SSL tensor.gi   Interaktion und
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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
##
InstallMethod( ConvertToNormalFormMonomialElement,
        "for a tensor qeaPow element",
        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 
    # PBW-type basis.

    if u![2] then return u; fi;

    eu:= ExtRepOfObj( u );
    fam:= FamilyObj( u );
    famqea:= fam!.qeaElementsFam;

    # `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:= [ ];
    cfts:= [ ];
    for i in [1,3..Length(eu)-1] do
        if eu[i] <> [ ] then #i.e., if it is not the zero tensor
            Add( tensors, eu[i] );
            Add( cfts, eu[i+1] );
        fi;

    od;
    if tensors = [ ] then
        # the thing is zero...
        res:= ObjByExtRep( fam, [ [], fam!.zeroCoefficient ] );
        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[k] );
            
            ei:= tensors[k][i]![1];
            c:= cfts[k];

            # we replace the tensor on position `k', and add the rest
            # to the end of the list.

            for j in [1,3..Length(ei)-1] do
                tt1:#
                tt1[i]:= ObjByExtRep( famqea, [ ei[j], ei[##  Spaces of elements of a dual space are handled by nice bases;
             
                ##  we use sparse vectors for that.
                    tensors[k]:= tt1;
                    cfts[k]:= c*ei[j+1];
                              
                    Add( tensors, tt1 );
                        := function(R, gens , zero java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
                fi;NiceVector :=function(V,vjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
            od;
        
            if Length( ei ) = 0  then
                # i.e., the tensor is zero, erase it
                Unbind( tensors[k] );
                Unbind( cfts          :  
            fi;

        od;
        s: Filtered(tensors,  - (x))
        cfts:= Filtered( cfts, x -> IsBound( x ) );
    od;

    # Merge tensors and coefficients, take equal tensors together.
    SortParallel( tensors, cfts );
[ ;
    len:= 0;
    for i in [1..Length(          for i in [1,3..Length)-1]do
        if len > 0 and tensors[i] = res[len-1] then
            [len]:= res[]cfts[;
            if res[len] = 0*res[len] then
                UnbindAdd cfs, evi]);
                Unbind( res[len] );
                len:= len-2;
            fi;
        else
            Add( res, tensors[i] );
            Add( res, cfts[i] );
            len:= len+2;
        fi;
    od;
    if          (nums cfs )

    res:= ObjByExtRep( fam, res );
    res![2]:= vec=[]
    return res;

end );

############################################################################
##
#M  \*( <qt1>, <qt2> )
#M  OneOp( <qt> )
##
InstallMethod( \*,
        "for qea tensor power elements",
        IsIdenticalObj, [ IsQEATensorPowElement, IsQEATensorPowElement ], 0,
        (qt1,qt2 )
    
    local   res,  e1,  e2,  i,  j,  mon,  k;
    
    res:= [ ];
    e1:=od;
    for i in [1,3..Length(e1)-1] do
        for j in [1,3..Length(e2)-1] do
            mon:= [ ];
            for k in [1..FamilyObj(qt1)!.degree] do
                Add( mon, e1[i][k]*e2[j][k] );
            od;
            Add( res, mon );
            Add(,e1[+]e2[+])java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        od;
    od;

    res:= java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 10
    res![2]:= false;
    return java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 0
    
end);

InstallMethod( OneOp,
        "for a qea tensor Power element",
        true             ev  ,cfs  ,  i  m;
        function( qt )        
    
    local   one,  res;    
    
    one:= One( 
      GeneratorsOfAlgebra( FamilyObj( qt )!.qeaElementsFam!.qAlgebra )[1]
               );
    
    res:= ObjByExtRep( FamilyObj( qt ), 
                  [ List( [1..FamilyObj(qt)!.degree], x -> one ), 
                   One( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    res![2]:= true;
    return res;

end)

############################################################################
##
#M  TensorPower( <U>, <d> )
##
##  returns the d-fold tensor product of U with itself.
##
InstallMethod( TensorPower,
        "for a quantized enveloping algebra and B:= ElementsFamily( FamilyObj( V ) )!.basisV
        true, [ IsQuantumUEA)1] 
        function( QA, deg               ( cfs,ev[+1 )

    local   qfam,  F,  fam,  type,  ggAdd  B evi  ;
    
    # In order to avoid constructing the same tensor powers twice
    # we store them in the elements family of the family of the 
    # quantized enveloping algebra.
    
    qfam:= ElementsFamily( FamilyObj( QA ) );
    if IsBound( qfam!.tensorPowers ) then
        if IsBound( qfamjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
            return qfam!.tensorPowers[deg];
        fi;
    lse
        qfam!.tensorPowers:=[];
    fi;
    
    # We first make the family of the tensor elements, 

    F:= LeftActingDomain( QA );
    
    fam:= NewFamily( "Add( m, cfs[] )
    type:= NewType( fam, IsMonomialElementRep );
    !monomialElementDefaultType=type;
    fam!.zeroCoefficient:= Zero( F );
    fam!.qeaElementsFam:= ElementsFamily( FamilyObj( QA           od;
    fam!.degree:= deg;
    
    gg:= GeneratorsOfAlgebra( QA );
    one:= One( gg[1] );
    gens:= [ ];
    for i in [1..deg] do
        for          return   (FamilyObj(  )java.lang.StringIndexOutOfBoundsException: Range [65, 27) out of bounds for length 65
            g:= java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 19
            [i]:=gg];  
            Add( gens, g java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
        od;
    od;
    
