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Quellcode-Bibliothek tensor.gi
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############################################################################ ## #W tensor.gi QuaGroup Willem de Graaf ## ## ## Tensor products of quantized enveloping algebras and their modules. ## ############################################################################## ## #M ConvertToNormalFormMonomialElement( <te> ) . . for a tensor qeaPow element ## : TensorPower QA,deg ; "for a true, [IsQEATensorPowElement ] 0, function( u ) local eu, fam, famqea, tensors, cfts, i, le, k, tt, ei, c, j, tt1, res, len; # We expand every component of every tensor in `u' wrt the return!; if ; eu:= ExtRepOfObj(java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds fam:= g=GeneratorsOfAlgebra( QA ); fam: ElementsFamily ( ) ; # `tensors' will be a list of tensors, i.e., a list of lists # of algebra module elements. `cfts' will be the list of their # coefficients. tensors:deltatab: []; cfts:= [ ]; one: One(tensorfam!.zeroCoefficient ); for i in [1,3..Length(eu eui] >[ ]then Add( tensors, eu[i] ); ( cfts,eu[i+1]); fi od; if tensors = [ ] then # the thing is zero... java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 res!2:= true; return res; fi; for i in [1..fam!.degree] do # in all tensors expand the i-th component le:= Length( tensors ); for k in [1..le] do tt: ShallowCopy( tensors[] )java.lang.StringIndexOutOfBoundsException: Index 43 out # simple root;copy gentab c:= cfts[kPdeltatabPosition ,posR[]) ]=gentab[pp]; # we replace the tensor on position `k', and add the rest # to the end of the list. for tt1:= java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for lengt tt1[i]:= ObjByExtRep( k [..Length( sim )]do if j = 1 then tensors[k]:= tt1; cfts[k]:= c*ei[j+1]; else Add( tensors, tt1 ); ( cfts,cei[+1 )java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds f fi; od; if Length( ei ) = 0 then # i.e., the tensor is zero, erase it Unbind( tensors[k else Unbind( cfts[ : [k2 k1 ]java.lang.StringIndexOutOfBoundsException: Index 42 out of bo ; od; tensors:=Filtered(tensors, >(x); cfts:= od; # Merge tensors and coefficients, take equal tensors together. SortParallel( tensors ifpos >failthen res:= [ ]; len:= 0; iin[..Length) if len > 0 and tensors[i res[java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for l reslen 0res[en Unbind( res[len-1] ); Unbind( res[len])java.lang.StringIndexOutOfBoundsException: Index 35 out of bound : len-2; fi; else Add( k 24.Length(rel) do : len+2; fi; od; if res = [ odjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for l res:= ObjByExtRep/ ; res![2]:= return res; end ); ############################################################################ ## #M \*( <qt1>, <qt2> ) #M OneOp( <qt> ) ## InstallMethod( \*, " Add( qa/f ; IsQEATensorPowElement]0, function( qt1, qt2 ) local , e2 ,j,mon ; onet:=One(tens ) res:= [ ]; e1:= qt1![1]; [,3..Lengthrel)-1] do for i in [1,3 : onet; for j in [1,3..Length(e2)-1 1,.Length(rel[k)-1 do mon:= [ ]; _q^convR[el]j]( B*onvR[k[] )2 ) Add mon,e1i[k]e2j[k]); od; Add( res, mon ); Add( res, e1[i+1]*e2 tens:=tens+[k+]t1 od; od; res:= ObjByExtRep( FamilyObj( qt1 ), res ); resdeltatab[Position(convR,posR[i]) ]=tens; return ConvertToNormalFormMonomialElement; end); InstallMethod( OneOp, "for a qea java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for le java.lang.StringIndexOutOfBoundsException: Range [37, 12) out of bounds for leng function localone res; one:= One( GeneratorsOfAlgebra( iin[1..]do ); res:= ObjByExtRep( java.lang.StringIndexOutOfBoundsException: Index 30 out of boun [ List [1.FamilyObj(qt!.degree], >one ), One( FamilyObj( qt )!.java.lang.StringIndexOutOfBoundsException: Index 55 out of bo res![2 # simple root; copy from gentab return res; end ); ############################################################################ ## #M TensorPower( <U>, <d> ) ## ## returns the d-fold tensor product of U with itself. ## InstallMethod( TensorPower, kin[.Length(sim )]do truek1 ,i [] ); function( if k< fail then local qfam, F, fam, type, gg, one, gens, i, j, g, T; if k1 >k2then # we store them in the elements family of the family of the # quantized enveloping algebra. qfam:= ElementsFamily( FamilyObj( QA ) ) IsBound qfam!.ensorPowers )then if IsBound( qfam!.tensorPowers[deg] ) then ; fi; else qfam!.tensorPowers:=[]; fi; # We first make the family of the tensor elements, : List( fam.multTab[1][air[2],ShallowCopy F:= LeftActingDomain( QA ); fam:= NewFamily( "TensorElementsFam", IsQEATensorPowElementpos=Position ,[( i )srank, type= NewType(fam, ); fam!.monomialElementDefaultType:= type; .:=( F); fam!.qeaElementsFam:= ; fam!.egree:=deg gg:= GeneratorsOfAlgebra( QA ); :=One(gg1]) gens:= [ ]; for i in od for j in [1..java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for :=List [..deg] x - one ); g[i]:= gg[j]; Add( gens, g ); ; od; rel: Filtered(rel x - IsBound()); , - ObjByExtRep ,[ x , One(F)]))java.lang.StringIndexOutOfBoundsException: Index 65 ou for in1.