2 "symalg, factoral g" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 12 " " } {TEXT 23 2 "- " }{TEXT 265 0 "" }{TEXT 266 14 "algebras with " } {HYPERLNK 17 "triangular" 2 "symalg, triangular" "" }{TEXT 272 19 " de composition (ie " }{H60 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 9 "simplify " } {TEXT 30 57 "- apply lie algebra simplification rules to an expression " }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 " " {TEXT -1 20 " simplify(expr)" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ simplify(expr, algname)" }}{PARA 258 "" 0 "" {TEXT -1 11 "Paramete rs:" }}{PARA 0 "" 0 "" {TEXT 256 8 " expr" }{TEXT 23 8 " - " } {TEXT -1 26 "any lie algebra expression" }}{PARA 0 "" 0 "" {TEXT 257 11 " algname" }{TEXT 23 5 " - " }{TEXT -1 117 "name of the lie al gebra if the expression to be simplified is built up of other generato rs than the default algebra's" }}{SECT 0 {PARA 259 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 8 "simplify" }{TEXT -1 80 " function is used to apply simplification r ules to a lie expression in order to " }{TEXT 30 51 "convert the expre ssion into a sum of hall monomials" }{TEXT -1 C"Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 } {PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 D1 264 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 266 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 269 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 272 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 273 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 274 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE E7ibmintelntmaplinputcourimathtimehyperlinkoutputnormaltextheadwarnbulletitemstorallowuserdefinrulesimplificatexpressionsalgebracallsequencrulenrulenalgnamdontprintparameterspecifitheyneedformexprreslieproductgeneratorpreviousdefaultusedoptionalparametpresechoresetinfodescriptossiblupsymalgsimplifexpressesultresultanyalgebraicvalidwithinexmaplnoneedbothwillautomaticalreversoptionisnmustpriorusesymalglistthnamelastelemwithusingturnoffveryfirstargumpassdeletalgebraexamplstoragusagparametrichelppageweloadpackagnewdefinitexpandsomeslhgegfggeneratorssimplifyygalsoworkindexmorecomplexmyparigjgkgsgseealsodeltaoverview1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT algebra as specified in the first argument to " }{TEXT 310 8 "KacMoody" } {TEXT -1 26 ". To override these names " }{TEXT 314 8 "KacMoody" } {TEXT -1 85 " expects n names for all three type s of algebras, in the form " }{TEXT 315 38 "[[plus-gens],[zero-gens], [minus-gens]]" }{TEXT -1 30 ". Along with these new names " }{TEXT 316 8 "KacMoody" }{TEXT -1 13 " also expects" }}{PARA 0 "" 0 "" {TEXT -1 139 " new weights as the fourth argument. Only the weights of \+ the positive algebra need to be specified in the same form as in gener ators: " }{TEXT 317 17 "[w1, w2, ..., wn]" }{TEXT -1 1 "." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" }{MPLTEXT 1 0 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 " " 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{PARA 7 " " 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " "" 0 "" {TEXT -1 5 " " }{TEXT 311 16 "- e[1],...,e[n] " }{TEXT -1 8 "for the " }{TEXT 296 9 "KMalgname" }{TEXT 297 5 "_plus" }{TEXT -1 35 " factor-algebra (and naturally for \+ " }{TEXT 295 9 "KMalgname" }{TEXT 298 10 "_free_plus" }{TEXT -1 17 " l ie-algebra and " }{TEXT 299 9 "KMalgname" }{TEXT 300 11 "_plus_ideal" }{TEXT -1 8 " ideal) " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 312 16 "- f[1],...,f[n] " }{TEXT -1 8 "for the " }{TEXT 302 9 "KMalgna me" }{TEXT 303 6 "_minus" }{TEXT -1 35 " factor-algebra (and naturally for " }{TEXT 301 9 "KMalgname" }{TEXT 304 11 "_free_minus" }{TEXT -1 17 " lie-algebra and " }{TEXT 305 9 "KMalgname" }{TEXT 306 12 "_minus_ ideal" }{TEXT -1 8 " ideal) " }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {TEXT 313 16 "- h[1],...,h[n] " }{TEXT -1 8 "for the " }{TEXT 307 9 "K Malgname" }{TEXT 308 5 "_zero" }{TEXT -1 9 " algebra," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " where " }{TEXT 309 9 "KMalgname" }{TEXT -1 77 " is the name of the Kac-MoodyJ" 2 "symalg, Overview" "" }}}}{MARK "10 11 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } G&%\"hG6$%\"kG%\"sG&%\"fG6$,&F*\"\"\"F+F5,&F/F5F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"hG6$%\"kG%\"sG&%\"eG6$%\"iG%\"j G,$&%\"fG6$,&F/\"\"\"F0F6,&F*F6F+!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"fG6$%\"iG%\"jG&%\"hG6$%\"kG%\"sG&%\"eG6$,&F*\"\"\"F +F5,&F/F5F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"hG6$% \"kG%\"sG&%\"fG6$%\"iG%\"jG,$&%\"eG6$,&F/\"\"\"F0F6,&F*F6F+!\"\"F8" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify((e[7,8]+f[5,6])&*h [6,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"fG6$\"#:!\"\"\"\"\"&% \"eG6$\"#6F(F)" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See \+ Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "delete" 2 "symalg, delete" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "delta" 2 "symalg, delta" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalgL F*F6F+!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"eG6$%\"i G%\"jG&%\"hG6$%\"kG%\"sG&%\"fG6$,&F*\"\"\"F+F5,&F/F5F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"hG6$%\"kG%\"sG&%\"eG6$%\"iG%\"j G,$&%\"fG6$,&F/\"\"\"F0F6,&F*F6F+!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"fG6$%\"iG%\"jG&%\"hG6$%\"kG%\"sG&%\"eG6$,&F*\"\"\"F +F5,&F/F5F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"hG6$% \"kG%\"sG&%\"fG6$%\"iG%\"jG,$&%\"eG6$,&F/\"\"\"F0F6,&F*F6F+!\"\"F8" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify((e[7,8]+f[5,6])&*h [6,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"fG6$\"#:!\"\"\"\"\"&% \"eG6$\"#6F(F)" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See \+ Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "delete" 2 "symalg, delete" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "delta" 2 "symalg, delta" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalgL the echo option turned off:" } {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "generato rs(lie, [x, y]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "store(DONTPRINT , [x&*y=y, lie]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simpli fy(x&*(x&*(x&*(x&*y))), lie);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"yG " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Also works with indexed generators (for more complex examples see " } {HYPERLNK 17 "symalg, simplify" 2 "symalg, simplify" "" }{TEXT -1 2 ") :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "generators(mypar, [e[i, j], f[k,s], h[l,n]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "use(mypar) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "store([e[i,j]&*f[k,s]=h[i+j,k- s],e[i,j]&*h[k,s]=f[i+j,k-s],f[i,j]&*h[k,s]=e[i+j,k-s]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"eG6$%\"iG%\"jG&%\"fG6$%\"kG%\"sG&% \"hG6$,&F*\"\"\"F+F5,&F/F5F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%#&*G6$&%\"fG6$%\"kG%\"sG&%\"eG6$%\"iG%\"jG,$&%\"hG6$,&F/\"\"\"F0F6,&M "with(symalg):" }} {PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }} {PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Define th e first set of generators and some rules for it:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "generators(sl2,[e,f,h]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "store([h&*e=2*e,h&*f=-2*f,e&*f=h]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG%\"eG,$F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"eG%\"hG,$F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG%\"fG,$F(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"fG%\"hG,$F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"eG%\"fG%\"hG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"fG%\"eG,$%\"hG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(e&*h-e&*(f&*e)+f&*h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"eG!\"%%\"fG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "WithNvarn7vars7ver7very777viraso7virasovo 77vo7wa777war777warnO3 7777777777777777777way7weO 7777777777777777777wei 77weig7weigh7weight'3777777777were777wg 77wheth 77wi 77width!7will+ 777777777!7wit 77nerat7777generato 77 generatorS 777777777 7777777 77 7!7 genhallmon7777 7!7or7file!7fine7fined 77first3777777777777fixed!7fly7fo77!7follow7777form77!7forward77777found 7!7fourth7free7func 77funct3 77777777777!7function77!7furth7fy7ge 77gebra 77gen7genbasi77777!7gene7777gener 77genera 77generat7777generato 77 generatorO 777777777 7777777 77!7 genhallmon7777!7or7id 77idea7ideal'07777 777 7!7idealgen7idealnam7ie777777ieparg7ificat7ified7ig77 7ight7 implementat7implfi7 important7includ 7!7index777indic7indicat!7ine 77infoO 7777777777777777777inputO 7777777777777777777ins7insetad7insid!7instead!7integ 777777integer7777intelS 7777777777777777777!7 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Define another set of generators, the default stays the same:" }{MPLTEXT 1 0 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "generators(another, [p, q, r , w]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(anotherG-%-anglebracketG6 &%\"pG%\"qG%\"rG%\"wG7&7&\"\"\"\"\"!F/F/7&F/F.F/F/7&F/F/F.F/7&F/F/F/F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "use();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$lieG-%-anglebracketG6%%\"xG%\"yG%\"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Change default: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "use(another);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(anotherG-%-anglebracketG6&%\"pG%\"qG%\"rG %\"wG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "use();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(anotherG-%-anglebracketG6&%\"pG%\"qG%\"rG%\" wG" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }} {PARA 