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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 ## 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:= ShallowCopy( tt ); tt1[i]:= ObjByExtRep( famqea, [ ei[j], ei[j+1]^0 ] ); if j = 1 then tensors[k]:= tt1; cfts[k]:= c*ei[j+1]; else Add( tensors, tt1 ); Add( cfts, c*ei[j+1] ); fi; od; if Length( ei ) = 0 then # i.e., the tensor is zero, erase it Unbind( tensors[k] ); Unbind( cfts[k] ); fi; od; tensors:= Filtered( tensors, x -> IsBound( x ) ); cfts:= Filtered( cfts, x -> IsBound( x ) ); od; # Merge tensors and coefficients, take equal tensors together. SortParallel( tensors, cfts ); res:= [ ]; len:= 0; for i in [1..Length(tensors)] do if len > 0 and tensors[i] = res[len-1] then res[len]:= res[len]+cfts[i]; if res[len] = 0*res[len] then Unbind( res[len-1] ); Unbind( res[len] ); len:= len-2; fi; else Add( res, tensors[i] ); Add( res, cfts[i] ); len:= len+2; fi; od; if res = [] then res:= [ [], fam!.zeroCoefficient ]; fi; res:= ObjByExtRep( fam, res ); res![2]:= true; return res; end ); ############################################################################ ## #M \*( <qt1>, <qt2> ) #M OneOp( <qt> ) ## InstallMethod( \*, "for qea tensor power elements", IsIdenticalObj, [ IsQEATensorPowElement, IsQEATensorPowElement ], 0, function( qt1, qt2 ) local res, e1, e2, i, j, mon, k; res:= [ ]; e1:= qt1![1]; e2:= qt2![1]; 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( res, e1[i+1]*e2[j+1] ); od; od; res:= ObjByExtRep( FamilyObj( qt1 ), res ); res![2]:= false; return ConvertToNormalFormMonomialElement( res ); end); InstallMethod( OneOp, "for a qea tensor Power element", true, [ IsQEATensorPowElement ], 0, 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( FamilyObj( qt )!.zeroCoefficient ) ] ); 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 integer", true, [ IsQuantumUEA, IsInt ], 0, function( QA, deg ) local qfam, F, fam, type, gg, one, gens, i, j, g, T; # 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( qfam!.tensorPowers[deg] ) then return qfam!.tensorPowers[deg]; fi; else qfam!.tensorPowers:=[]; fi; # We first make the family of the tensor elements, F:= LeftActingDomain( QA ); fam:= NewFamily( "TensorElementsFam", IsQEATensorPowElement ); type:= NewType( fam, IsMonomialElementRep ); fam!.monomialElementDefaultType:= type; fam!.zeroCoefficient:= Zero( F ); fam!.qeaElementsFam:= ElementsFamily( FamilyObj( QA ) ); fam!.degree:= deg; gg:= GeneratorsOfAlgebra( QA ); one:= One( gg[1] ); gens:= [ ]; for i in [1..deg] do for j in [1..Length(gg)] do g:= List( [1..deg], x -> one ); g[i]:= gg[j]; Add( gens, g ); od; od; Sort( gens ); gens:= List( gens, x -> ObjByExtRep( fam, [ x , One(F) ] ) ); for i in [1..Length(gens)] do gens[i]![2]:= true; od; T:= AlgebraByGenerators( F, gens ); SetOne( T, gens[1]^0 ); qfam!.tensorPowers[deg]:= 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, deg, gentab ) local T, tensorfam, R, B, g, fam, sim, posR, convR, deltatab, one, rank, s, i, pp, k, k1, k2, pair, rel, pos, cf, qa, tens, onet, t1, j; T:= TensorPower( QA, deg ); # We store the table in the family of tensor elements: tensorfam:= ElementsFamily( FamilyObj( T ) ); if IsBound( tensorfam!.deltaTable ) then return tensorfam!.deltaTable; fi; R:= RootSystem( QA ); B:= BilinearFormMatNF( R ); g:= GeneratorsOfAlgebra( QA ); fam:= ElementsFamily( FamilyObj( QA ) ); sim:= SimpleSystemNF( R ); posR:= PositiveRootsNF( R ); convR:= fam!.convexRoots; deltatab:= [ ]; one:= One( tensorfam!.zeroCoefficient ); rank:= Length( CartanMatrix(R) ); s:= Length( posR ); # first we do the F elements for i in [1..s] do pp:= Position( sim, posR[i] ); if pp <> fail then # simple root; copy from gentab deltatab[ Position( convR, posR[i] ) ]:= gentab[ pp ]; 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( sim ) ] do k1:= Position( convR, posR[i] - sim[k] ); if k1 <> fail then k2:= Position( convR, sim[k] ); if k1 > k2 then pair:= [ k1, k2 ]; else pair:= [ k2, k1 ]; fi; rel:= List( fam!.multTab[pair[1]][pair[2]], ShallowCopy ); # see whether F_i is in there... pos:= Position( rel, [ Position( convR, posR[i] ), 1 ] ); if pos <> fail then break; fi; fi; od; # F_i is in `rel'; we get it out cf:= rel[ pos+1]; Unbind( rel[pos] ); Unbind( rel[pos+1] ); rel:= Filtered( rel, x -> IsBound(x) ); for k in [2,4..Length(rel)] do rel[k]:= -(1/cf)*rel[k]; od; Add( rel, [ pair[1], 1, pair[2], 1 ] ); Add( rel, 1/cf ); qa:= _q^( -convR[k1]*( B*convR[k2] ) ); Add( rel, [ pair[2], 1, pair[1], 1 ] ); Add( rel, -qa/cf ); # Now apply delta to the elements of rel, and add... tens:= Zero( T ); onet:= One( tens ); for k in [1,3..Length(rel)-1] do t1:= onet; for j in [1,3..Length(rel[k])-1] do qa:= GaussianFactorial( rel[k][j+1], _q^(convR[rel[k][j]]*( B*convR[rel[k][j]] )/2) ); t1:=(1/qa)*t1*( deltatab[rel[k][j]]^rel[k][j+1] ); od; tens:= tens+rel[k+1]*t1; od; deltatab[ Position( convR, posR[i] ) ]:= tens; fi; od; # Add K-elements from gentab: Append( deltatab, gentab{[rank+1..3*rank]} ); # then we do the E elements for i in [1..s] do pp:= Position( sim, posR[i] ); if pp <> fail then # simple root; copy from gentab deltatab[s+2*rank+Position( convR, posR[i] )]:= gentab[ 3*rank+pp ]; else # find a `definition' for E_{\alpha} # 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 break; fi; fi; od; # E_i is in `rel'; we get it out cf:= rel[ pos+1]; Unbind( rel[pos] ); Unbind( rel[pos+1] ); rel:= Filtered( rel, x -> IsBound(x) ); for k in [2,4..Length(rel)] do rel[k]:= -(1/cf)*rel[k]; od; Add( rel, [ pair[1], 1, pair[2], 1 ] ); Add( rel, 1/cf ); qa:= _q^( -convR[k1]*( B*convR[k2] ) ); Add( rel, [ pair[2], 1, pair[1], 1 ] ); Add( rel, -qa/cf ); # Apply delta to rel... tens:= Zero( T ); onet:= One( tens ); for k in [1,3..Length(rel)-1] do t1:= onet; for j in [1,3..Length(rel[k])-1] do qa:= GaussianFactorial( rel[k][j+1], _q^(convR[rel[k][j]-s-rank]*( B*convR[rel[k][j]-s-rank] )/2) ); t1:=(1/qa)*t1*( deltatab[rel[k][j]+rank]^rel[k][j+1] ); od; tens:= tens+rel[k+1]*t1; od; deltatab[ s+2*rank+Position( convR, posR[i] ) ]:= tens; fi; od; tensorfam!.deltaTable:= deltatab; return deltatab; end; QGPrivateFunctions.make_generic_comulmap:= function( arg ) local U, d, twisting, f, finv, T, tfam, one, R, rank, s, posR, g, gens, i, p, fgens, nice, x, tens, j, ten, dt, map, images, noPosR, tendeg, imgs; 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 ); tfam:= ElementsFamily( FamilyObj( T ) ); one:= One( tfam!.zeroCoefficient ); R:= RootSystem( U ); rank:= Length( CartanMatrix(R) ); s:= Length( PositiveRoots(R) ); posR:= PositiveRootsInConvexOrder( R ); g:= GeneratorsOfAlgebra( U ); gens:= [ ]; for i in [1..rank] do p:= Position( posR, SimpleSystemNF( R )[i] ); gens[i]:= g[ p ]; gens[ rank+i ]:= g[ s + 2*i -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; # 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:= [ ]; 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 # an F tens:= [ ]; for j in [1..d] do ten:= List( [1..j], x -> One(g[1]) ); Append( ten, List( [j+1..d], x -> g[s+2*p] ) ); ten[j]:= x; Add( tens, ten ); Add( tens, one ); od; elif p <= 2*rank then # a K p:= p-rank; tens:= [ List([1..d], ii -> g[s+2*p-1] ), QuantumParameter(U)^0 ]; elif p <= 3*rank then # a K^-1 p:= p-2*rank; tens:= [ List([1..d], ii -> g[s+2*p] ), QuantumParameter(U)^0 ]; else # an E p:= p-3*rank; tens:= [ ]; for j in [1..d] do ten:= List( [1..j], x -> g[s+2*p-1] ); Append( ten, List( [j+1..d], x -> One(g[1]) ) ); ten[j]:= x; Add( tens, ten ); Add( tens, one ); od; fi; Add( fgens, tens ); fi; od; if nice then # apply f<x>f<x>...