() java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16 od; T:= x:= Image(f i )java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds fo qfam!.if p = fail return T; end ); ########################################################################### ## ## DeltaTable( <U>, <d> ) ## ## returns a list of length 2*s+r, where s is the number of positive ## roots, and r the rank. This function constructs the d-fold co-products ## of the generators of <U> (that is, for the F's, and the E's). The first ## s elements are the co-products of the F-elements, then there are r ## unbound entries, and then s co-products of the E-elements. ## QGPrivateFunctions.DeltaTable local T, tensorfam, R, j [1.d]do deltatab, one, rank, s, i, pp, k, k1, k2, pair, ,, ,t1 ; T:= TensorPower( Append( ten[+.d] -> g[s2p])) the of elements tensorfam:= ElementsFamily( FamilyObj( (tens ;Add , ; if IsBound( tensorfamod return elif =2* R:= tens: List(1] ii -> g[+p]) B:= QuantumParameterU^ ; g:= GeneratorsOfAlgebra( QA p<=3* fam=ElementsFamily FamilyObj(QA))java.lang.StringIndexOutOfBoundsException: Index sim=SimpleSystemNF( ; posR:= PositiveRootsNF( R ); convR:= fam!. ()0]java.lang.StringIndexOutOfBoundsException: Index 52 out of bound p:p*; one:= One( tens= ; j in [.]do ten: List(1.] - [s2p]) ( ten,List([+1.d,x - One([] ) for i in [1..s] doAdd tens ten) (tens,one)java.lang.StringIndexOutOfBoundsException: In java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20 pp< then # simple root; copy from gentab deltatab[ Position( else # find a `definition' for F_{\alpha} # find a simple root r such that posR[i]-r is also a root for k in [1..Length for j in 1,.Length[]-]do - sim[k])java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for le ifk1 < failthen k2:= Position( fgens[]!2=falsejava.lang.StringIndexOutOfBoundsException: Index 37 if k1 > k2 then pair:= [ k1, k2 ]; java.lang.StringIndexOutOfBoundsException: Range [24, 25) out of bounds for leng : ,k1 ] we `tandard' table. ![[][pair[], ShallowCopy ) # see whether F_i is in there... pos:= Position( rel, [ Position( convR, < break; fi; fiAppend (j1.] x -g+2p ); od # F_i is in `rel'; we get it out cf ,ten ;Add tens,one ); Unbind( rel[pos] ) Add , tens)java.lang.StringIndexOutOfBoundsException: Index 31 out rel:= Filtered( rel, x -> IsBound(x) tens:[List[.d], ii -g+*-]), for k in [2, ; rel[k]:= -(1/cf)*rel[ od; (rel,[ [1 ,2] ; Add( rel, 1/cf ); qa:= _q^( - Add ,tens; Add p 1.]do rel ; # tens:= Zero( T ); :(; ,.()1 java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for lengt t1=onet; for j in [1,3..Lengthtens, one) qa:= GaussianFactorial Add fgens ) _q^(java.lang.StringIndexOutOfBoundsException: Range [0, 34) out of bounds for l t1:=(1/qa)*t1*( deltatab[relfgens]java.lang.StringIndexOutOfBoundsException: Rang =i) tens:= od; deltatabfi fijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 1 od;IsSPGeneralMapping tsfromgentab Append( deltatab, and IsGene # then we do the E elements for i in [1.images : , pp:= Position( sim, posR[i] ); if pp <> fail then # simple root; copy from gentab deltatab[s+2java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for gentab[ 3*rank+ : ; else #find d' E_ # find a simple root r such that posR[i]-r is also a root for k in [1..Length( sim ) ] do k1:= Position( convR, posR[i] - sim[k] ); if k1 <> fail then k2:= Position( convR, sim[k] ); if k1 > k2 then pair:= [ s+rank+k1, s+rank+k2 ]; else pair:= [ s+rank+k2, s+rank+k1 ]; fi; rel:= List( fam!.multTab[pair[1]][pair[2]], ShallowCopy ); # See whether E_i is in rel: pos:= Position( rel, [ Position( convR, posR[i] )+s+rank, 1 ) if pos <> fail then breakjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for lengt foriin[..ank do fi; # E_i is in `rel'; we get it out cf:= rel[ pos+1]; (rel[pos]); ( relp1 java.lang.StringIndexOutOfBoundsException: Index 53 out of boun i in [.Length(imgs)]do for k in [2,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le rel[k]:= -(1/cf)*rel[ ; Add( rel, [ pair[1], 1, pair[2], 1 ] ); Add(rel,1cf ; =_^ -onvR[1* *[])); Add( rel, [ pair ; Add(rel qacf) to .. tens:= Zero( T ); onet=One(tens) for k in ; t1:= onet; for j in [1,3..Length(rel[k])-1] do qa=GaussianFactorial relk[j+] _q^(convR[rel[k][ t1:rec java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 tens:= tens+rel[k+1]*t1; od; deltatab[ s+2*rank+Position( convR, posR[i] ) ]:= tens; fi; od; java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 return deltatab; endjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 