0 "" 0 "" {HYPERLNK 17 "directsum" 2 "symalg, directsum" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "facrst defined algebra becomes the f irst default algebra in the session." }}{PARA 15 "" 0 "" {TEXT -1 9 " \+ Calling " }{TEXT 258 3 "use" }{TEXT -1 66 " without any arguments show s the current default algebra settings." }}}{SECT 0 {PARA 259 "" 0 "ex amples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need \+ to load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Wa rning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "War ning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "Define the first set of generators, this \+ becomes the default:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gene rators(lie,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$lieG-%-angl ebracketG6%%\"xG%\"yG%\"zG7%7%\"\"\"\"\"!F.7%F.F-F.7%F.F.F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "use();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$lieG-%-anglebracketG6%%\"xG%\"yG%\"zG" }}}{PARA 0T257 65 " - sets an algebra to be the default algebra during the sess ion" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 12 " use( )" }}{PARA 0 "" 0 "" {TEXT -1 18 " u se(algname)" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 " " 0 "" {TEXT 256 11 " algname" }{TEXT 23 3 " - " }{TEXT -1 28 "a pr eviously defined algebra" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 102 " The algebra must be d efined prior to its use in one of the following algebra constructing f unctions: " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "directsum" 2 "symalg, directsum" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "triangular" 2 "symalg, triangular" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simple" 2 "symalg, simple" "" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 76 " The fiU1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 3 "use" }{TEXT V7ibmintelntmaplinputcourimathtimehyperlinkoutputnormaltextheadwarnbulletitemwtcalculatweightlieexpresscallsequencexpralgnamparameteranynamealgebrauseinfodescriptfunctcaseexpresssumhomogenoualltermwithsameweightwillreturnemptlistresultalsoworkparametricusedfirstdefingeneratorsymalgnospecifihedefaultnotehatsimplifisimplifybeforexamplweneedloadpackagnewdefinitsimplifexpandstraightforwardxamplseealgenbasigenhallmonymalgoverview7777777777 7pa7pack7zero 77777zerog777zg 777777 7enerat7777generato 77 generatorS 777777777 77777777 7 7!7 genhallmon7777 7!77{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 258 "" 0 1 0 0 0 0 0 1 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 W nglebracketg7ning777nnot7no+7777777777non 7!7nops7normalS 7777777777777777777!7nota!7note 777777!7now777ns7ntS 7777777777777777777!7ntprint7numb 77od7ody7off 7777offg 7ok7omogenou7once7777only777777ons 77op7opera7operat7operator777!7 operatorg7option 77ical# 7777777 7genn7genna7gennam777gens7 genweight7gg7777ght7ghts7give!7given7777great7gs7gsg7gson7gsong7gsow7gsowg7had7hal7hall 77han7har 7hat7have77777having7he 77777!7headS& 7777777777777777777!7triangular#!7777777 !7true7trueg 77ts 77turn 777ty7type77777typical# 7777777 71803 } EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "generators(lie, [x,y,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wt (x&*(z&*y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "wt((x&*z)&*y + x&*z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Parametric weights:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "generators(lie2, [p[i],q[j],r[k]], [[i,0,0], [0,j,0],[0,0,k]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "wt((p[1]&*q[1 ])&*(q[2]&*r[5])-r[4]&*((q[3]&*r[1])&*p[1]), lie2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%\"\"\"\"\"$\"\"&" }}}}{SECT 0 {PARA 260 "" 0 "seeal so" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "genbasis " 2 "symalg, genbasis" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "genhallmon" 2 "symalg, genhallmon" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "s ymalg, Overview" "" }}}}{MARK "9 1 6" 3 }{VIEWOPTS 1 1 0 1 1 `s:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "generators(lie, [x,y,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wt (x&*(z&*y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "wt((x&*z)&*y + x&*z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Parametric weights:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "generators(lie2, [p[i],q[j],r[k]], [[i,0,0], [0,j,0],[0,0,k]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "wt((p[1]&*q[1 ])&*(q[2]&*r[5])-r[4]&*((q[3]&*r[1])&*p[1]), lie2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%\"\"\"\"\"$\"\"&" }}}}{SECT 0 {PARA 260 "" 0 "seeal so" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "genbasis " 2 "symalg, genbasis" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "genhallmon" 2 "symalg, genhallmon" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "s ymalg, Overview" "" }}}}{MARK "9 1 6" 3 }{VIEWOPTS 1 1 0 1 1 ` 2 "Wt " }{TEXT -1 36 " also works with parametric weights." }}{PARA 15 "" 0 "" {TEXT -1 51 " The algebra to be used first has to be defined in " } {HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 61 " If no algebra name is specified, then t he generators of the " }{HYPERLNK 17 "default algebra" 2 "symalg, use " "" }{TEXT -1 10 " are used." }}{PARA 15 "" 0 "" {TEXT -1 33 " Note t hat the expression is not " }{HYPERLNK 17 "simplified" 2 "symalg, simp lify" "" }{TEXT -1 33 " before its weight is calculated." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" }{MPLTEXT 1 0 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 " " 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{PARA 7 " " 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "A straight forward e xample with default weightb0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 2 "wt" }{TEXT 256 46 " - calculates the weight of a lie expression" }}{PARA 257 " " 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 14 " wt(expr)" }}{PARA 0 "" 0 "" {TEXT -1 23 " wt(expr, algname) " }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 10 " expr " }{TEXT 23 5 " - " }{TEXT -1 18 "any lie expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 13 " algname " }{TEXT 23 2 "- " }{TEXT -1 26 "name of the algebra \+ to use" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 10 " Function " }{TEXT 259 2 "wt" }{TEXT -1 143 " calculates the weight of an expression. In the case when the exp ression is a sum, it has to be homogenous (all terms with the same wei ght) or " }{TEXT 260 2 "wt" }{TEXT -1 45 " will return an empty list ( [] ) for result." }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 261c3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 d7^ibmintelntmaplinputcourimathtimehyperlinkoutputhelpnormaltextheadwarnbulletitemidealuseddefincallsequencidealnamidealgendealalgnamparameteridealnamnamedefinedlistgeneratoralgebraotherdefaultinfodescriptpossiblnewcanlatersessionceallprocedursymalgsymalgoverviewpackagsupportfunctionalitableworkwithargumentuseassumbasichowevthirdargumsespecifineedalreadsomeconstructfunctexamplweloadsymalgdefinitsimplifyexpantypicalusaglieanglebracketgxgygournowokelylgseealsoaldirectsumfactoralgsymakacmoodsimpltriangulartr7tri7triangu7 triangular'#77777777 !7true7trueg 77ts 77turn 777ty7type77777typical# 7777777 777{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 }{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 "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 e 2 "A_" }{TEXT 270 9 "KMalgname" }{TEXT 293 2 " " }{TEXT -1 45 "is the Cartan matrix for the KacMoody algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " Once defined by " }{TEXT 271 8 "KacMoody" }{TEXT -1 197 ", any of the above algebras can be used as n eeded. Note that if there are no other algebra defined prior to the d efinition of the Kac-Moody algebra, then it will b e set as the " }{HYPERLNK 17 "default algebra" 2 "symalg, use" "" } {TEXT -1 17 " for the session." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 63 " As the second argument it is advisable \+ to pass a matrix in to " }{TEXT 256 8 "KacMoody" }{TEXT -1 40 ", but a list of lists is also accepted. " }}{PARA 15 "" 0 "" {TEXT -1 45 " As an optional third and fourth argument to " }{TEXT 294 8 "KacMoody" } {TEXT -1 145 " the default generator names and their weights can be ov erriden. For a Kac-Moody algebra with an n x n Cartan matrix the defa ult generators are:" }}{PARA 0Kstorag7storeg7straight77777strict7 substitut7 successful7such7sum777sums7support7777sy777sym7777!7syma7symal 77symalgS 77 7777 77 7777 7 77 7777!7 symalgsimpl7 symalgsimplif7symalgwt7symbasi 7!7syntax7take7temp7tempg7tempt7term777!7tes!77ma7make777malg777malgnam7mann7map7777ogniz 77 recommend7refor71 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 29 " The functions available are:" }}{PARA 268 "" 0 "" {TEXT -1 31 " " }{HYPERLNK 17 "&<" 2 "symalg, &operators" "" }{TEXT -1 9 " " }{HYPERLNK 17 "&<=" 2 "symalg, &operators" "" }{TEXT -1 8 " " }{HYPERLNK 17 "&<>" 2 "symalg, &operators" "" }{TEXT -1 8 " " }{HYPERLNK 17 "&=" 2 "symalg, &operators" "" }{TEXT -1 10 " " } {HYPERLNK 17 "&>" 2 "symalg, &operators" "" }{TEXT -1 9 " " } {HYPERLNK 17 "&>=" 2 "symalg, &operators" "" }{TEXT -1 5 " " }} {PARA 267 "" 0 "" {TEXT -1 31 " " } {HYPERLNK 17 "char" 2 "symalg, char" "" }{TEXT -1 7 " " } {HYPERLNK 17 "delete" 2 "symalg, delete" "" }{TEXT -1 5 " " } {HYPERLNK 17 "delta" 2 "symalg, delta" "" }{TEXT -1 6 " " } {HYPERLNK 17 "directsum" 2 "symalg, directsum" "" }{TEXT -1 3 " " } {HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" }{TEXT -1 2 " " } {HYPERLNK 17 lexicograp7lexicographical 77lg7777lgebra7lhs7lhsg7li 77libnam!7lieC) 777777777777777!7 lieandspec7 lieandspecg7lieg 777777liepar7liezero7liezerog7lificat7lify7777list3777777777777loadS 7777777777777777777!7locat!7long7lues7ly7ma7make777malg777malgnam7mann7map7777typical7777777or7cketG6$%$lieG%$sl2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " expr1 := (h&*f)&*e-y&*x+y&*f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ex prG,(-%#&*G6$-F'6$%\"hG%\"fG%\"eG\"\"\"-F'6$%\"yG%\"xG!