    Sort( gens );

    gens:= List( 
    for i in [1..Length(gens)] do
        [i![]:=true;
    od;

    T:= AlgebraByGenerators( F, gens od
    SetOne( T, gens[1]^0 );
    qfam!tensorPowersdeg: T
    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:= function( QA, end )
    
    local   T,  tensorfamjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            deltatab,  one,  rank,  s,  i,  pp,  k,  k1,  InstallMethod( \,
            rel,  pos,  cf,  qa,  tens,  onet,  t1,  j;

    T:= TensorPower( QA, deg "for twodual ", IsIdenticalObj,
    
    # We store the table in the family of tensor elements:
    tensorfam:= ElementsFamily(         [ IsDualElement, IsDualElement  IsDualElement ], 0,
    if IsBound( tensorfam!.deltaTable ) then
        tensorfam.eltaTable
    fi;
    
    R:= RootSystem##
    B:= BilinearFormMatNF( R );
##############################################################################
    fam:= ElementsFamily( FamilyObj( QA ) )####    local     ,         c2  ,  pos  
   = SimpleSystemNF R);
    
    posR:= PositiveRootsNF( R );
    convR:= fam!.        true     =  d1 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    deltatab:= [ ];
    one:= One( tensorfam!.zeroCoefficient )    # Catch trivial case:

    rank:= java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    s    # PBW-type basis.
    
# first we do the  elements    fam: FamilyObj(u)
    
    for i in [1..s] do
        
        pp:= Position( sim, posR[i] );    # of algebra module elements. `cfts' will be the list of their
        if pp < fail then
            # simple root; copy from gentab=[if[i >[]then#i.e., if it is not the zero tensor
            Add ,eu[1 ;


        else
            # find a `definition' for F_{\alpha}

            # find a simple root r such that posR[i]-r is also a root
forkin .Length   java.lang.StringIndexOutOfBoundsException: Range [43, 44) out of bounds for length 43
                 
                if > 
                    k2:= Position( convR, sim[k] );
                    if  >java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
                        pair=[k1 k2 ]
                    else
                        pair:= [ k2, k1 ];
fijava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
rel   Add
                    
                    # see whether F_i is in there...
                    pos:=             cfts[k];
                    if pos <> fail then
                        break;
                    fijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                    
                fi;c1pos:llowCopy )
            od;

            # F_i is in `rel'; we get it out                                 j  then
            cf:= rel[ pos+1];
             rel[pos])Unbind(rel+1] ;
                else                    
        
            for k in [2,4..                    Unbind( m1[posm1[Addcfts *eij+1 ;
                [k]: -1cf*rel[]java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
            java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
                
Add ,[pair[1
            Add( rel, 1/cf );
                
            qa:=  _q^( -convRUnbind(java.lang.StringIndexOutOfBoundsException: Range [24, 27) out of bounds for length 12
Add rel,[pair],1,pair[1,1 ]);
            Add( rel, -qa/cf );
            
             Now apply delta to the elements of rel, and add...

           tens:  T)
            onet:=One tens );
            
            for k in [1,3..Length(rel)-1] do
                : onet;
                java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
qa= []j1,
                         _q^(convR[rel##
                    
                od" qea tensor power",
                : rel]t1;
            ;
            
            deltatab[ Position( convR,java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

            
        
    od;

    # Add K-elements from gentab:   in[,3.Length(1)1]do
    Append(         for j in [1.Length(e2)-1] do
            mon: [];
    # then  we do the E elements
    
    for i in [1..s] do
        
        pp:            odjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
        then
            ;copy fromgentab
            
            deltatab[s+*rank+Position( convR,posR[i]        IsScalar,0
                                gentab[ 3*rank+p]java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

        else
            # find a `definition' for E_{\alpha}

            # find a simple root r such that posR[i]-r is also a root
forkin[[.sim)] do
k1=Position( convRconvR posR[i]- sim    ;
                if k1 <> fail then
                   (FamilyObj  )!java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 63
                    
                    if k1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                        pair:= [ s+rank+                        
                   java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
ank+, s+ank+]
                    function QA,deg)
            
                    rel:= List( fam!.java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
                    # See whether E_i is in rel:
                    pos 
                                  ]);
                    if pos <> fail then
                        ;
                    ;
                fi;
            ;
            
            
            # E_i is in `rel'; we get it out
            :=rel pos+1] llowCopy ExtRepOfObj(   ))
            Unbind( rel[pos] ); Unbind( rel[pos+1] );
            rel     We first make the family of the tensor elements, 
        
            for k in [2,4..java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 4
                rel[]:=-(1/f*rel[k] returnObjByExtRep(FamilyObj( d ), ed );
            fam.monomialElementDefaultType:=type;
                
            ( rel,[pair1, 1end)
            , 1/f)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            qa:=  _q    
            Add(rel pair2,1,pair[1], 1 ] );
            Add( rel, -java.lang.StringIndexOutOfBoundsException: Range [0, 25) out of bounds for length 23
            
             delta  rel.

            tens:= Zero( )
    =One();
            
            for k in[,3.Lengthrel)1] do
                t1:= onet;
 forjin[1,..ength(elk)-1]do
                    qa:= GaussianFactorial( rel od
        (gens ;
                    t1:    := List(gens x >    ed 
   od;
                tens:=  
            od;

           deltatab[s2rankPosition(convR,posR[]) ]llowCopy ExtRepOfObj( );

        fijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        
       od;
   
    tensorfam!.deltaTable:= deltatab;
    
    return deltatab;
            
end;