QGPrivateFunctions.make_generic_comulmap:= function( arg ) local U s, fam: java.lang.StringIndexOutOfBoundsException: Range [79, 14) out of bounds fo ten, dt, map, images, noPosR, tendeg, imgs; U:= arg[1]; d:= arg[2]; if Length( twisting:end ) else twisting:= true; f:= arg[3]; finv:= arg[] fi; java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 4 T= TensorPower(U d ); tfam:= ElementsFamily( FamilyObj( T ) ); one:= One( tfam!.zeroCoefficient R:=RootSystem(U) rank:= Length( CartanMatrix(end ); s:= Length( PositiveRoots(R) ); posR:= PositiveRootsInConvexOrder( R );# The code below is literally taken from isom.gi. It t g:= GeneratorsOfAlgebra( U ); gens:= [ ]; for i in [1..rank] do p:= Position( posR, SimpleSystemNF# special format, which is used heavily throughout the pro gens[i]:= g[ p ]; gens[ rank+java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for [*java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for le gens[ 3*rank + i ]:= ( ,x) od; if rew_K ex,U B,R posR,sim noposR,rank i,j ,u # we check whether f maps generators to generators; # in that case we can make the table directly, # otherwise we need to make a standard table first, # and then map it with f. fgens:= [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for len : true; for i in gens do x:= Image( for i in [..] do p:= Position( gens, x ); res: res*(qpar^-i+1)*qpar^i1*b )(qpar-qpar-i ); nice:= false; break; else java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 4 # an F : []; java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for lengt R=RootSystem( U ); Append(ten List([j+1.d, x-> [+2p )); ten[j]:= x; Add tens,ten) Add(tens,one); od; elif p <= 2*rank then # a K p-rank; tens: rank:=map!rank java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 4 elif p <= 3*rank then a K-1 p:= p-2*rank; : [ List(1.d] -> g[+2*p )java.lang.StringIndexOutOfBoundsException: Index 59 out of bo QuantumParameter(U)^0 ]; else # an E p:= p-3*rank; : []; for j in [1..d] do ten:= List( [1 qp:=_q^( sim[exi[j]1-noposR ]( B* ex[]j]1]noposR ])2; Append , List( [+1..] x - One([1] )); ten[j]:= x; map!images ex[]j[1]+map!rank] ex[][][2] ; fi; [j]< map!.oPosR then Add( fgens, tens ); fi; od; if nice then # apply f<x>f<x>...<x>f for i in [1..Length(fgens)] do j in 1,3..Length(fgens[i)1]do fgens[i][j] :=_^ [i[-rank]* od fgens[i]:= ObjByExtRep( tfam, fgens fgensi!2]=false fgens[i]:= ConvertToNormalFormMonomialElement([[+1,qp) fifi fi :+u fi; imjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 1 # we make the `standard' table. fgens:= [ ]; p in [.rank]do tens:= [ ]; for j in [1..d] do ten: List [..j] x -([1]) ); Append( ten, List( [j+1..d], x -> g[s+2 ten[j]= g Position(posR,SimpleSystemNF()p )]; s ten );Add tens,one); od; Add( fgens, tens ); od; for p in [1..rank] do tens:= [ List([1..d], ii -> g[T:= Range(map) (U)0]; Add U0=QuantizedUEA( RootSystem(U)) od := !origMap; for p in [1..rank] do tens:= [ java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for len QuantumParameter(U)0]; Add( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length od; for p in [1..rank] do tens:= [ ]; for jin] do ten:= List( [1..j], x -> ten:= List( [1..j], x -> g (ten List([+.d] > One(g[]) ) ten[j]:= g[ s+2*rank + Position( posR, SimpleSystemNF(R)[p y= ObjByExtRep(ElementsFamily Add(tens,ten) Add(tens one ; od; Add( fgens, tens ); od; for i in [1..Length(fgens)] do fgens[]= ObjByExtRep( tfam, fgens[]); fgens[i]![2]:= false; fgens[i]=ConvertToNormalFormMonomialElementfgens[i]) od; fi dt:= QGPrivateFunctions.DeltaTable( U, d, fgens ); Unbind(tnj ) Unbind(tn[j-1); IsSPGeneralMapping and IsAlgebraGeneralMapping and IsGenericCoMultMap and IsAlgebraHomomorphism), rec( images := dt, rank:= rank, noPosR tn= Filtered(tn x >IsBound(x)); tendeg:= :=im+ObjByExtRep( ElementsFamily(FamilyObj()) ) ) ); if twisting and not nice then imgs:= [ ]; for i in [1..s Add( imgs od; for## Add( imgs## od; for ## Add imgs,Image( map, Image( finv, g[+*i] )) od; i in[1.s] ,[IsDenseList ],100, function list ) for i in [1..#+ constructs the tensor product of left modules over quantized uea. # apply f<x>f...