\"\"-F'6$F1F,F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(expr1, Ds);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%#&*G6$-F%6$%\"eG%\"fG%\"hG\"\"\"- F%6$-F%6$F)F+F*!\"\"-F%6$%\"xG%\"yGF," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "With defined rules for one of the alg ebras:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "generators(spec, [ p,q,r]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%%specG-%-anglebracketG6% %\"pG%\"qG%\"rG7%7%\"\"\"\"\"!F.7%F.F-F.7%F.F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "store([p&*q=r,q&*r=p,r&*p=q, spec]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"pG%\"qG%\"rG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"qG%\"pG,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"qG%\"rG%\"pG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"rG%\"qG,$%\"pG"simple" 2 "symalg, simpl e" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "triangular" 2 "symalg, triangular" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "use" 2 "symalg, use" "" }}}}{MARK "7 2 " 49 }{VIEWOPTS 1 1 0 1 1 1803 } enerators(lie,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$lie G-%-anglebracketG6$%\"xG%\"yG7$7$\"\"\"\"\"!7$F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "### our default is now lie, so it's OK to r ely on it" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ideal(L, [x&*(x&*(x&*( x&*y))),(((x&*y)&*y)&*y)&*y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" LG-%-anglebracketG6$-%#&*G6$%\"xG-F)6$F+-F)6$F+-F)6$F+%\"yG-F)6$-F)6$- F)6$F0F2F2F2" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Al so: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "directsum" 2 "sy malg, directsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "syma lg, factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 p1 "" {TEXT -1 36 "Warning, new definition for simp lify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expan d" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "An \+ example of the typical usage:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "generators(lie,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$lie G-%-anglebracketG6$%\"xG%\"yG7$7$\"\"\"\"\"!7$F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "### our default is now lie, so it's OK to r ely on it" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ideal(L, [x&*(x&*(x&*( x&*y))),(((x&*y)&*y)&*y)&*y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" LG-%-anglebracketG6$-%#&*G6$%\"xG-F)6$F+-F)6$F+-F)6$F+%\"yG-F)6$-F)6$- F)6$F0F2F2F2" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Al so: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "directsum" 2 "sy malg, directsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "syma lg, factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 p0 " it is po ssible to define a new ideal that can be later used in the maple sessi on. Once an ideal is defined all procedures in the " }{HYPERLNK 17 "sy malg" 2 "symalg, Overview" "" }{TEXT -1 75 " package that support func tionalities for ideals are able to work with it. " }}{PARA 15 "" 0 "" {TEXT -1 6 " When " }{TEXT 261 5 "ideal" }{TEXT -1 35 " is called with two arguments, the " }{HYPERLNK 17 "default algebra" 2 "symalg, use" "" }{TEXT -1 195 " is assumed to be the basic algebra of the generator s of the ideal. However, with a third argument the basic algebra to u se can be specified (the specified algebra need to be already defined \+ in " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 68 " or some other algebra constructing function in the symalg package )." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }} {PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" } {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(sym alg):" }}{PARA 7 "" r0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 5 "ideal" }{TEXT 256 5 " - " }{TEXT 30 21 "used to define ideals" }}{PARA 257 "" 0 " " {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ ideal(idealname, [idealgens])" }}{PARA 0 "" 0 "" {TEXT -1 44 " i deal(idealname, [idealgens], algname)" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 13 " id ealname" }{TEXT 23 4 " - " }{TEXT -1 31 "name of the ideal to be defi ned" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 15 " idealgens " } {TEXT 23 2 "- " }{TEXT -1 32 "list of generators for the ideal" }} {PARA 0 "" 0 "" {TEXT 262 14 " algname " }{TEXT 23 3 " - " } {TEXT -1 49 "algebra of the generators (if other than default)" }} {SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 " " 0 "" {TEXT -1 4 " In " }{TEXT 260 5 "ideal" }{TEXT -1 13s 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE " Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 ttextK777777777777777777tg 77th7777777their7them7777ther7thes77777they777777third77777thos7thrown7timeS" 7777777777777777777!7ting7tion 77tionalit7tionship7too7tors777tr7tri7triangu7 triangular#!7777777 !7true7trueg 77ts 77turn 777ty7type77777typical7777777or7uate7ules 77ult7um 77unct7unction7 unevaluat7unles7up777777urier 77us7usag3 77777777777!7useK<77777777777777777!7used;77777777777777user7777uses7using777!7utput 77va7valid77777validit7valu  77var 77variabl77!77trueg 77ts 77turn 777ty7type77777typical7777777or7ncluding user " }{HYPERLNK 17 "defined rules" 2 "symalg, store" "" }{TEXT -1 1 ")" }{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }} {PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" } {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(sym alg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simp lify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expan d" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "An \+ example of the typical usage:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "generators(lie, [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$li eG-%-anglebracketG6$%\"xG%\"yG7$7$\"\"\"\"\"!7$F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "generators(sl2, [e,f,h]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$sl2G-%-anglebracketG6%%\"eG%\"fG%\"hG7%7%\"\"\" \"\"!F.7%F.F-F.7%F.F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " directsum(Ds, [sl2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#DsG-%-angl ebran }}{PARA 15 "" 0 "" {TEXT -1 23 " The first argument t o " }{TEXT 337 9 "directsum" }{TEXT -1 39 " is the name of the directs um algebra. " }}{PARA 15 "" 0 "" {TEXT -1 24 " As the second argument \+ " }{TEXT 339 9 "directsum" }{TEXT -1 119 " expects the name of two alg ebras in a list (they can appear without order). If there is only one entry in the list, " }{TEXT 341 9 "directsum" }{TEXT -1 46 " assumes it's the other algebra to use is the " }{HYPERLNK 17 "default algebra " 2 "symalg, use" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 5 " T he " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 93 " f unction recognizes if an algebra is a directsum and applies special si mplification rules: " }}{PARA 0 "" 0 "" {TEXT -1 85 " - t erms that contain generators from both algebras are simplifed to zero " }}{PARA 0 "" 0 "" {TEXT -1 161 " - all other terms are s implfied according to the rules that apply to the particular algebra o f the generators contained in the term (iz 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 9 "directsum" } {TEXT 256 5 " - " }{TEXT 30 38 "used to define a directsum of algebr as" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 38 " directsum(dirsumalgname, [alg1])" }}{PARA 0 " " 0 "" {TEXT -1 44 " directsum(dirsumalgname, [alg1, alg2])" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 17 " dirsumalgname" }{TEXT 23 3 " - " }{TEXT -1 29 "name of the directsum-algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 258 18 " alg1, alg2 " }{TEXT 23 2 "- " }{TEXT -1 19 "prede fined algebras" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Descripti on:" }}{PARA 15 "" 0 "" {TEXT -1 6 " With " }{TEXT 330 9 "directsum" } {TEXT -1 72 " it is possible to define a directsum of two already exis ting algebras. "{1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0|able 77abov 77accept7accord 77 according7acmood7act7advisabl 77aff7affin7after7again7age7al7777alg777777 !7alge7algeb777algebr7algebraK77 77777 7 7777 77 7 7 7!7 algebraic7algn 77algnam3777777777777algnamen7all' 77777777!7allow7777!7integ 777777integer7777intelS 7777777777777777777!7along7alread77777alsoO 777777777777777777!7alway!7ame777ames7among 77ampl 77angl777 anglebracket7 anglebracketg/ 77777777777angular7anoth7anotherg7answ7ant!7any' 777777777aple7appear 77appera7appl777appli7ar  77arameter7args!7argum3 777777777777argument 777777map7777typical7777777or7n the generators" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 259 14 " gennames " }{TEXT 23 3 " - " }{TEXT -1 39 "list of \+ names to use for the generators" }}{PARA 0 "" 0 "" {TEXT 260 14 " w eight " }{TEXT 23 3 " - " }{TEXT -1 33 "weight of the positive gene rators" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 6 " With " }{TEXT 261 8 "KacMoody" }{TEXT -1 247 " it is possible to define a Kac-Moody algebra with a special s et of simplification rules (set up from the Cartan matrix). Along wit h the Kac-Moody algebra the following algebras are defined automatical ly by the function with given default names (" }{TEXT 264 9 "KMalgname " }{TEXT -1 75 " is the name of the Kac-Moody algebra as specified in \+ the first argument): " }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ " }{TEXT 272 2 "- " }{TEXT 262 9 "KMalgname" }{TEXT 273 5 "_plus" } {TEXT 285 2 " " }{TEXT -1 41 "is a factor algebra with