.make_generic_comulmap#


    ##  returns a list of length 2*s+r, where s is the number of positive
            s,  posR,  g,  gens,  i,  p,  fgens,  nice,  x,end ;
            ten

    U:= arg[1];
    d:= arg[2];
    if Length( arg ) = 2 then
        twisting:= false;
    else
        twisting:= true;
        f:= arg[3];
        finv:= arg[4];
    fi;
    
    T:= TensorPower( U, d        [IsDualElement ] 0,
    
    tfam:= ElementsFamily( FamilyObj(        function( d java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    one:= One( tfam!.zeroCoefficient )       ed,i

    R:= java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    rank:= Length( CartanMatrix(R) );
    s:  ())java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
       : PositiveRootsInConvexOrder  )
    
    g:= GeneratorsOfAlgebra( Ued: ed[]
    gens:= [ ];
od;
        p:= Position( posR, SimpleSystemNF( R )[i] );
        gens[i]:= g[ p ];
        gens[rank+]=g  +2i-1];
        gens[ 2*rank + i ]:= g[ s+2*i ];
        gens[ 3*rank + i ]:= g[ s+2*rank + p ];
    od;
        
    if twisting then
        # we check whether f maps generators to generators;
        
        # otherwise we need to make a standard table first,
        # and then map it with f.
        
        fgens=  [];
        nice:= true;
        for i in gens do
            
            x:= Image( f, i );
            p:= Position( gens, x );
            if p = fail then
  
                nice:= false;
                break;
            else
                
                if p <= rank then
    return ObjByExtRep( FamilyObj( d ) [ )
                    tens:= [ ];
                    for j in [1..d] do
                        ten:=List([.j] x >One([1))
                        Append( ten, List( [j+1..d],  x -> g[s+2*p] ) );
                        ten[j]:= x;
                        
                    od;
                elif p <= 2*rank then        "for dual element", true,
                    # a K
                    p:= p-rank;
                    tens:= [ List([1..d], ii -> g[s+2*p-1
                            QuantumParameter(U)^0 ];
                elifp< 3rank then
                    # a K^-1
                    p:= p-2*rank;
                    tens:= [ List([1..d], ii -> g[s+2*p] ),
                            QuantumParameter(UInstallMethod(< 
                else
                    #
                    p:= p-3*rank;
                    tens:= #
                    ##
                        ten##
( [j+1..d], x->Oneg[1) ) );
                        ten[j]:= x;
                        Add( tens, ten ); Add( tens,return d1  d2)
                    od;
                fi
                    
                (fgens  ;
            fi;
        od
        
        if nicetrue,   ,0,
            # apply f<x>f<x>...<x>f
            for i in [1..Length(fgens)] do
                for j in [1,3..Length(       B cf, res, ,i  pos;
        :res=+[*pos]
                ;
                         [i:ObjByExtRep( fgensi  )
                fgens[][2: false;
                java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            od;
 
    
   
if nottwistingnot then
        # we make the `standard' table.
        fgens: [];
        java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
            tens:tt1=( tt );
             j []
if   then
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
                if res[len *[len 
             ens  ten); (tens,one ;
            ;
            Add( fgens,             fi;
odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
tens=[List([..],ii->[on= function(u )
                    res=ObjByExtRep( fam, res );
            Add( fgens, tens     res![]:= true;
             od;
        for p in [1..rank
            tens:= #M  \*( <qt1>, <qt2> )
                     QuantumParameterU^ ;
            Add fgens, tens );
        od;
        for p in ( qt1, qt2 )
            tens:= [ ];
            for jin[.d]do    
                :=List([1..j] x > g+2p1] ;
                  List([+.d,x >One([[])))java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
                     i returncfs*(f)!basisVdual;
                Add( tens, ten ); Add( tens, one );
            od;            
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4


        =;
                : sim 
i!] falsejava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
            IsBound.[  java.lang.StringIndexOutOfBoundsException: Range [50, 51) out of bounds for length 50
        od;
    fi;
    
   java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 54

map:= Objectify(TypeOfDefaultGeneralMapping        [i: s+ ;
#
                  and: (QA);
                  and 
                  type=NewType fam, IsMonomialElementRep )
                  rec(
                        :=dt
                      rank:= rank,
                      :=s
                                        tendeg=d
                      ) )bgg=(QA ;
    
      notthen

        imgs:for=ff>(x[2;# i.e., corresponding basis vecs
        i                                             .
            Add , ,Image    : 
g]= j]java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  1.g]do
      (imgs, (map,(             Add( imgs, Image( map, Image( finv
        od
        fori in 1.rank 
    ( imgs,Image(map Image(finvimu=List(vv  >u^ ;
        od;
        od
             T=AlgebraByGenerators(F gens )
        ;

         .()]do
                    for

            ##  DeltaTable( <U>, <d> )
            ##  returns a list of length 2*s+r, where s is the number of positive
            for j in [1,3..Length(x            U that ##  s elements are the co-products of the F-elements, then there are r
                ##  of the generators of <U> (that is, for the F's, and the E's). The first
                Add##
            odjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
             ObjByExtRep(,ten );
            ten![2            deltatab,one  rank,s  , pp,k  k1, k2,pair,java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

            imgs
        od
        
        map=Objectify( TypeOfDefaultGeneralMapping( U, T,
                      IsSPGeneralMapping
                      and IsAlgebraGeneralMapping
                      and IsGenericCoMultMap    return (U Mstr,imgs);
                      and IsAlgebraHomomorphism),
                      rec(
                          imagesfi
                          : rank,
                         :    
                          tendeg:= d
                           );
   fi
returnmapjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
end;