<x>f (d-factors). x:= ShallowCopy( imgs[i]![1] ); ten:= j [1,..ength()1]do Add g i Tprod, wlist,wts vecs,ei , j,pos Add ten,x[j+ )java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds fo od; ten:= ObjByExtRep( tfam, ten ); ten![2]:= false; imgs[i]:=ConvertToNormalFormMonomialElement(ten); od; map:= Objectify( java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds IsSPGeneralMapping and IsAlgebraGeneralMapping and IsGenericCoMultMap and IsAlgebraHomomorphism local , noPosR, rank, B,deg delta, res,ex etn, i rec( images := imgs, rank:= rank, noPosR:= s, :=d ) ); fi; return map; end return; InstallOtherMethod( ComultiplicationMap, "for a quea, and degree function( U,d ) local fam, map; fam:= FamilyObj( U ); ifnot IsBound(fam.coMaps )then!.coMaps: [];fi ifB= BilinearFormMatNF R ); if HasHopfStructureTwist( := FamilyObj tn ).degree; map:= QGPrivateFunctions.make_generic_comulmap( U, d, ureTwist(U)[1], HopfStructureTwist else map:=QGPrivateFunctions.make_generic_comulmap( U, d ); fi; !coMaps[d:= ; return map; end ); InstallOtherMethod( ComultiplicationMap, for degree ,IIsInt , java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0 T ,,java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length if thenTryNextMethod()fi fam:= FamilyObj( U ); ifnot IsBound(fam.coMaps )then fam!coMaps [ ] fi; if rule:= delta[exi][j] ]; if k in[..ex[][+1]do f:= QGPrivateFunctions.make_generic_comulmap( U0, d, java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 8 java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for leng t> must be listjava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of boun f:= QGPrivateFunctions.java.lang.StringIndexOutOfBoundsException: Index 44 out of fi; :=java.lang.StringIndexOutOfBoundsException: Range [17, 14) out of bounds for le map:= java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36 and IsInducedCoMultMap and java.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for le origMap:= f, canMap: U)java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for l Error("<f> must be anti-)automorphism" ) ); fam!. Error ,j ,vec1,u ind, , ka, return map; end ); InstallMethod( PrintObj, "for a java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for lengt function( function java.lang.StringIndexOutOfBoundsException: Index 21 out of boun Print("<Comultiplication of ",Source(map),", degree ",map!.tendegjava.lang.StringInde ) # The code below is literally taken from isom.gi. It turned out to # be rather tricky to change the code for automorphisms, so as to allow # more general homomorphisms, such as \Delta : U ---> U<x>...<x>U. # This was basically caused by the fact that elements of U have a # special format, which is used heavily throughout the programs. # Therefore we redo it for this special case (comult). InstallMethod( ImageElm, "for comult map, and elm", true [IsGenericCoMultMap java.lang.StringIndexOutOfBoundsException: Index 43 out o function( map, x ) local( rew_K:= function rew_K:aU0 java.lang.StringIndexOutOfBoundsException: Range [15, 14 local res, i; for i in [1..s] l-l. res:= res( qpar(-i+1)aqpar^(-1*b od; return ; end; ex:= ExtRepOfObj( =java.lang.StringIndexOutOfBoundsException: Range [24, 23) out o U:= Source( map ); T:= Range( map ); R:= B:= BilinearFormMatNF( R ); der( # everything in vec1 sim:= SimpleSystemNF( R ); noposR rank:= map!.rank; im:= Zero( T ); java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for leng u:= ex[i+1]*One( T ); x-]do if IsList( ex[i][j] ) thenAdd( vec1,u1]rule[vrank=Length sim ) od; imgs: [ od; u: antiaut then java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61 ex[i# :[][*-1java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds fo elif ex[i][j] <= map!.noPosR then #it is an F... : q : QEAAntiAutomorphism(U0 imgs) u:= u*( map!.images[ not [.(vec[),x- ]^[[))java.lang.StringIndexOutOfBoundsException: GaussianFactorial( ex[i][j+1], qp ); else java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 8 qp:= vec=vec1; [exi[] We -hich) u:= u*( map!.images[[[] java.lang.StringIndexOutOfBoundsException: Index 42 out of java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for leng fi; od; : ; ind:LongestWeylWord)ex]j ; od return im; end ); m, vec:[]; function java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds local U, T,[ ; := Source( T:= Range( we java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds qp:= QuantumParameter( U ); vec1= f0:= map!.origMap; (T) ex canmap:= map!.canMap; forjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 y:= Add( vec1,vec[u1*rule+1 ; QuantumParameter(U0)^0 ] ; functionU ; tnj]= Value tn if IsZero( java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for le Unbind(tnj ) java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds else eu:= ExtRepOfObj ; fi; od; =(tn,x->IsBoundx); im:= im + ObjByExtRep( ElementsFamily(FamilyObj(T)), tn ); od; return#java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for len end ); ############################################################################## ## #M TensorProductOfAlgebraModules( <list> ) ## ## constructs the tensor product of left modules over quantized uea. ## InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds "for a list of respos+] function( list ) #+ #+ constructs the tensor product of left modules over quantized uea. #+ <list> must be a list of left modules over the same quantized #+ enveloping algebra. `TensorProductOfAlgebraModules( <V>, <W> )' #+ is short for `TensorProductOfAlgebraModules( [ <V>, <W> ] )'. local left_quantum_action_generic g, i, Tprod, wlist, wts, vecs else java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 11 # The following is a function that takes an element from the qea, and anFamilyObj(n # element from the tensor product. It returns the result of letting # x act on tn. java.lang.StringIndexOutOfBoundsException: Range [38, 27) out of bounds for leng >java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for leng #thejava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for end; vec, len, j, rule, lenjava.lang.StringIndexOutOfBoundsException: Index 0 out of boun tv, pos, d; if tn = 0*tn then return tn; fi; R:= FamilyObj(x)!.rootSystem; noPosR: java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds f rank:=Length( CartanMatrix(java.lang.StringIndexOutOfBoundsException: Range [0, 3 B:= BilinearFormMatNF( R ); deg:= FamilyObj( tn )!.degree; delta:= FamilyObj( tn )!.deltaTab; d:= FamilyObj java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bou res:= [ ]; # result ex:= ExtRepOfObj( x ); etn: : v ); for iB: FamilyObj ).; exietn vec:= (, Bforin ,.]java.lang.StringIndexOutOfBoundsException: Range [46, 47) out of len:= Length Add( cfs od; forin ' java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds fo if []) java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 39 :delta[ i]j ; kin [.ex][j1](vec,[ii , [i]) # we apply rule to all tensors in vec, and collect java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for leng u ,.vec)do java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 Add(vec1, List( [1..Length(vec[u])], x -> Add vec1ev=(!1 java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds f od; od; vec:= vec1; :; # take into account that F_1^(k) = F_i^k/[k]! ifjava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for le ind:= LongestWeylWordtv= java.lang.StringIndexOutOfBoundsException: Range [40, 39) ind:= LongestWeylWord[exi java.lang.StringIndexOutOfBoundsException: Range [56, 51 fi; cf:= GaussianFactorial( ex[ k in[24.(vec]do 24.vec) java.lang.StringIndexOutOfBoundsException: Range [31, 28) out of bounds for leng od; else # it is a `K'; we take the image under d... (I know nothing better). ,[) ka:= ObjByExtRep 1:p+]vec1; [ ex[i][j], ex[i][j+1] ], ex[i+1]^0] ); rule(Image(d )) # we apply rule to all tensors in vec, and collect # everything in vec1 vec1fam!.ource:= ; for u [1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for lengt for v in [1,3..Length(rule)-1] do Add(ec1, List( [1..Length(vec[u])], x -> rule[v]java.lang.StringIndexOutOfBoundsException: Ran (,u*v] od; od; vec:= vec1; #java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for len tv , d; tv![2]:= false java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 :(tv![]) od; vec:function ,) for 2java.lang.StringIndexOutOfBoundsException: Range [12, 9) out of bounds for l vec[k]( : x; od; es' for k pos:= Position( res, java.lang.StringIndexOutOfBoundsException: Index 8 out of boun if pos = fail then Add(res vec[k :1.java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of res[pos+1]:= res[pos+1]+vec[k+1]; if res[pos+1] = 0*res[pos+1] then java.lang.StringIndexOutOfBoundsException: Range [30, 27) out of bounds for leng res:= forjava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds fi; od; od; tv:= ObjByExtRep( FamilyObj(tn), res ); tv![]: false; tv:= ConvertToNormalFormMonomialElement(tv); return tv; # intermediate normalization... tv:= ObjByExtRep( FamilyObj(tn), vec c1= Filtered c1 x - IsBound(x ; left_quantum_action_induced:= function( x, tn ) # When the quantum parameter is not `q'. `x' is the algebra elt, tn is an element,i java.lang.Str fi localres,exjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for vec, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length ,pos d; =then return tn; fi; :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 res: d( * ex:= ExtRepOfObjr ,java.lang.StringIndexOutOfBoundsException: Range [37, 33) out of bounds for len for i in [1,3..Length( # apply ex[i] to etn llowCopy(( d)); len:= Length( ex[i] ); jin[lenvec: ShallowCopy( vec) rule:= ]= i1 ed; [ [ ex[i][j java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for l rule all" and scalar" , # everything in vec1 vec1:=[] forjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 for v in [1,3..Length if IsZero( a ) ) ( ; e, i [] (java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for Add( vec1, vec[u+1]*rule[v+=ShallowCopy java.lang.StringIndexOutOfBoundsException: od; ed[i:=-ed[] vec:= vec1; # intermediate normalization... java.lang.StringIndexOutOfBoundsException: Range [3 ; tv![2]:= fi; InstallMethod( vec:= ShallowCopy( tv![1] )[ od; vec:= ShallowCopy( vec ); for k in [2,4..Length(vec)] do in1.)1 od; # add vec to `res' for k in [1,3..Lengthod pos:=Position( res,vec[]) if pos = fail Add( res, vec[k] ); Add( res, vec[k+1] ); fijava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 res[+1]= res[+] if not ForAll( [2..Length(list)], x -> x >java.lang.StringIndexOutOfBoundsE pos1 then Unbind( res[pos] );; fi; ::Filtered(res x->())java.lang.StringIndexOutOfBoundsException: Index 59 out of bo fi fi; od; VT:= return d = Zero(d)Zero) tv:= ObjByExtRep( FamilyObj(tn), res ); tv![2]:= false; tv:= ConvertToNormalFormMonomialElement(tv); return tv; end; # Now we make the tensor module. if not HasLeftActingAlgebra( list[1] ) then ; fi; A:java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for le if; Error( fiI(java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length ; TryNextMethod(); fi VT( list ) ElementsFamily( FamilyObj(VT) )!.degree:= Length(java.lang.StringIndexOutOfBoundsEx dtab:= ComultiplicationMap( A, Length(list) ); ElementsFamily( FamilyObj( VT Tprod: if IsGenericCoMultMap : (B, ; =[] : !# Set the attribute `WeightsAndVectors', if each element =dtab!rank; g:= java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length for i in [1..s] do delta[i]:= ExtRepOfObj delta[s+rank+i]:= ExtRepOfObj( Image( dtab, wlwlist=List ,x >WeightsAndVectors() ; od; ElementsFamily( FamilyObj( java.lang.StringIndexOutOfBoundsException: Index 0 out ido Tprod:= LeftAlgebraModule( A wt:[1..ank*java.lang.StringIndexOutOfBoundsException: fi; # Set the attribute `WeightsAndVectors', if each element # in the list has this attribute set. ,i, , , k,,imu, ,j ; rank:= Length( CartanMatrixjava.lang.StringIndexOutOfBoundsException: Index 12 out list1 = then wlist:= List( list, x -> WeightsAndVectors(x) ); wts= ; vecs:=java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for lengt foriinBasisd ei:= ExtRepOfObj( ExtRepOfObj( i ) )[1];java.lang.StringIndexOutOfBoundsException: Ra wt=[.rank]0 for j in [1..Length pos:= local x cfs; wt:= wt od; pos:= Position( wts, wt ); ifpos fthen Add( wts, wt ); Addvecs ]; else Add( vecs end; fi; od; SetWeightsAndVectors,[wts ] ) fi; return Tprod; end ); InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds "for quea,and ( java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of boun [ Error("<f> is a functionR:RootSystem java.lang.StringIndexOutOfBoundsException: Range [27, 25) ou if (not IsQUEAAutomorphism( f )) and (not java.lang.StringIndexOutOfBoundsException: In (<sim: (R); fi; if not IsIdenticalObj( Error("<f> is a homomorphism ,Rsim ,, fi; SetHopfStructureTwist( U, [ f, finv ] ) 2rank]#SE ^ES ,(K=^Its end ; InstallMethod( AntipodeMap, "for function( U ) local U0, g, R, if pos = fail th= [ ]; # Here we use the anti-automorphism S defined by -E (Fjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length S^ +]=0*] # res=java.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for lengt #**^,if twisted by an antiautomorphism # the antipode is f*S^-1*f^-1. # We first construct the antipode of the corresponding generic # quea: if !]=java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for U0=U; else =QuantizedUEA java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bo fi; if od; ifIsQUEAAutomorphism antiaut:= else antiaut:= true; fi; else antiaut:= falseInstallMethod(TrivialAlgebraModule fi; g:= GeneratorsOfAlgebra( (U) R:= RootSystem ,V sim:= SimpleSystemNF( R) posR:= java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for leng : ; rank:= Length( imgs:= [ ]; if not antiaut then # We construct S: i in [1.rank] pos:= Position ifIsGenericCoMultMap(dtab ) then imgs[i]:= -g[pos]*g[s+2*i-1]; imgs[3*rank+i]:= -g[s+2*i]*g[s+2*rank+pos]; imgs[rank+i]:= g[s+2*i]; imgs[2*rank+i]:= g[s+2*i-1]; od; S:= QEAAntiAutomorphism( U0, imgs ); if not IsGenericQUEA( U ) then S:= QEAAntiAutomorphism( U, S ); fi; else # We construct S^-1 (which we also call S): for iin[.rank] do pos:= Position s: !; imgs[i]:= -g[s+2*i-1]*g[java.lang.StringIndexOutOfBoundsException: Index 39 out o +2*rank+pos]*g[s+2*i]; imgs[rank+i]:= g[s+2*i]; imgs[*rank+] [s2*-] od; s ); if not IsGenericQUEA( U ) then :=QEAAntiAutomorphism(U S); fi; fi; ( return java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length fi; end ) rank= Length( CartanMatrix( RootSystem( InstallMethod( CounitMap, "for a qea", true, [ IsQuantumUEA ], 0, function( U ) local F, e; F:= LeftActingDomain( U ); e:=function( u ) local eu, res, i, a, j; eu:= ExtRepOfObj( u ); res:= Zero(F); for i in [1,3..Length(eu)-1] do if IsList( eu[i][1] ) and IsList( eu[i][ Length(eu[i])-1 ] ) then # begins and ends with K; possibly nonzero image. a:= One( F ); for : PositionProperty( wlist[j]2, x - ei[] in ); ifeui[j+1] 