positive weigh ts" }}{PARA 0 "" 0 "" {TEXT -1 19 " " 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 8 "KacMoody" } {TEXT 256 5 " - " }{TEXT 30 33 "used to define Kac-Moody algebras" } }{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 34 " KacMoody(KMalgname, Cmatrix)" }}{PARA 0 "" 0 "" {TEXT -1 56 " KacMoody(KMalgname, Cmatrix, [gennames], [weight]) " }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 13 " KMalgname" }{TEXT 23 4 " - " }{TEXT -1 43 "name of the Kac-Moody algebra to be defined" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 13 " Cmatrix " }{TEXT 23 4 " - " }{TEXT -1 95 "Cartan matrix for the Kac-Moody algebra that describes the rela tionships betwee0 0 }{PSTYLE "N ormal" -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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 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symalg,wtsymalg,symbasis$Mathematics/Packages/symalg/symbasissymalg,symbasis$Mathematics/Packages/symalg/symbasissymalg,symbasis$Mathematics/Packages/symalg/symbasissymalg,symbasis$Mathematics/Packages/symalg/symbasissymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,triangular&Mathematics/Packages/symalg/triangularsymalg,Overview$Mathematics/Packages/symalg/Overviewsymalg,OverviewMathematics/Symbolic Algebrasymalg,Overview$Mathematics/Packages/symalg/Overviewsymalg,OverviewMathematics/Symbolic Algebrasymalg,Overview$Mathematics/Packages/symalg/Overviewsymalg,OverviewMathematics/Symbolic Algebrasymalg,Overview$Mathematics/Packages/symalg/Overviewsymalg,OverviewMathematics/Symbolic Algebrasymalg,Overview$Mathematics/Packages/symalg/OverviewverviewMathematics/Symbolic Algebralg/directsumsymalg,directsumsymalg,&operators&Mathematics/Packages/symalg/&operatorssymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,KacMoody$Mathematics/Packages/symalg/KacMoody symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt symalg,wtMathematics/Packages/symalg/wt"" {XPPMATH 20 "6#,(-%#&*G6$%\"xG%\"yG!\"\"F(\" \"\"%\"qG\"\"#" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See \+ Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "directsum" 2 "symalg, directsu m" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "symalg, factoralg " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "simple" 2 "symalg, simple" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "store" 2 "symalg, store" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "use" 2 "symalg, use" "" }}}}{MARK "8 3 4" 46 } {VIEWOPTS 1 1 0 1 1 1803 } cketG6%%%plusG%%z eroG%%minsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify((x& *p)&*(g&*q), T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"qG!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify((y&*(x&*p))-(g&*q), T);" }}{PARA 11 "" 1 MPLTEXT 1 0 4 "x-1;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG%\"xG,&F(\"\"\"!\"\"F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"xG%\"hG,&F'!\"\"\"\"\" F+" }}}{EXCHG {PARA 0 "h&*y = " 0 "" {MPLTEXT 1 0 4 "y+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG%\"yG,&F(\"\"\"F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"yG%\"hG,&F'!\"\"F*\"\"\"" }}}{EXCHG {PARA 0 "h&*p = " 0 "" {MPLTEXT 1 0 2 "0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG%\"pG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"pG%\"hG\"\"!" }}}{EXCHG {PARA 0 "h&*q = " 0 "" {MPLTEXT 1 0 2 "0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG% \"qG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"qG%\"hG\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"TG-%-anglebracketG6%%%plusG%%z eroG%%minsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify((x& *p)&*(g&*q), T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"qG!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify((y&*(x&*p))-(g&*q), T);" }}{PARA 11 "" 1 , [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%%plusG-%- anglebracketG6$%\"xG%\"yG7$7$\"\"\"\"\"!7$F-F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "generators(zero, [g,h],[[0,0],[0,0]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%%zeroG-%-anglebracketG6$%\"gG%\"hG7$ 7$\"\"!F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "generators(m ins, [p,q],[[-1,0],[0,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%%min sG-%-anglebracketG6$%\"pG%\"qG7$7$!\"\"\"\"!7$F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "triangular(T, [plus, zero, mins], [g&*x=2*x ,g&*y=2*y,g&*p=-2*p,g&*q=-2*q]);" }}}{EXCHG {PARA 0 "x&*p = " 0 "" {MPLTEXT 1 0 4 "x+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"xG %\"pG,&F'\"\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"pG% \"xG,&F(!\"\"F*\"\"\"" }}}{EXCHG {PARA 0 "y&*q = " 0 "" {MPLTEXT 1 0 4 "y-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"yG%\"qG,&F'\"\" \"!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"qG%\"yG,&F(! \"\"\"\"\"F+" }}}{EXCHG {PARA 0 "h&*x = " 0 "" { -1 6 ", the " } {HYPERLNK 17 "simplification" 2 "symalg, simplify" "" }{TEXT -1 61 " r ules are retrieved in an interactive manner and then saved." }}{PARA 15 "" 0 "" {TEXT -1 79 " With the optional third argument the already \+ set up rules can be passed in to " }{TEXT 333 10 "triangular" }{TEXT -1 141 " in a list. However, triangular checks the rules, and prompts for any that is not found in the list. For the valid syntax of the r ules see " }{HYPERLNK 17 "store" 2 "symalg, store" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }} {PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }} {PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "An exampl e of the typical usage:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g enerators(plusmaplO$ 7777777777777777777match7mathO 7777777777777777777matric7matrix 7777matrixg 77may7me 77mes7min7mins7minsg7minu 777minusg777mixed7mod  7!7monomial 77mood 77 7more777most7mple7777 mplificat 77mpt7insetad7insid!7instead!7integ 777777 7integer7777intelS 7777777777777777777!7multipli!7 multiplicat!7mus7must7777mws 77mynam7mynamesg7mypar7na7nameG47777777777777 77 77natural7ne 7777ned 77nee7needK777777777777777777negat777nerator7newO, 7777777777777777777ng77!77triangu7 triangular#!7777777 !7true7trueg 77ts 77turn 777ty7type77777typical# 7777777 7or7 symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/deltasymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsumsymalg,directsum%Mathematics/Packages/symalg/directsum"" 0 "" {TEXT -1 14 "Typical usage:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "genera tors(lie, [x,y,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "symbasis([2, 1,2],lie);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(-%#&*G6$%\"xG-F%6$F'-F %6$%\"zG-F%6$F,%\"yG-F%6$F'-F%6$F/-F%6$F,-F%6$F,F'-F%6$F'-F%6$F,-F%6$F 'F--F%6$F/-F%6$F'F4-F%6$F,-F%6$F'F<-F%6$F,-F%6$F'-F%6$F/F6" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "char" 2 "symalg, char" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "genhallmon" 2 "symalg, genhallmon" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "wt " 2 "symalg, wt" "" }}}}{MARK "9 1 2" 5 }{VIEWOPTS 1 1 0 1 1 1803 } 0 "> " 0 "" {MPLTEXT 1 0 13 "wi th(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition fo r simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 9 "symbasis " } {TEXT 256 5 " - ?" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence: " }}{PARA 0 "" 0 "" {TEXT -1 22 " symbasis(weight)" }}{PARA 0 "" 0 "" {TEXT -1 31 " symbasis(weight, algname)" }}{PARA 256 "" 0 " " {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 10 " weight" }{TEXT 23 5 " - " }{TEXT -1 18 "a list of intege rs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 13 " algname " } {TEXT 23 2 "- " }{TEXT -1 26 "name of the algebra to use" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 3 " ?" }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Exam ples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg packa ge:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wi th(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition fo r simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 0 0 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA7`ibmintelntmaplinputcourimathtimehyperlinkoutputnormaltextheadwarnbulletitemsymbasicallsequencweightalgnamparameterlistintegrsnamealgebrauseinfodescriptexamplexamplesweneedloadsymalgpackagewithnewdefinitfosimplifexpandtypicalusaggeneratorsliexgzgygseealsoalsochargeneratorgenhallmonoverviewwt7{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0XT 262 8 "symalg.m" }{TEXT -1 66 " file on your computer. T o load it, set the environment variable " }{TEXT 263 7 "libname" } {TEXT -1 41 " so that it includes the directory where " }{TEXT 264 8 " symalg.m" }{TEXT -1 86 " is to be found. Use the with(symalg) command \+ to load all the functions in he package." }}}{SECT 0 {PARA 266 "" 0 "s eealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "with " 2 "with" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "mod" 2 "mod" "" }}}} {MARK "3 2 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 } 17 " will result in" }}{PARA 259 "" 0 "" {TEXT -1 16 " 3 * z * y" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 258 2 " " }{TEXT 259 18 " so instead, use" }}{PARA 261 "" 0 "" {TEXT -1 19 " z \+ &* (3 * y)" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 260 2 " " } {TEXT 261 31 " which gives the correct form" }}{PARA 263 "" 0 "" {TEXT -1 19 " 3 * (z &* y)" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 44 " To use a symalg package, please locate \+ the " }{TEyG%\"zG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Turn the characteristic off:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "char();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3characteristic~offG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%# &*G6$-F%6$-F%6$-F%6$%\"xG%\"zG%\"yGF/F.\"\"$-F%6$-F%6$F)F.F/!\"$-F%6$F /F.!