      ;
"fora quea, degree", true, [IsGenericQUEA,IsInt], 0,
        function
    
    local   fam,  map;

fam=FamilyObj  ;
    
    if notforiin[.s]do
     IsBound fam!.oMapsList !multTabpair1][air2] ShallowCopy;

    if                    =Position rel,[Position(convR posR[]) 1 ] ;
        map:= QGPrivateFunctions.make_generic_comulmap( U,break;
                            HopfStructureTwist( U )[1], 
                            HopfStructureTwist( U )[2] );
    else
        map:= QGPrivateFunctionsod;
          # F_i is in `rel'; we get it out
    
fam!c[]= map;
    rel=Filtered( rel,x ->IsBoundx );

end);


InstallOtherMethod( ComultiplicationMaprelk: (1cf*rel[];
        "            
        (U  java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    
    local   T,  fam,  U0, f, map;

    if IsGenericQUEA( U ) then TryNextMethod(Add ,-/cf)
                # Now apply delta to the elements of rel, and add...
            : 
    ifnotIsBound    !coMaps= ;fijava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
IsBoundfam.d)then !d;fi

    U0:=t1=onetjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    if HasHopfStructureTwist( U ) then
       : .( ,d,
                            HopfStructureTwist( U ) _q^convR[elk[j]*(B*[rel[][] /2));
                            HopfStructureTwist U [].origMap );
    else
        f:= QGPrivateFunctions.make_generic_comulmap( U0, d );


    T:=           [Position ,posR[])]=tensjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
    map:     Add K-elements from gentab: 
                      (,gentab[1.*] ;
and
                  and IsInducedCoMultMap
                  and IsAlgebraHomomorphism   [.Length(   
                  (
                      origMap:= f,
                      canMap:= CanonicalMapping( U ),
                      tendeg= d
                      ) );    
    
    fam!.coMaps[d]:= map;
                        if

end ) ;



InstallMethod( PrintObj,
        "for a java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 15
        function java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    
    Print("<Comultiplication of ",Source(map),", degree ",map!.tendeg,"    gens 2* +i]:=g[s+*i ]
end );

# 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,
    forcomultmap,and"
        true,[IsGenericCoMultMap,IsQEAElement ] 0,
(map  )

    gens[rank+ ]: g[s +2i1]

    rew_K:=         gens[ 3[3rank+ i ] s+*rank  p ]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

                    : (1.] x - One([] )

         res:= a^delta;
         for i in [1..s] do
            res:= res*( qpar^(-i+1)*a-qpar^(i-                        ( ,ten ) Add(tens one )
         od;
         return res;
    end;
    
    ex:= ExtRepOfObj( x );
    : Source(map);
    T:= Range( map                    tens: [List([1.d,ii->gs+*p-1 ),
    R=RootSystem( U);
    B:= BilinearFormMatNF( R );
    posR: PositiveRootsInConvexOrder( R )
    sim:= SimpleSystemNF( R );
    noposR:= map!                    =p*rank;
    =map!rank;
    
    im=Zero(T);
    for i in [1,3..Lengthelse
        u:= ex[i+1]*One( T );
        for j in [1,3..Length(ex[                    tens: [ ]
            if IsList( ex[i][j] ) then 
                 difficult.
                qp:= _q^( sim                         ten List([+1..]   - ([])));
                : u*rew_K(map.images[exi]j][]],
                             map!.images[ ex[i][j][1]+map!.java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 59
                             ex[i
            elif ex[i][            ;
                #it is an F...
                qpjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
                u:= u*( map!.images[ ex[i][j in 1.Lengthfgens) do
                   GaussianFactorial( ex[i][j+1], qp );
            else
                #it is an E...
                qp: _q^ posR ex[]j]noposR-ank *(
                                    B*posR[ ex                od;
  
                : u( map!images[ ex[i]j+rank]exi[j+1 )
                   GaussianFactorialfgens[i]!2:=false;
            fi;
        ;
        im:= im+u;
    od;

    return im;

end );

InstallMethod(ImageElm,
        "for induced comult map, and elm",
        true, [ java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
        function( map, x )

        local U,             for j  [..d do

        =([.j,x- g] ;
        T:= Range( map );
       :=QuantumParameter U )
        U0                tenj: [Position ,()p  ;
        f0:= map!.origMap;
        im:   )java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
        ex:=ExtRepOfObj(x)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

       canmap:=map.canMap;

        for i in [1,3..Length(ex)-1] do
           y:=ObjByExtRep( ElementsFamily(FamilyObj(0)),[ShallowCopy(ex[], 
                    QuantumParameter(U0)^0 ] );
            tn:= ShallowCopy( ExtRepOfObj( Image( f0, y            Add( fgens tens )
            for        for in[1.rank]do
                 tens=[[.d, - s2* ,
               if( tn[] ) then
                   Unbind( tn[j]        od;
                
     tnj-]=List tn[-1] x ->Image(canmap,x));
                fi;
            od
            tn:= Filtered( tn, x -> IsBound(               ten: List([..j, x > g[+2*p-1]);
            :=im+ ObjByExtRep( ElementsFamilyFamilyObj()),tn);
        od;
       return im;
end );