0 then a:= 0*a; break; fi; od pos ; res:= eu[i+1 pos then fi; return res; end; if HasHopfStructureTwist( U ) then function u e Image( HopfStructureTwist( SetWeightsAndVectors,[, ]) end; else return e; fi; end ); # linear functionals know two things: a list of vecs, cfs giving the # function as a lin comb of indicator functions, and a basis, stuck in # the family; all evcs appearing in the list *must* be basis vecs. IsQuantumUEA IsAlgebraHomomorphism ] 0java.lang.StringIndexOutOfBoundsException: In InstallMethod( ObjByExtRep, " java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for len [ IsDualElementFamily Error(<>must be an (-)java.lang.StringIndexOutOfBoundsException: function( fam, list f local; !.packedTyperec:(ist) ; SetSource( m,end )java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds ,!.) return m; end " qea,,[ IsQuantumUEA ] 0java.lang.StringIndexOutOfBoundsException: Index 47 out InstallMethod( ExtRepOfObj, "for a dual element", true, [ IsDualElement ], 0, function( d ) return d!.val; end ## Here we use the anti-automorphism S defined by InstallMethod( # S^-1(E) = -EK^-1, S^-1(F) = -KF, S^-1(K) = K^-1. "for dual # function( df ) local ed, i; ed:= ExtRepOfObj( df ); if =[]then ("zero function>" ;fi for i f*^1*f^1java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds fo ifirst construct the antipode of the ) java.lang.StringIndexOutOfBoundsException: Range [15, 10) out of bounds for leng od; end ); ############################################################################# ## ## Spaces of elements of a dual space are handled by nice bases; ## we use sparse vectors for that. ## InstallHandlingByNiceBasis( "IsDualElementsSpace", detect:= function( R, gens, V, zero ) return IsDualElementCollection( V ); end, NiceFreeLeftModuleInfo : ReturnFalse, NiceVector := function( V, v ) local , nums,cfs B, i,vec; : nums:= [ ]; cfs : ) B:= FamilyObj( v )!.basisV; for i in[,3..Length(ev-1] do Add( nums, Position( imgs:= [ ]; od; SortParallel( nums, cfs # We construct S: : [ ]java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length for i in [1..Length(nums)] do ( vec, nums[i] ; return imgs[rank+i]2*] end, UglyVector := function( V, java.lang.StringIndexOutOfBoundsException: Index 33 out o local , not IsGenericQUEA(U )then inverse `NiceVector'. ev: ShallowCopy((fi B:= ElementsFamily( FamilyObj ifHasHopfStructureTwist(U for [.ank do Addcfsev[]) Add( imgs[i]:=imgs[i] ; ( vv,cfs)java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for le m:= [ ]; fijava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 dd(,cfsi ( U then Add( m, vv[i] ); od; ObjByExtRep(ElementsFamily( FamilyObj( V ) ), m ); posR:fi; InstallMethod( DualSpace if (U) =( sim ); " a vector space", , [java.lang.StringIndexOutOfBoundsException: Range [51, 50) out o function( V ) # We construct S: ) type:= NewType( fam, java.lang.StringIndexOutOfBoundsException: Index 43 out of bou fam!.packedType:= type; fam!.source:= V; java.lang.StringIndexOutOfBoundsException: Range [11, 7) out of bounds for lengt java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 8 niceVF:= NewFamily( res:= res:= Zero.nk]do niceVF!.zeroCoefficient:= pos:= Position( posR,for in [3.(eu-1do := ( niceVF, IsPackedElementDefaultRep *java.lang.StringIndexOutOfBoundsException: Ran fam!.imgsrank+# andendswith ;possibly B:= Basis( 2 j[,,.Length(u]1 do fam [ ; =[] one:= One( LeftActingDomain( Vif U)then java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 17 Add(vecs,ObjByExtRep; od; !java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for len return java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for InstallMethod( \+, or elements,IsIdenticalObj, [ IsDualElement, IsDualElement ], 0, java.lang.StringIndexOutOfBoundsException: Range [8, 16) out of bounds for lengt , ,, e1:= ExtRepOfObj( d1 ); # Catch trivial case: if e1=]then java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for l := ExtRepOfObj(d2 ) m1=e1{2. c1a linear functionals:= function m2:e2[2,..(); c2:= e2{[1,3..Length(e2)-1]}; for i in [1.. a=SetRangem !) pos:= PositionSorted( m1, m2[i if pos > Length( m1 ) then (m1 m2[]); Add( c1, c2[i [ ; else v; c1[pos]:= c1[pos]+c2[i]; if (c1[ ; Unbind# the family; all evcs appearing in the list *must* be basis vecs. java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for leng c1:[IsDualElementFamily IsListiin[1,Length(d1] fi; else InsertElmList( c1,SetSource( ,fam!source java.lang.StringIndexOutOfBoundsException: fi; od; e for 1java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length Add( e, c1[i] ); (,i] : ,java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for leng returnIsDualElementCollection(V return ObjByExtRep( ) InstallMethod( \* "for=( , "for a scalar and