\"\"" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: \+ " }}{PARA 0 "" 0 "" {HYPERLNK 17 "mod" 2 "mod" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }}}}{MARK "8 1 4" 1 } {VIEWOPTS 1 1 0 1 1 1803 } erators(lie,[x,y,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a := (3*(x&*z)&*y)&*(y&*z)+z&*y:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "char(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/characteristicG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(a);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#&*G6$%\" symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple 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symalg,ideal!Mathematics/Packages/symalg/idealsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,KacMoody$Mathematics/Packages/symalg/KacMoodysymalg,Overview$Mathematics/Packages/symalg/Overviewsymalg,OverviewMathematics/Symbolic Algebrasymalg,OverviewMathematics/Symbolic Algebrasymalg,OverviewMathematics/Symbolic 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4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 6 "delete" } {TEXT 256 48 " - allows the deletion of rules for an algebra" }} {PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 15 " delete( )" }}{PARA 0 "" 0 "" {TEXT -1 23 " del ete(DONTPRINT)" }}{PARA 0 "" 0 "" {TEXT -1 47 " delete(algname1, \+ algname2, ..., algnamen)" }}{PARA 0 "" 0 "" {TEXT -1 47 " delete( DONTPRINT, algname1,..., algnamen)" }}{PARA 256 "" 0 "" {TEXT -1 11 "P arameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 11 " algname " }{TEXT 23 21 "1, ..., algnamen - " }{TEXT -1 40 "names of algebras with rules defined in " }{HYPERLNK 17 "store" 2 "symalg, store" "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 13 " DONTPRINT" }{TEXT 23 19 " - " }{TEXT -1 51 "optional parameter, if present \+ no output is printed" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Des cription:" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 262 6 "Delete" } {TEXT -1 86 " removes a set of rules defined for an algebra. These rul es are typically used in the " }{HYPERLNK 17 "simplification" 2 "symal g, simplify" "" }{TEXT -1 67 " process of an expression (for side effe cts please see note below)." }}{PARA 15 "" 0 "" {TEXT -1 49 " The alge bra must be defined prior to its use in " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 60 " and for successful deletion i t must have rules declared in " }{HYPERLNK 17 "store" 2 "symalg, store " "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 35 ".When no algebra \+ is specified then " }{TEXT 260 6 "delete" }{TEXT -1 54 " attempts to d elete the rules for the default algebra." }}{PARA 15 "" 0 "" {TEXT -1 102 " When using the DONTPRINT option to turn off the echo, it has to \+ be the very first argument passed to " }{TEXT 259 6 "delete" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 46 " Any number of algebra names \+ can be passed to " }{TEXT 261 6 "delete" }{TEXT -1 1 "." }}{PARA 15 " " 0 "" {TEXT -1 61 " Note that once the rules were applied (ie an expr ession was " }{HYPERLNK 17 "simplfied" 2 "symalg, simplify" "" }{TEXT -1 154 " using the rules) then deleting the rules for the algebra and \+ simplifying the expression again will not produce a different answer t han in the first case." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symal g package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new defin ition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new defini tion for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Define the first set of generators and some rules for it: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "generators(sl2,[e,f,h]): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "store([h&*e=2*e,h&*f=-2*f,e&*f= h]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"hG%\"eG,$F(\notatesliealgebraproductexamplnoteusingconstantinsidexpressalwayindicatpartplacbracketaroundconstanttermmultipliwillresultinsteadgivecorrectformpleaslocatfileyourcomputloadsetenvironmvariabllibnamincluddirectorfoundwithcommandallheeealsoalsomod0 0 0 } {CSTYLE "" -1 329 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 330 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 332 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "No rmal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -k0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 9 "genbasis " } {TEXT 256 76 " - generates the basis for the given weight in the spec ified factor algebra" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequenc e:" }}{PARA 0 "" 0 "" {TEXT -1 22 " genbasis(weight)" }}{PARA 0 " " 0 "" {TEXT -1 31 " genbasis(weight, algname)" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 10 " weight" }{TEXT 23 5 " - " }{TEXT -1 18 "a list of intege rs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 13 " algname " } {TEXT 23 2 "- " }{TEXT -1 33 "name of the factor algebra to use" }} {SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 " " 0 "" {TEXT -1 10 " Function " }{TEXT 259 8 "genbasis" }{TEXT -1 103 " is used to generate the basis for a factor algebra. The algebra the refore first has to be defined in " }{HYPERLNK 17 "factor algebra" 2 " symalg, factoralg" "" }{TEXT -1 24 ". There is support for " } {HYPERLNK 17 "defined characteristics" 2 "symalg, char" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 260 8 "Genbasis" } {TEXT -1 45 " also works with algebras that have negative " } {HYPERLNK 17 "weights" 2 "symalg, wt" "" }{TEXT -1 103 " defined. Howe ver, mixed weights (weights with positive and negative elements in the m) are not allowed." }}{PARA 15 "" 0 "" {TEXT -1 39 " If no algebra na me is specified, then " }{TEXT 261 8 "genbasis" }{TEXT -1 25 " will at tempt to use the " }{HYPERLNK 17 "default algebra" 2 "symalg, use" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examp les:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg packag e:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wit h(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for \+ expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Example with a KacMoody algebra:" }}{EveryfirstargumpassanynumbcanoncewereappliieexpressionsimplfisimplifyagainwillproducdifferanswhancaseexamplweneedloadpackagnewitiondefinitionexpandsomeslhgegfgstoreglieliegfailgseealsoalsooverviewWWWWWWWWxhWČWi1 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"xG6$%\"iG%\"sG&%\"yG6$%\" jG%\"tG,&*&F/\"\"\"&F-6$,&F*F3F/F3,&F+F3F0F3F3F3*&F+F3&F(F5F3!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"yG6$%\"jG%\"tG&%\"xG6$%\" iG%\"sG,&*&F*\"\"\"&F(6$,&F/F3F*F3,&F0F3F+F3F3!\"\"*&F0F3&F-F5F3F3" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(x[1,3]&*y[2,5], W2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"yG6$\"\"$\"\")\"\"#&%\"xGF &!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " - Virasovo:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "generators(Virasovo, [e[i],k]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "store([e[i]&*e[j]=(j-i)*e[i+j]+delta[i,-j]*(((i^3)-i) /6)*k,k&*e[i]=0, Virasovo]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&* G6$&%\"eG6#%\"iG&F(6#%\"jG,&*&,&F-\"\"\"F*!\"\"F1&F(6#,&F*F1F-F1F1F1*( &%&deltaG6$F*,$F-F2F1,&*$)F*\"\"$\"\"\"#F1\"\"'F*#F2FAF1%\"kGF1F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$&%\"eG6#%\"jG&F(6#%\"iG,&*&,& F*\"\"\"F-!\"\"F1&F(6#,&F-F1F*F1F1F2*9acmoodmatrixgganglebracketgplusgzerogminusgplusegminufgseealsogeneratorgenhallmonidealkacmoodyoverviewxW|WWdWeWWWWWWWWWWWWWWdWČWf -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 11 "isgenerator" }{TEXT 256 64 " - checks the validity of a generator in the specifie d algebra" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }} {PARA 0 "" 0 "" {TEXT -1 22 " isgenerator(var)" }}{PARA 0 "" 0 " " {TEXT -1 41 " isgenerator([var1, var2,..., varn])" }}{PARA 0 " " 0 "" {TEXT -1 31 " isgenerator(var, algname)" }}{PARA 0 "" 0 " " {TEXT -1 50 " isgenerator([var1, var2,..., varn], algname)" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 22 " var, var1,...,varn" }{TEXT 23 3 " - " } {TEXT -1 24 "generators to be checked" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 258 8 " algn" }{TEXT 23 17 "ame - " }{TEXT -1 64 "name of algebra in which the generators (vars) are to be checked" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "Isgenerator" }{TEXT -1 183 " is \+ a boolean function, with which it is possible to check whether a given generator is valid in an algebra. The algebra to be used first needs \+ to be defined in one of the following: " }{HYPERLNK 17 "directsum" 2 " symalg, directsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "sy malg, factoralg" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "ideal" 2 "symalg, \+ ideal" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, gener ators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMood y" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simple" 2 "symalg, simple" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "triangular" 2 "symalg, triangular" "" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 33 " If no algebra name is \+ specified " }{TEXT 259 11 "isgenerator" }{TEXT -1 45 " checks the gene rator in the default algebra." }}{PARA 15 "" 0 "" {TEXT -1 23 " The fi rst argument to " }{TEXT 260 11 "isgenerator" }{TEXT -1 70 " is a sing le generator or a list of generators that are to be checked." }}{PARA 15 "" 0 "" {TEXT -1 6 " The " }{TEXT 261 11 "isgenerator" }{TEXT -1 100 " function provides support for parametric generators. It employs a rather strict matching facility:" }}{PARA 0 "" 0 "" {TEXT -1 118 " \+ - generators that were declared with integer indices eval uate to true only in the case of an exact match" }}{PARA 0 "" 0 "" {TEXT -1 255 " - generators that were declared with names as indices evaluate to true even if there are integers substituted in for the names BUT they evaluate to false \+ if different names are used for indices" }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 7 "amples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg package:" }{MPLTEXT 1 0 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 " " 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{PARA 7 " " 1 "" {TEXT -1 34 "Warning, new definition for expand" }}}{PARA 0 "" 'G {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "char(2):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "simplify(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%#&*G6$-F%6$-F%6$-F%6$%\"xG%\"zG%\"yGF/F.