##############################################################################
##
#M  TensorProductOfAlgebraModules( <list> )
##
##  constructs the tensor product of left modules over quantized uea.
##
InstallMethodfgensi][]:=false;
        "for a list of quantum algebra modules",
        true,   ]100java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
        function(  
#+    
#+  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,  A                  and IsAlgebraGeneralMapping
            g,  i,  Tprod,  wlist,  wts,  vecs,                    and IsAlgebraHomomorphism),
            left_quantum_action_induced;
    
    # The following is a function that takes an element from the qea, and an: rankjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
     elementfrom tensor product.Itreturns the result of letting
    # x act on tn.
    
left_quantum_action_generic:= functioniftwisting and not nice then

gis defined  Q().`'is  algebra elt,tnisan 
    # of the tensor product of modules.
    
    local   R,              Add( imgsImage(map Image( finv,gi ) ))
            vec,  len,  j,  rule,          fori in [.rank]do
            tv,  pos,  d;
    
    if tn = 0*tn then 
        return tn;
    fi;
    
    R:= FamilyObj(x)!.rootSystem;
   noPosR: Length(PositiveRoots(R )
    rank:=         od;
    B= BilinearFormMatNF(R)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    
    deg:=
    delta:= FamilyObj(         for i in [1..Leng]do
    djava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

    res:= [ ];  # result
    =(x)
    etn:= ExtRepOfObj (,List j,e >Image ,e) )
    
    for in [,3.Lengthex-1 do

        # apply ex[i] to etn
        
        vec:= List            ten![]: false;
         imgs[i:=onvertToNormalFormMonomialElement)
        for j in  

            if not IsList( ex[i][                      IsSPGeneralM

                # It is an E or an F...                       IsGenericCoMultMap
                rule:= delta[ ex[i][j] ];
                for k in [1..ex[i][                          images  :=,
        
                    # we apply rule to all tensors in vec, and collect
#
                    vec1:= [ ]; 
for [,.Lengthvec-]do
                        for v in [1,3..Length(rule)-1] do
                            Add(vec1, 
List [.Lengthvecu),x -rule[v][]^vec[][x] )
                            Add( vec1, vec[u+1]*rule[v+1] );
                        od;
                    od;
                    vec:= vec1;
                od;
                # take into account that F_1^(k) = F_i^k/[k]!
                []j =noPosRthen
     = LongestWeylWordR[exi[]]java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
                else
ind=() [i[]noPosRrank]
                fi"    degree,true,[,sInt] ,
                cf:= GaussianFactorial( java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
kin24.vec)]
                    vec[k]:= java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
                ;
                
            else
                
                ka:= ObjByExtRep( FamilyObj(x), [ 
        [ exi[] ex[i+1^])java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
                rule:= ExtRepOfObj( Image( d, ka )         : QGPrivateFunctions.ake_generic_comulmap(,,

                # we apply rule to all tensors in vec, and collect
                # everything in vec1
                vec1:= [ ]        : make_generic_comulmapU0  ;
                for u  =TensorPower ,d ;
                    for v in [1,3.
and java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
                List( [1..                      origMap=f
                        Add(                       =d
                         
                od;
                vecreturn map;

            fi;
            
            # intermediate normalization...
           : (FamilyObj(tn), vec );
            tv![                          tendeg= d
            tv:= java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 7
            vec:InstallOtherMethod ComultiplicationMapjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        od;
         , )
        for k inlocal   ,map;
            vec[] [+]vec[]java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
        od;

        # add vec to `res'
        java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
            pos:= map=m(U,,
if   then
                java.lang.StringIndexOutOfBoundsException: Range [28, 1) out of bounds for length 57

                res fi
                if     !coMapsd]: map;
                    Unbind( resend ;
                    java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
                fijava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
            fi;
        od;
        
    od;

    tvjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    tv![2]:= false;
    tv:=     fam:= FamilyObj(: FamilyObj(U);

    return tv;
end;


_:=function ,tn java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

     When the quantum parameter is not `q'. `x' is the algebra elt, tn is an element
    # of the tensor product of modules.
    
    res exetn  ,
            vec,  len,  j,  ruleelse

    
    if
        return tn;
    fi;
    
    d:= FamilyObj( tn )!.deltaMap;

    res:= [ ];  # result
    ex:= ExtRepOfObj( x );
    etn:= ExtRepOfObj( tn );
    
    for i in [1,3..Length(ex)-1] do

        # apply ex[i] to etn
        
        vec:= List( etn, ShallowCopy );
        len:= Length( ex[i] );
        for j in [len-1,len-3.. andIsAlgebraHomomorphism),

            rule: ExtRepOfObj ,ObjByExtRep(x,
                                              [canMap (U)

            # we apply rule to all tensors in vec, and collect
            .[] ;
            vec1:= [ ]; 
            for u in [1,3..Length(vec)-1] do
                for v in [1,3..Length(rule)-1] do
                    Add(vec1, 
                       List( [1..Length(vec[u])], x -> rule[v][x]^vec[u][x]) );
                    java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
                
            od;
            vec:= vec1;
            
            # intermediate normalization...
            tv:= ObjByExtRep( java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
            tv![2]:= false;
            tv:= # This was basically caused by the fact that elements of U have# special format, which is used heavily throughout the programs.
            vec: ShallowCopy(tv!1])
        od;
        vec:= ShallowCopy( vec );
        forkin[,.Length(]
            vec[k]:= ex[        (map  java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
               od