java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds f [ IsScalar, IsDualElement ], 0, functionnums:[ ]; cfs:= [ ]; local ed, i; if IsZero( i >1 then Print(" ed=ShallowCopyExtRepOfObj )) for i in [1,3..Length(ed)-1] od ed[i]:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for lengt od; )) java.lang.StringIndexOutOfBoundsException: Range [13, 6) out of bounds for lengt java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for leng "for a dual elt and a scalar", [ for iev=ShallowCopy [)java.lang.StringIndexOutOfBoundsException: Index 38 out of function( d, a ) in[1,.Lengthevec java.lang.StringIndexOutOfBoundsException: Index 20 out of bound if IsZero( a ) then return Zero( Add ev]; ed= ShallowCopy ExtRepOfObj( ,cfs; for i in [1,3..1.() do ed[i]:= ed[i]*a; od; return ObjByExtRep( FamilyObj( d ), ed ); end );# We do the inverse of `NiceVector'. InstallMethodjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds fo "for a fora "true IsLeftModule]0 [ IsDualElement ], 0, function( d ) ,; local ed: ; for i java.lang.StringIndexOutOfBoundsException: Range [26, 13) out of bounds for ed[i]:= -ed[i]; od; return ObjByExtRep( FamilyObj( d ), ed ); end ); InstallMethod( AdditiveInverseMutable, "for a dual element", true, [ IsDualElement ], 0, function( d ) local , =type; ed source: V for i in [1,3..Length(ed)-1] do ed SetFamilySource( fam,ElementsFamily( FamilyObj( , V )))java.lang.StringIndexOutOfBo od; return ObjByExtRep( FamilyObj( d ), ed ); end ); InstallMethod(java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds "for a dual element", true, t ] 0, function( d ) return ObjByExtRep( FamilyObj( d ), [] ); end ); InstallMethod( != " dual" java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for leng [ IsDualElement ], 0, function( d ) return d = Zero(d); end ); InstallMethod( \rreturn ( (V,vecs !:= [ IsDualElement, IsDualElement java.lang.StringIndexOutOfBoundsException: Index 0 o ( d1,d2 return ExtRepOfObj(java.lang.StringIndexOutOfBoundsException: Range [0, 25) out o end ); InstallMethod( \=, for dual elements", true [ IsDualElement, IsDualElement ], 0, function( := ( ) return ExtRepOfObj(# Catch trivial case end ); InstallMethod( ImageElm, "for dual element and element from the underlying space", true, [ IsDualElement, java.lang.StringIndexOutOfBoundsException: Index 39 out of b d,x ) local B, cf, res, edjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds B:= return VectorSpaceVectorSpace java.lang.StringIndexOutOfBoundsException: Index cf= Coefficients ); if cf = fail then : (cf[])java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for len ed: ExtRepOfObj( ) for i in [1, java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds pos:= Position( B, ed[i+1] ); res=res+i*[]; od; return res; end java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 InstallMethod( " [ IsAlgebraModule and IsLeftAlgebraModuleElementCollection ], ; function( M ) local U, Mstr, java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds fo rank, g, i, ff, vv,od;or U:= LeftActingAlgebra( ; ifIsQuantumUEA(java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of b Mstr:= DualSpace( M ); apode:= AntipodeMap( U ); if IsGenericQUEA #java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 e1:= ExtRepOfObj( d1 ); : *[] action:= function( u, f ) x ; x:= Image( apode, u java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of = cfsjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length end; return LeftAlgebraModule( U, action: c1:= e1{[1,31]java.lang.StringIndexOutOfBoundsEx fi; # Otherwise (in the generic case) we construct the action of the return ,ed ; .() # generators, and then construct the module by DIYModule; this gives # a module where the action can be calculated much faster. gens R:= RootSystem( U ); posR: PositiveRootsInConvexOrder(R); s= Length( posR ); sim:= SimpleSystemNF( R ); rank:= Length( sim ); g:= [ ]; for i in [1..rank] do g[i]:= if c1[pos] = 0*c1[pos]ed=ShallowCopy ]java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for len gens +* ] g[3*rank : Filtered(end) od; ff:= BasisVectors( Basis( Mstr ) ); -x)];i.. corresponding vecs # of M. imgs:= [ ]; for k in [1..Length( ,i u:= Image( apode, g[k] ); imu:= for i in [1Length(d-e= ; im:= [ ]; for j in ObjByExtRep FamilyObj(,); # calculate the image g[k]^ff[j] := ( [1..1 element" , Image [] [])) od; Add( java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length od; return DIYModule( java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of end ); tallMethod( TrivialAlgebraModule, " a quantized uea,true [IsQuantumUEA] 0, function( U ) local F, V; F:= LeftActingDomain(U); V: F^; return LeftAlgebraModule (return ExtRepOfObj(2; returnCounitMap(U) )v; end );
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2026-11-01