\"\"\"-F%6$-F%6$F)F. F/F0-F%6$F/F.F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Parametric examples" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ - affine:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "generators(aff ine,[e[i],f[i],h[i]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "store(DO NTPRINT, [h[i]&*e[j]=2*e[i+j],h[i]&*f[j]=-2*f[i+j],e[i]&*f[j]=h[i+j]+i *delta[i,-j]*k,k&*e[i]=0,k&*f[i]=0,k&*h[i]=0,e[i]&*e[j]=0,f[i]&*f[j]=0 ,h[i]&*h[j]=2*i*delta[i,-j]*k, affine]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(e[3]&*(h[2]&*e[5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sim plify(e[3]&*(h[2]&*f[5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%\"hG 6#\"#5!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(e[3] &*(h[2]&*f[-5]));" }}{PARA 11 "" 1 "" {XPPMA" 0 "" {TEXT -1 32 "An example of the ty pical usage:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "KacMoody(G,[ [2,-1,-1],[-1,2,-1],[-1,-3,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"GG-%-anglebracketG6%%'G_plusG%'G_zeroG%(G_minusG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "### our default generators for G" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "use();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"GG-%-anglebracketG6+&%\"eG6#\"\"\"&F)6#\"\"#&F)6#\"\"$&%\"hGF*&F3F -&F3F0&%\"fGF*&F7F-&F7F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "### the matrix" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "eval(A_G);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"#!\"\"F)7%F)F(F)7% F)!\"$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "To override default names:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "KacMoody(MyNames,[[2,-3],[-3,2]],[[x,y],[g,h],[p,q]],[[1,0],[0,1 ]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(MyNamesG-%-anglebracketG6%% -MyNames_plusG%-MyNames_zeroG%.MyNames_minusG" }}}{EXCHG {P;toralg" 2 "symalg, factoralg" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "simple" 2 "symalg, simple" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT 32 2 " , " }{HYPERLNK 17 "triangular" 2 "symalg, triangular" "" }}}}{MARK "8 \+ 1 10" 4 }{VIEWOPTS 1 1 0 1 1 1803 } 1 "" {XPPMATH 20 "6#/%$lieG-%-anglebracketG6%%\"xG%\"yG%\"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Change default: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "use(another);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(anotherG-%-anglebracketG6&%\"pG%\"qG%\"rG %\"wG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "use();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(anotherG-%-anglebracketG6&%\"pG%\"qG%\"rG%\" wG" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }} {PARA 0 "" 0 "" {HYPERLNK 17 "directsum" 2 "symalg, directsum" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "fac!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"rG%\"pG%\"qG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"pG%\"rG,$%\"qG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "directsum(lieandspec, [lie,spec]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%+lieandspecG-%-anglebracketG6$%$lieG%%specG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "expr2 := q&*(p&*q)-y&*x+r&*( (x&*y)&*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr2G,(-%#&*G6$%\"q G-F'6$%\"pGF)\"\"\"-F'6$%\"yG%\"xG!\"\"-F'6$%\"rG-F'6$-F'6$F1F0F1F-" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(expr2, lieandspec );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"pG\"\"\"-%#&*G6$%\"xG%\"yGF %" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }} {PARA 0 "" 0 "" {HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "ideal" 2 "symalg, ideal" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "isgenerator" 2 "symalg, isgenerator" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simple" 2 "symalg, simple" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "symalg" 2 "symalg, Overview" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "use" 2 "symalg, use" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " wt" 2 "symalg, wt" "" }}}}{MARK "9 1 4" 10 }{VIEWOPTS 1 1 0 1 1 1803 } 1 "" {XPPMATH 20 "6#>%&expr2G,(-%#&*G6$%\"q G-F'6$%\"pGF)\"\"\"-F'6$%\"yG%\"xG!\"\"-F'6$%\"rG-F'6$-F'6$F1F0F1F-" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(expr2, lieandspec );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"pG\"\"\"-%#&*G6$%\"xG%\"yGF %" }}}}{SECT 0 {PARA 260 "" 0 "seealso" {TEXT -1 10 "See Also: " }} {PARA 0 "" 0 "" {HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "ideal" 2 "symalg, ideal" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "isgenerator" 2 "symalg, isgenerator" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYsetting7sg77777 sgenerator7show7si 77side7sim 77simp77777simpl/47777777777!7 simplealgnam7simplfi 77simpli7simplifOW 7777777777 77777777!7simplifi7777 simplificat 777777777gebra 77gen7genbasi77777!7gene7777gener 77genera77 7generat7777generato 77 generatorS 777777777 7777777 77 7!7 genhallmon7777 7!7or7simplify7simulat7sing7sl 7777smallest 77sn7some777777spec 77specg7special777777specifi/77777777777ssibl7ssion7st7stabl7stay7stor''777777 77!7 symalgsimplif7symalgwt7symbasi  7!7syntax7take7temp7tempg7tempt7term777!7tes!7ariabl77!7erato 77 generatorO 777777777 7777777 7 7!7 genhallmon777 7!7ommend7refor7 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT " }{TEXT 274 2 " - " }{TEXT 263 9 "KMalgname" }{TEXT 275 6 "_minus" }{TEXT 286 1 " " } {TEXT -1 41 "is a factor algebra with negative weights" }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{TEXT 276 2 "- " }{TEXT 265 9 "KMalgname" }{TEXT 277 5 "_zero" }{TEXT 287 2 " " }{TEXT -1 34 "is a \+ lie algebra with zero weights" }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ " }{TEXT 278 2 "- " }{TEXT 266 9 "KMalgname" }{TEXT 279 10 "_free_plus" }{TEXT -1 6 " and" }{TEXT 267 1 " " }{TEXT 288 9 "KMalg name" }{TEXT 283 11 "_free_minus" }{TEXT 289 2 " " }{TEXT -1 48 "are \+ the lie algebras for the two factor algebras" }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{TEXT 280 2 "- " }{TEXT 269 9 "KMalgname " }{TEXT 281 11 "_plus_ideal" }{TEXT -1 4 " and" }{TEXT 268 1 " " } {TEXT 290 9 "KMalgname" }{TEXT 284 12 "_minus_ideal" }{TEXT 291 1 " " }{TEXT -1 42 "are the ideals for the two factor algebras" }}{PARA 0 " " 0 "" {TEXT -1 19 " " }{TEXT 282 2 "- " }{TEXT 292jsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgeneratorsymalg,isgenerator'Mathematics/Packages/symalg/isgenerator symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,simple"Mathematics/Packages/symalg/simple symalg,delete"Mathematics/Packages/symalg/delete symalg,delete"Mathematics/Packages/symalg/delete symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta symalg,delta!Mathematics/Packages/symalg/delta -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 6 "simple" } {TEXT 256 5 " - " }{TEXT 30 30 "used to define simple algebras" }} {PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 43 " simple(simplealgname, Cmatrixtype[n])" }}{PARA 0 " " 0 "" {TEXT -1 55 " simple(simplealgname, Cmatrixtype[n], [genna mes])" }}{PARA 0 "" 0 "" {TEXT -1 53 " simple(simplealgname, Cmat rixtype[n], [weight])" }}{PARA 0 "" 0 "" {TEXT -1 65 " simple(sim plealgname, Cmatrixtype[n], [gennames], [weight])" }}{PARA 256 "" 0 " " {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 17 " simplealgname" }{TEXT 23 3 " - " }{TEXT -1 26 "name of the simple algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 18 " Cm atrixtype " }{TEXT 23 2 "- " }{TEXT -1 70 "type of the Cartan matrix for the simple algebra (A, D, E, B, C, F, G)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 18 " n  " }{TEXT 23 2 "- " } {TEXT -1 30 "dimension of the Cartan matrix" }}{PARA 0 "" 0 "" {TEXT 263 18 " gennames " }{TEXT 23 2 "- " }{TEXT -1 55 "names of th e generators to use insetad of default names" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 18 " weight " }{TEXT 23 2 "- " }{TEXT -1 34 "weights of the positive generators" }}{SECT 0 {PARA 258 "" 0 "info " {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 6 " With " } {TEXT 265 6 "simple" }{TEXT -1 109 " it is possible to define simple a lgebras of type An, Dn, En, Bn, Cn, F4, G2. After having defined one, the " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 80 " function applies a special simplification to expressions within these algebras." }}{PARA 15 "" 0 "" {TEXT -1 23 " The first argument to " } {TEXT 260 6 "simple" }{TEXT -1 118 " is the name of the simple algebra . If no other algebras were defined previously, the simple algebra is \+ set to be the " }{HYPERLNK 17 "default algebra" 2 "symalg, use" "" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 24 " As the second argument " }{TEXT 261 6 "simple" }{TEXT -1 135 " expects the type of the algeb ra (ie A[3], etc). A matrix of the specified type is generated and ma de available with the default name " }{TEXT 268 2 "A_" }{TEXT 267 13 " simplealgname" }{TEXT -1 8 ", where " }{TEXT 269 13 "simplealgname" } {TEXT -1 38 " is the name of the simple algebra. " }}{PARA 15 "" 0 " " {TEXT -1 46 " There are two kinds of optional arguments to " }{TEXT 276 6 "simple" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 67 " \+ - to override the default names for the generators" }}{PARA 0 "" 0 "" {TEXT -1 69 " - to override the default weig hts for the generators" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " By default the names of the generators are " } {TEXT 278 12 "e[k1,...,kn]" }{TEXT -1 5 " and " }{TEXT 280 5 "h[i1]" } {TEXT -1 113 ", where n is the dimension of the Cartan matrix. To ove rride them, for example, we could pass [x, y] to " }{TEXT 279 6 "simple" }{TEXT -1 46 " and in this case the generators would become " }{TEXT 281 12 "x[k1,...,kn]" }{TEXT -1 6 " and y" }{TEXT 282 4 "[i1 ]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 51 " The default the \+ weights for the generators are " }{TEXT 288 11 "[k1,...,kn]" }{TEXT -1 5 " for " }{TEXT 289 12 "e[k1,...,kn]" }{TEXT -1 5 " and " }{TEXT 290 9 "[0,...,0]" }{TEXT -1 14 "(n zeros) for " }{TEXT 285 5 "h[i1]" } {TEXT -1 127 ", where n is the dimension of the Cartan \+ matrix. To override them, for example, we could pass [[k1+k2,k3]] to " }{TEXT 284 6 "simple" }{TEXT -1 130 " if we had n = 3, the zero wei ghts are automatically generated by simple to correspond w ith this new set of weights. " }}{PARA 0 "" 0 "" {TEXT -1 114 " It is also possible to make use of both optional arguments to override b oth the default names and the weights." }}}{SECT 0 {PARA 259 "" 0 "exa mples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need t o load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Wa rning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "War ning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "An example of the typical usage:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simple(Gs, D[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#GsG-%-anglebracketG6$&%\"eG6&%#k1G%#k2G%# k3G%#k4G&%\"hG6#%#i1G7$7&F+F,F-F.7&\"\"!F6F6F6" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "simplify(h[1]&*(e[1,1,0,1]&*h[3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"eG6&\"\"\"F&\"\"!F&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "simple(Gson, E[6], [q,r]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%%GsonG-%-anglebracketG6$&%\"qG6(%#k1G%#k2G%#k3G%#k 4G%#k5G%#k6G&%\"rG6#%#i1G7$7(F+F,F-F.F/F07(\"\"!F8F8F8F8F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "simplify(q[1,0,1,0,1,1]&*q[0,1,0,1, 0,0], Gson);" }}symalg,generators&Mathematics/Packages/symalg/generatorssymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmonsymalg,genhallmon&Mathematics/Packages/symalg/genhallmon. Even if the character istic was disabled previously this will automatically enable it." }} {PARA 15 "" 0 "" {TEXT -1 82 " The enabling of the characteristic will set the characteristic to its last value." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We ne ed to load the symalg package:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition for simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 " Warning, new definition for expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 25 "A simplification example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "generators(lie,[x,y,z]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a := (3*(x&*z)&*y)&*(y&*z)+z&*y:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "char(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/characteristicG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(a);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#&*G6$%\" "" {MPLTEXT 0 21 0 "" }{TEXT -1 4 "char" }{TEXT 256 44 " - sets the characteristic for the session" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 13 " \+ char( )" }}{PARA 0 "" 0 "" {TEXT -1 13 " char(n)" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 5 " n" }{TEXT 23 4 " - " }{TEXT -1 36 "an integer, the \+ characteristic value" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Des cription:" }}{PARA 15 "" 0 "" {TEXT -1 10 " Function " }{TEXT 258 4 "c har" }{TEXT -1 120 " sets a default characteristic value for the maple session and enables/disables the usage of the defined characteristic. " }}{PARA 15 "" 0 "" {TEXT -1 18 " In the case when " }{TEXT 260 4 "ch ar" }{TEXT -1 78 " is called with zero arguments it enables/disables t he default characteristic." }}{PARA 15 "" 0 "" {TEXT -1 6 " When " } {TEXT 259 4 "char" }{TEXT -1 170 " is called with an integer argument \+ the default characteristic is set to that value 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 4 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 3 "" 0  7ibmintelntmaplinputcourimathtimehyperlinkoutputnormalextheadwarnbulletitemcharsetscharacteristicsesscallsequencparameterintegvaluinfodescriptfuncthardefaultenabldisablusagdefincasecharwithzeroargumentheargumsetevencharactisticpreviouswillautomaticallastexamplweneedloadsymalgpackagnewdefinitsimplifexpandsimplificatgeneratorliecharacteristicgygzgturnoffoffgxgygfseealsoalsomodoverview 7{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 "" 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07ibmintelntmaplinputcourimathtimehyperlinkoutputnormaltextheadwarnutputbulletitemgeneratorallowdefinitalgebracallsequencalgnamgennamgenweightparameternamethdefingennamesvariablwillactmostbasicbuildblockexpressionsnumbinteglisteachcorrespondinfodescriptpossiblnewcanlaterusedmaplesessonceallfunctionsymalgoverviewpackagableworkwithadvisablusesuchameassignvalueselsewherduringhenfirstsetdefaultfunctionnospecifithemamongargumentcaseexpectalgebraorderdecidlexicographicaloperatorsmallestetcweightpassgeneratoroptionalthirdargumdefaultarbuiltusertheshavethrownintogeneratorscontainneperintegerparametricmussamelengthpossibleweightwtrecommendusindextheyspecialsimplificatrulestordependvalurecognsimplifsymalgsimplifmwssymalgwtaccordingexamplimplementatvirasovonoteexistancconstructordirectsumfactoralgsymalgidealkacmoodsimpltriangularsomerequirirectsumbefo8ecified, then the generators of the " } {HYPERLNK 17 "default algebra" 2 "symalg, use" "" }{TEXT -1 10 " are u sed." }}{PARA 15 "" 0 "" {TEXT -1 52 " For the definition of valid hal l monomials see the " }{HYPERLNK 17 "overview" 2 "symalg, Overview" " " }{TEXT -1 1 "." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Exa mples:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg pack age:" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w ith(symalg):" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new definition f or simplify" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition fo r expand" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "A straight forward example with default weights:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "generators(lie, [x,y,z]):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "genhallmon([1,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%#&*G6$-F%6$-F%6$%\"xG%\"zG%\"yGF--F%6$-F%6$-F%6$F+F -F,F--F%6$-F%6$F2F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA(on" } {TEXT 256 61 " - generates the set of hall monomials for the given w eight" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 " " 0 "" {TEXT -1 24 " genhallmon(weight)" }}{PARA 0 "" 0 "" {TEXT -1 33 " genhallmon(weight, algname)" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 10 " \+ weight" }{TEXT 23 5 " - " }{TEXT -1 18 "a list of integers" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 13 " algname " }{TEXT 23 2 "- " }{TEXT -1 26 "name of the algebra to use" }}{SECT 0 {PARA 258 " " 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 10 " Function " }{TEXT 259 10 "genhallmon" }{TEXT -1 106 " generates all t he valid hall monomials for the given weight from the generators of th e specified algebra." }}{PARA 15 "" 0 "" {TEXT -1 51 " The algebra to \+ be used first has to be defined in " }{HYPERLNK 17 "generators" 2 "sym alg, generators" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 61 " I f no algebra name is sp7ibmintelntmaplinputcourimathtimehyperlinkoutputnormaltextheadwarnbulletitemisgeneratorcheckvaliditgeneratorspecifialgebracallsequencvarvarnalgnamparameteralgnamenamevarsinfodescriptbooleanfunctwithpossiblwhethgivenvalidusedfirstneeddefinfollowdirectsumsymalgfactoralgsymalgidealgeneratorkacmoodsimpltriangularnogeneratordefaultfirstargumsinglelistprovidsupportparametricemployrathstrictmatchfacilitweredeclarintegindicevaluatetrueonlycaseexactevaluatevenintegersubstitutbuttheyfalsdifferexamplamplweloadpackagnewdefinitsimplifexpandstraightforwardexamplelietruegindexdicematrixgganglebracketgplusgerogminusguseeghgffgffalsegsghgsgeneratoreealsoalsostoroverview {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 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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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -F"" {TEXT -1 0 "" }{TEXT 259 13 " DONTPRINT" }{TEXT 23 9 " - " }{TEXT -1 73 "optional parameter, if present, the rules are not echoed when they 're set" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 71 " By storing rules for an algebra it is p ossible to set up user defined " }{HYPERLNK 17 "simplification" 2 "sym alg, simplify" "" }{TEXT -1 46 " rules for the lie expressions in the \+ algebra." }}{PARA 15 "" 0 "" {TEXT -1 175 " A rule has a form expr = r esult, where expr is a lie-product of two generators in the specified \+ algebra, and the result is any algebraic expression valid within that \+ algebra." }}{PARA 15 "" 0 "" {TEXT -1 71 " For exmaple, there is no ne ed to define rules for both x&*y and y&*x, " }{TEXT 261 5 "store" } {TEXT -1 107 " will automatically define the reverse of the expression and echo it if the DONTPRINT option isn't present." }}{PARA 15 "" 0 " " {TEXT -1 49 " The algebra must be defined prior to its use in " } {HYPERLNK 17 "+0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 5 "store" }{TEXT 257 79 " - allows user defined rules