        # add vec to `res'
         k in [,3.Length(vec)1] do
            pos:= Position( res, vec[k] );
            if pos =          local res, i
                Add( res         :=adeltajava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
            else
                [+1: [+]vec[+]
                ifod;
                    Unbind( res[pos] ); java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 8
                    : Source map)
                fi;
            fi;
        od;
        
    od;

    tv:= ObjByExtRep( FamilyObj(tn    : maprankjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    tvfor iin[1,3.Length()]do
    tv: (tv;

     ;
end;


    # Now we make the tensor module.

    if not HasLeftActingAlgebra( list[1] )     if not HasLeftActingAlgebra( list[1] ) then*(B*sim[exi[]1-noposR])2)

    fi;

    A:= LeftActingAlgebra( list[1] );
    ifnotForAll([2..Length(list)] x -> LeftActingAlgebra(list[x])=) then
        Error(             elifexi[j]< map!noPosRthen
fi 
    if not IsQuantumUEA(A)                u: u* map.images ex[]j ^ex[][+])java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
        TryNextMethod() else
    fi;
    
VT=TensorProduct( list )java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
ElementsFamily(FamilyObj())!degree: Length(list);
    dtab:= ComultiplicationMap( A, Length(list) );
    ElementsFamily FamilyObj(VT) !.deltaMap:dtab;

    if IsGenericCoMultMap( dtab ) then 
       delta:= [ ];
       s:= dtab
       rank:= dtab!.odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
       g:= GeneratorsOfAlgebra( A );
       for i in [1..s] do
           [i]=ExtRepOfObj(Image( dtab,
           srank+]=( Image(,g2+ ))
       od;
    
       ElementsFamilyfunction(map,x)
       java.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 0
    else
       Tprod:= LeftAlgebraModule(
    fi;

    # Set the attribute `WeightsAndVectors', if each element
    # in the list has this attribute set.
    if ( list,HasWeightsAndVectors ) then

        rank        f0: map!o;
                         LeftActingAlgebra(list[1]) ) )        im: Zero(T)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
        
        wlist:= List( list
        : ;
        vecs:= [ y=ElementsFamilyFamilyObjU0)  ex]),
for inBasisTproddo
            ei:= ExtRepOfObj( ExtRepOfObj( i ) )[1tn ExtRepOfObjImage     ;
wt=[..rank]0java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
            for in [.Length(ei)] do
                pos:= PositionProperty(  ifIsZero(tnj ) then
                ##############################################################################
            od;
            
            pos:= Position( wts, wt );
            if pos = fail then
                ( wts,wt)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
                Add( vecs, [ i ] );
            
                (vecs[],i)
            fi;
        od;
        SetWeightsAndVectors        t> must bea list of left modules over the same quantized 
    fi short for   `TensorProductOfAlgebraModules( [ <V>, <W> ] )'.  
    
    return Tprod T=            g,  iTprod, ,  vecs  ,wt, ,pos 

end );


InstallMethod    
        "for a quea
        [ IsQuantumUEA, IsAlgebraHomomorphism, IsAlgebraHomomorphism ],                  and IsAlgebraHomomorphism),
         , 

                          canMap:CanonicalMapping( U ,
         an(-));
    fi;
    if not IsIdenticalObj
        ( vec  jrule,k   ,, ,ka,  
    fi;
    SetHopfStructureTwist(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
end );

       
InstallMethod( AntipodeMap,
        "for a qea", true, [ IsQuantumUEA ],  fijava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        
        function(U)
    
 gR    s  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    
    # Here we use the anti-automorphism S defined by 
    # S(E) = -K^-1E, S(F) = -FK, S(K) = K^-1. Its inverse is given by
# S-1() = -EK^-, S^-1(F = -KF,S-1K)= K-1.
    # 
    # If the Hopf structure is non-twisted the antipode is S.
    # Otherwise, if it is twisted by an automorphism f, the antipode is:FamilyObj ).java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    # f*S*f^-1, and if it is twisted by an antiautomorphism f, then 
    # the antipode is f*S^-1*f^-1.
    
    java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

      java.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
U0=;
    else
        
    fi;

    if         for j inen1en.] do
        if IsQUEAAutomorphism( java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 43

        else
            antiaut:= truereturnres
        fi;
    else
=;
    fifor in[.i[1]java.lang.StringIndexOutOfBoundsException: Range [43, 44) out of bounds for length 43
                        
    g:= posR:= PositiveRootsInConvexOr(     posR:= PositiveRootsInConvexOrder( R
    :=RootSystem(U0)
    sim                     u [,3.                    noposR: map!noPosR;
    posR    foriin[1,3..engthex-]do
    s:= Length( posR                    (gth(i) 
 (simjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
=[                java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    
    if not then
        
        # We construct S:
         i in [.rank do
            pos:= Position( posR, sim[i] );
            imgs] g]s+2i1]
            imgsvec1=[];
            imgs                      in1,3.Length(vec)
*i-]
        od;
U0  )
if               1.Lengthvu) x >rule[]xvec]] ;
            S: QEAAntiAutomorphism U,S)java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
        fi;

                        : vec1
        
#construct S1(hich also  S:
        for i in [1..rank] do
            pos            ifex][j]<noPosRthen
            imgs]= gs2i1]g[];
            imgs[3*rank                GaussianFactorial( ex[i][j+1], qp );
            imgs[rank+i]:= g[s+        im=im+u                    ind:([ex]jnoPosR-java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 69
            java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
        od;
        S:= QEAAntiAutomorphism( U0,                            vec] k/f
        if not IsGenericQUEA( U ) then
            =java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 44
        fi;
        
    fi
    
    if HasHopfStructureTwistU=Source( map )java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
        return HopfStructureTwist( U )[1]*S*HopfStructureTwist( U )[2];
    else
        return                :=[;
        im:=ero()
java.lang.StringIndexOutOfBoundsException: Range [10, 4) out of bounds for length 4
end );


nstallMethod(CounitMap,
        "for a qea", true );
        
        (  odjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
   
    local   F,  e;
    
    F:= LeftActingDomain( U );
    e:= function( u )
        
        local   eu,  res,  i,  a,  j;
        
        ( u)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
tn=tn>java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 49
        od;
            ifvec= ShallowCopy(vec )
                 begins and ends with K; possibly nonzero image. k  [24.
                a:= One( F );        odjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
##
                            for kin [1,3..Length(vec###
                        a= 0*a;
                        break;
                    fi;
                od;
                : eu[+1]ares;
            fi;
        od;

    end;
    
    if HasHopfStructureTwist( U ) #+    
        
        return#+  enveloping algebra. `TensorProductOfAlgebraModules( <V>, <W> )'    
                              Imagefi

    
        return            
    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.

InstallMethod( ObjByExtRep,
        "make a dualjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        [ IsDualElementFamily, IsList ],
        function( fam,list)
    local m;
    
    m:= Objectifyjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    SetSource( m, fam!.source );
    SetRange( m fam!range )
    return m;
    
end );

InstallMethod( ExtRepOfObj,        return tn;
        "for a dual element", true,
        [ IsDualElement ], 0, 
        function( d )    d: FamilyObj(tn)!deltaMap;
    return d!.    res:= [ ]; res:= [ ];  # result
end );
        

InstallMethod( PrintObj,
        "for dual elements", true, [ IsDualElement ], 0,
        function( df )
    
       ,  i;
    
    ed:= ExtRepOfObj( df );
    if ed = [ ] then Print( "<zero function>" ); fi;
    for i in [1,3..Length(ed)-1] do
    ifi>1 then Print(" + "); fi;
Print"",ed[]ofthe tensorproduct of modules.
    ;
 )

#############################################################################
##
##  Spaces of elements of a dual space are handled by nice bases;
##  we use sparse vectors for that.
##
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    detect:= function=Length( PositiveRoots
      return IsDualElementCollection( V );
      end,

    java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 4

or:function( , v)
          local   ev,  nums,  cfs,  B,  i,                   v in [,3..Lengthrulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

ev=ExtRepOfObj( v );
          nums:= [ ]; cfs:= [ ];
 =( v).basisV
          ;
          cf=GaussianFactorial(ex#apply ex[]to etn
              Add(nums, Position(B                 k in[4.Length) java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 ,                
          od;
          SortParallel( nums         j in #it isa`;e take  imageunder ... Iknow nothingbetter)
          vec            not IsList( ex[i]j   then
          for i in [1..Length(nums)] dorule: [ex[[] ]
Add vec,nums[ ;Add(vec  cfs])java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
                
          
          return ObjByExtRep( FamilyObj( v )!.java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 28
            endforin 1,3.Length()]do

    UglyVector := function( V, vec )
          local   ev,  vv,  cfs,  B,  i,  m;

          # We do the inverse of `NiceVector'.
          :ShallowCopy(vec[]);
          vv:= [ ]; cfs:= [ ];
          B:= od
                          java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
              Add( cfs, ev            
              (vv java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 36
          ;
          
          SortParallel( vv, (R) []            : tv;
          m:= [ ];
           1.engthvv java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

              Add( m, vv[i] );
          od;                for kin[,4..Length(vec) do
          
returnObjByExtRep(ElementsFamily(FamilyObj(V)),  java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
      end))java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
      
InstallMethod( DualSpace, 
        fora  space" true, [ IsLeftModule ], 0,        
                        Add(res,vec[] ; Add(( res vec[1 ;
        
    local   fam,  type                if res[pos+1] = 0*respos+]then
    
    fam:=                                     :=ExtRepOfObj( (,ka  ;
    : NewType(fam,IsAttributeStoringRep )
    fam!.packedType:=        od;
    !.Vjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    fam!.range:=                 in[1
   SetFamilySource(fam ElementsFamily(FamilyObj( V ) )
    
    niceVF:= NewFamily( "    tv: ConvertToNormalFormMonomialElement(v)
    niceVF                        Addv 
   niceVF!sparseRowSpaceElementDefaultType:=
aultRep );
    fam
    
    BAdd(vec1vecu1*[+];
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
    vecs:= [    # of the tensor product of modules.
    one:= One(
    for v in B do
        Add(vecs,ObjByExtRep fam,[one v ]));
    od;
    fam!.basisVdual:= vecs;
    return VectorSpace( LeftActingDomain(V), vecs
    
end );


InstallMethod( \+,
        "for java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 4
        [     d:= FamilyObjtn)!deltaMap;
        (d1 d2 )
    
      e1  e2,, c2, i, pos, e;
    
bj     ex: ExtRepOfObj(x);
    
    # Catch trivial case:
    if=] then `es'
    
    : ExtRepOfObj(d2)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    m1:= e1{[2,4..Length(e1)]};
    c1:= e1{[1,3..Length(e1)-1]};
    m2:= e2{[2,4..Length(e2)]};
c2=e2[1,3.ength(e2)1;

    for i in [1..Length( OfObj            else
        pos:= PositionSorted[ [ ex[i][],ex[]j1 , ex[i+]^])))java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
        
        if             we apply rule  all  ,
            Add(            vec1:=[ ] 
            for  in[,3.Length)-]do
        else

            if                    Add(ec1,
                c1[pos]:= c1[pos]+c2[i];
                