for simplifications of expressi ons in an algebra" }}{PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence: " }}{PARA 0 "" 0 "" {TEXT -1 39 " store([ rule1,rule2, ..., rulen ])" }}{PARA 0 "" 0 "" {TEXT -1 81 " store([ rule1,rule2, ..., rul en, algname],...,[rule1, .., rulen, algname] )" }}{PARA 0 "" 0 "" {TEXT -1 94 " store(DONTPRINT, [ rule1, rule2, ..., rulen, algnam e],...,[rule1, ..., rulen, algname] )" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT 256 22 " rule1,.., rulen - " }{TEXT -1 124 "the rules for the specified algebra (they to need be in the form: expr = res, where expr is a lie-product of two generator s)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 11 " algname" } {TEXT 23 11 " - " }{TEXT -1 71 "a previously defined algebra, \+ if not specified, default algebra is used" }}{PARA 0 "" 0  1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 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0>79ibmintelntmaplinputcourimathtimehyperlinkoutputhelpnormaltextheadwarnbulletitemfactoralguseddefinfactoralgebracallsequencfactoralgnamidealcutoffwtalgideaalparameternamepredefinhighestweightexpresswillsimplfigeneratbasiinfodescriptwithpossibldefinefunctexpecthavethmakeupalreadgeneratorsymalgsecondargumlisttheycanappearwithoutorderonlyentrheassumidealusesdefaultusebothsimplifgenbasineestablsetcalculattakelongoptionalthirdefincutoffsimplifidurinsesswayanylaterdoesnregeneratresultfastperformancoverallsomemoreexamplfactoesweneedloadpackagnewdefinitxpandtypicalusaglielieganglebracketgxgyglgnglebracketgfganglebracketnonomogenouwtygfhomogenouexpressseealsoalsodirectsumkacmoodsimploverview 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 7Y3{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 " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 0 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 260 "Courier" 1 10 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 "Courier" 1 10 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 '37 " assumes it's the id eal and uses the " }{HYPERLNK 17 "default algebra" 2 "symalg, use" "" }{TEXT -1 17 " for the algebra." }}{PARA 15 "" 0 "" {TEXT -1 6 " Both \+ " }{HYPERLNK 17 "simplify" 2 "symalg, simplify" "" }{TEXT -1 5 " and \+ " }{HYPERLNK 17 "genbasis" 2 "symalg, genbasis" "" }{TEXT -1 428 " nee d to have the stable set calculated for the factor-algebra, which can \+ take a long time. With the optional third argument it is possible to d efine a cut off weight, which is the highest weight for which a basis \+ will be calculated or for which an expression will be simplified durin g the session. This way for any later calculations the stable set doe sn't have to be regenerated which results in a faster performance over all." }}{PARA 15 "" 0 "" {TEXT -1 47 " For some more examples on facto r-algebras see " }{HYPERLNK 17 "genbasis" 2 "symalg, genbasis" "" } {TEXT -1 1 "." }}}{SECT 0 {PARA 259 "" 0 "examples" {TEXT -1 9 "Exampl es:" }}{PARA 0 "" 0 "" {TEXT -1 35 "We need to load the symalg packa?efined algebra" }}{PARA 0 "" 0 "" {TEXT 329 15 " \+ ideal " }{TEXT 23 5 " - " }{TEXT -1 16 "predefined ideal" }} {PARA 0 "" 0 "" {TEXT 336 15 " cutoffwt " }{TEXT 23 5 " - " } {TEXT -1 104 "highest weight for the expressions that will be simplfie d or generated a basis for in the factor-algebra" }}{SECT 0 {PARA 258 "" 0 "info" {TEXT -1 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 6 " With " }{TEXT 330 9 "factoralg" }{TEXT -1 157 " it is possible to def ine a factor-algebra. The function expects to have the algebra and th e ideal that make up the factor-algebra to be already defined in " } {HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 5 " and \+ " }{HYPERLNK 17 "ideal" 2 "symalg, ideal" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 24 " As the second argument " }{TEXT 334 9 "factor alg" }{TEXT -1 129 " expects the name of an algebra and an ideal in a \+ list (they can appear without order). If there is only one entry in t he list, " }{TEXT 335 9 "factoralg" }{TEXT -1 +1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 9 "factoralg" } {TEXT 256 5 " - " }{TEXT 30 30 "used to define factor-algebras" }} {PARA 257 "" 0 "" {TEXT -1 17 "Calling sequence:" }}{PARA 0 "" 0 "" {TEXT -1 39 " factoralg(factoralgname, [ideal])" }}{PARA 0 "" 0 " " {TEXT -1 49 " factoralg(factoralgname, [ideal], cutoffwt)" }} {PARA 0 "" 0 "" {TEXT -1 44 " factoralg(factoralgname, [alg, idea l])" }}{PARA 0 "" 0 "" {TEXT -1 54 " factoralg(factoralgname, [al g, ideal], cutoffwt)" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 17 " factoralgname" } {TEXT 23 3 " - " }{TEXT -1 26 "name of the factor-algebra" }}{PARA 0 " " 0 "" {TEXT -1 0 "" 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}!7ibmintelnthyperlinkcouritimenormalheadingheadbulletitemfixedwidthintroductsymalgpackagusagcallsequencfunctargsfodescriptfunctionavailabloperatorchardeletdeltadirectsumfactoralggenbasigeneratorgenhallmonidealymalgisgeneratorkacmoodsymalgsimplsimplifstorsymbasitriangularusewtoverloadnoncommutatmultiplicatnotatesliealgebraproductexamplnoteusingconstantinsidexpressalwayindicatpartplacbracketaroundconstanttermmultipliwillresultinsteadgivecorrectformpleaslocatfileyourcomputloadsetenvironmvariabllibnamincluddirectorfoundwithcommandallheeealsoalsomodd op(0,GENERATORS[2]) = op(0,Rhs)) then RETURN(0); ### on lhs we have h[i], on rhs we have e[k1,..,kn] elif( op(0,GENERATORS[2]) = op(0,Lhs) ) then ### rhs has to be a true generator if( member(RI,TrueGens) and (LI[] > 0 and not(LI[] > nops(RI))) ) then RETURN(add(RI[j] * M[LI[],j],j=1..nops(RI)) * op(0,GENERATORS[1])[RI[]]); notatesliealgebraproductexamplnoteusingconstantinsidexpressalwayindicatpartplacbracketaroundconstanttermmultipliwillresultinsteadgivecorrectformpleaslocatfileyourcomputloadsetenvironmvariabllibnamincluddirectorfoundwithcommandallheeealsoalsomodd op(0,GENERATORS[2]) = op(0,Rhs)) then RETURN(0); ### on lhs we have h[i], on rhs we have e[k1,..,kn] elif( op(0,GENERATORS[2]) = op(0,Lhs) ) then ### rhs has to be a true generator if( member(RI,TrueGens) and (LI[] > 0 and not(LI[] > nops(RI))) ) then RETURN(add(RI[j] * M[LI[],j],j=1..nops(RI)) * op(0,GENERATORS[1])[RI[]]); "\"\"\"\"!F.7%F.F-F.7%F.F.F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Indexed generators with u ser defined weights:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "gene rators(liezero, [h[1],h[2]],[[0,0],[0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(liezeroG-%-anglebracketG6$&%\"hG6#\"\"\"&F)6#\"\"#7$ 7$\"\"!F1F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Parametric generators 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07rotheracmoodflywithinthosneedcheckwhethervalidalreadisgeneratorloadexpandeightlielieganglebracketgxgygzgsergeneratorliezeroliezeroghgmorevirasovoliepariepargegigjgfgkgsglgngseealsoalsotriangularuildblockexpressionsnumbinteglisteachcorrespondinfodescriptpossiblnewcanlaterusedmaplesessonceallfunctionsymalgoverviewpackagableworkwithadvisablusesuchameassignvalueselsewherduringhenfirstsetdefaultfunctionnospecifithemamongargumentcaseexpectalgebraorderdecidlexicographicaloperatorsmallestetcweightpassgeneratoroptionalthirdargumdefaultarbuiltusertheshavethrownintogeneratorscontainneperintegerparametricmussamelengthpossibleweightwtrecommendusindextheyspecialsimplificatrulestordependvalurecognsimplifsymalgsimplifmwssymalgwtaccordingexamplimplementatvirasovonoteexistancconstructordirectsumfactoralgsymalgidealkacmoodsimpltriangularsomerequirirectsumbefo8(&%&deltaG6$F-,$F*F2F1,&*$)F-\"\" $\"\"\"#F1\"\"'F-#F2FAF1%\"kGF1F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%#&*G6$%\"kG&%\"eG6#%\"iG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% #&*G6$&%\"eG6#%\"iG%\"kG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "simplify((e[1]&*(e[2]&*e[3]))&*(e[-2]&*e[-4]), Virasovo);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"eG6#\"\"!\"#'*%\"kG!$!G" }}}} {SECT 0 {PARA 261 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 " " 0 "" {HYPERLNK 17 "&<" 2 "symalg, &operators" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "char" 2 "symalg, char" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "directsum" 2 "symalg, directsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" }{TEXT 32 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " simple" 2 "symalg, simple" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "store" 2 "symalg, store" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg , Overv:iew" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "triangular" 2 "symalg, \+ triangular" "" }}}}{MARK "8 27 0 0" 25 }{VIEWOPTS 1 1 0 1 1 1803 } \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% #&*G6$&%\"eG6#%\"iG%\"kG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "simplify((e[1]&*(e[2]&*e[3]))&*(e[-2]&*e[-4]), Virasovo);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"eG6#\"\"!\"#'*%\"kG!$!G" }}}} {SECT 0 {PARA 261 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 " " 0 "" {HYPERLNK 17 "&<" 2 "symalg, &operators" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "char" 2 "symalg, char" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "directsum" 2 "symalg, directsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "factoralg" 2 "symalg, factoralg" "" }{TEXT 32 2 ", " }{HYPERLNK 17 "generators" 2 "symalg, generators" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "KacMoody" 2 "symalg, KacMoody" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " simple" 2 "symalg, simple" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "store" 2 "symalg, store" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "symalg" 2 "symalg , Overv:ARA 0 "> " 0 "" {MPLTEXT 1 0 18 "### the generators" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "use(MyNames);" }}{PARA 11 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