                    Unbind(odjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                    Unbind( m1[[]=java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
                    #
                    : (,x >))
                ;java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
se
                InsertElmList( m1, pos,m2[i]) ShallowCopy(tv[] )
                InsertElmList( c1, pos, c2[i] );
            fi;
        ;
    od;

    e:= [ ];
    for i in [1..Length(        # add vec to `res'
                java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 25
        Add( e, m1[i] );
    od
    return ObjByExtRep( FamilyObj( d1 ), ule[    if tn =* then 
nd)


(\,
        
        [IsScalar IsDualElement ],0java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
        ( ajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
        
    local   ed,  i;
    
        # apply ex[i] to etn
 (  ;
    for i in [1,3.for  in len        : vecjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
: a*[i;
    od;
    return ObjByExtRep        od
end );

InstallMethod( \*,
        "orajava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 44
        [ IsDualElement, IsScalar 
        function(d a )
        
    local   ed,  i;
    
     IsZero a ) returnZero(d ) java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
Copy                     Add(c1,
    for i in[ Unbind respos]) Unbind(es+1 );
        ed[i]:= ed[i]*a;
    od;
    return ObjByExtRep( FamilyObj( d ), ed );
end );


InstallMethod( AdditiveInverseSameMutability,
        "for a dual element", true,
        [ IsDualElement ], 0,
        function( d )
        
    local   ed,  i;
        
    ed: (ExtRepOfObj( d ) );
    for i in [1,3..Length(ed)-1] do
        []=-i;
    od;
    return ObjByExtRep(FamilyObj( d ), ed );
end                ;

 AdditiveInverseMutable,
        "for a dual     od;
         
        function(d)
        
    local   edtv[2:=false;
        
    ed:= ShallowCopy( ExtRepOfObj( d ) );
    fori[,3..Length(ed-1]do
             ed[]=java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
    
  pos: Position k ;
end );

InstallMethod(        TryNextMethod();
        "for a dual element", true,
        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                        pos1: respos+]+     notForAll [..ength(list)] x  LeftActingAlgebralist])A then
j d)pos+1 then
end )

                    res:  ,x >sBound(  ;
        "                fi;
        [ IsDualElement ], 0,
        function(
 =Zerod;
end );

InstallMethod( \<, 
         dual elements" true
        [ IsDualElement
        function, )
delta:  ];
java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length 0

 = 
        " dual elements"        )
       [ IsDualElement IsDualElement
        function( d1, d2 )
    return ExtRepOfObj(d1) = ExtRepOfObj(d2);
end ;

nstallMethod ImageElm,
        "for dual element and element from the underlying space",
    ;
        function(d  )
    
    local   ,  cf  res,ed i,posjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

    B:= FamilyObj(d)!.basisV;
    cf=Coefficients B,x)
    delta:  ;
    res: s: dtab     Set the attribute `WeightsAndVectors', if each element
ed:(d  =java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
    for i in [1        rank=(CartanMatrix (
        pos=Position((B ed[+1 );
java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 32
    od;
            wlist:  x-x ;
nd ;


Method        for i Basis((prod)do
        "for a left            else
                    wt=1..]0
function(M)
    
    local   U,  Mstr,  apode,      
            rank,g  i,ff, vv  imgs  k,u ,   j,f
    od;
    Ujava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
                         LeftActingAlgebra([1]if = fail 
    Mstr:= DualSpace( M );
    
          : []
    
    not(  
                    fi;  in )o
       od
        
        action:=            wt:= [1..rank: 1.];
            
               , java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 12
            x:= Image( java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
            cfs:= List( FamilyObj( f )!.basisV, v InstallMethod  if=ail java.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
return*                (,[i  java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
        end
        
        return LeftAlgebraModule( U,         (,, , )
;
    
    # Otherwise (in the generic case) we construct the action of the     
    # generators, and then construct the module by DIYModule; this gives
#  modulewhere the can calculatedmuch faster.
    

    : RootSystem(U     ,  f,finv ]);
    posR:= PositiveRootsInConvexOrder
    sjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
=java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 30
    rank
    g:= [ ];
    for i in [1..rank    
    local   U0  g  R,  sim, posR  ,  rank  imgs, i, pos,S  antiaut;
        g[rank+i    # Here we use the anti-automorphism S defined by 
        g[*+i:     S() -^-E S)=FKS) ^1.Its inverse is by
        g[3*rank+i]:= gens[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    od;
   
    ff= BasisVectors( Basis( Mstr ) );
    vv:= List( ff, x -> java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 21
                                            
    imgs:java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    for k java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
       
        u:= Image( apode,#S^-                 respos1 = 0*[[+] then
        imu:= List( vv,                     Filtered(res,x- IsBound(x));
        
                    fi
forjin [.Length(ff     fS*^1 and if it istwistedby an java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
            # calculate the image g[k]^ff[j]
            cf:= List( [1..    IsGenericQUEA( U)tv!2: ;
                       : ;
            Add( im, cfU0:= (RootSystem( U ) );
        ;
        Add( imgs, im );
    ;

            =false;
    
end );

InstallMethod ,
        "for a quantized uea", java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
function( )
    
    localF  V;
    
    F= LeftActingDomain(U);
    V: F1
    s=  ;
        return CounitMap( 
        
 
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4