<x>f for i in [1..Length(fgens)] do for j in [1,3..Length(fgens[i])-1] do fgens[i][j]:= List( fgens[i][j], uu -> Image( f, uu ) ); od; fgens[i]:= ObjByExtRep( tfam, fgens[i] ); fgens[i]![2]:= false; fgens[i]:= ConvertToNormalFormMonomialElement(fgens[i]); od; fi; fi; if not twisting or not nice then # we make the `standard' table. fgens:= [ ]; for p in [1..rank] do tens:= [ ]; for j in [1..d] do ten:= List( [1..j], x -> One(g[1]) ); Append( ten, List( [j+1..d], x -> g[s+2*p] ) ); ten[j]:= g[ Position( posR, SimpleSystemNF(R)[p] ) ]; Add( tens, ten ); Add( tens, one ); od; Add( fgens, tens ); od; for p in [1..rank] do tens:= [ List([1..d], ii -> g[s+2*p-1] ), QuantumParameter(U)^0 ]; Add( fgens, tens ); od; for p in [1..rank] do tens:= [ List([1..d], ii -> g[s+2*p] ), QuantumParameter(U)^0 ]; Add( fgens, tens ); od; for p in [1..rank] do tens:= [ ]; for j in [1..d] do ten:= List( [1..j], x -> g[s+2*p-1] ); Append( ten, List( [j+1..d], x -> One(g[1]) ) ); ten[j]:= g[ s+2*rank + Position( posR, SimpleSystemNF(R)[p] ) ]; Add( tens, ten ); Add( tens, one ); od; Add( fgens, tens ); od; for i in [1..Length(fgens)] do fgens[i]:= ObjByExtRep( tfam, fgens[i] ); fgens[i]![2]:= false; fgens[i]:= ConvertToNormalFormMonomialElement(fgens[i]); od; fi; dt:= QGPrivateFunctions.DeltaTable( U, d, fgens ); map:= Objectify( TypeOfDefaultGeneralMapping( U, T, IsSPGeneralMapping and IsAlgebraGeneralMapping and IsGenericCoMultMap and IsAlgebraHomomorphism), rec( images := dt, rank:= rank, noPosR:= s, tendeg:= d ) ); if twisting and not nice then imgs:= [ ]; for i in [1..s] do Add( imgs, Image( map, Image( finv, g[i] ) ) ); od; for i in [1..rank] do Add( imgs, Image( map, Image( finv, g[s+2*i-1] ) ) ); od; for i in [1..rank] do Add( imgs, Image( map, Image( finv, g[s+2*i] ) ) ); od; for i in [1..s] do Add( imgs, Image( map, Image( finv, g[s+2*rank+i] ) ) ); od; for i in [1..Length(imgs)] do # apply f<x>f...<x>f (d-factors). x:= ShallowCopy( imgs[i]![1] ); ten:= [ ]; for j in [1,3..Length(x)-1] do Add( ten, List( x[j], e -> Image( f, e ) ) ); Add( ten, x[j+1] ); od; ten:= ObjByExtRep( tfam, ten ); ten![2]:= false; imgs[i]:=ConvertToNormalFormMonomialElement(ten); od; map:= Objectify( TypeOfDefaultGeneralMapping( U, T, IsSPGeneralMapping and IsAlgebraGeneralMapping and IsGenericCoMultMap and IsAlgebraHomomorphism), rec( images := imgs, rank:= rank, noPosR:= s, tendeg:= d ) ); fi; return map; end; InstallOtherMethod( ComultiplicationMap, "for a quea, and degree", true, [IsGenericQUEA,IsInt], 0, function( U, d ) local fam, map; fam:= FamilyObj( U ); if not IsBound( fam!.coMaps ) then fam!.coMaps:= [ ]; fi; if IsBound( fam!.coMaps[d] ) then return fam!.coMaps[d]; fi; if HasHopfStructureTwist( U ) then map:= QGPrivateFunctions.make_generic_comulmap( U, d, HopfStructureTwist( U )[1], HopfStructureTwist( U )[2] ); else map:= QGPrivateFunctions.make_generic_comulmap( U, d ); fi; fam!.coMaps[d]:= map; return map; end ); InstallOtherMethod( ComultiplicationMap, "for a quea, and degree", true, [IsQuantumUEA,IsInt], 0, function( U, d ) local T, fam, U0, f, map; if IsGenericQUEA( U ) then TryNextMethod(); fi; fam:= FamilyObj( U ); if not IsBound( fam!.coMaps ) then fam!.coMaps:= [ ]; fi; if IsBound( fam!.coMaps[d] ) then return fam!.coMaps[d]; fi; U0:= QuantizedUEA( RootSystem(U) ); if HasHopfStructureTwist( U ) then f:= QGPrivateFunctions.make_generic_comulmap( U0, d, HopfStructureTwist( U )[1]!.origMap, HopfStructureTwist( U )[2]!.origMap ); else f:= QGPrivateFunctions.make_generic_comulmap( U0, d ); fi; T:= TensorPower( U, d ); map:= Objectify( TypeOfDefaultGeneralMapping( U, T, IsSPGeneralMapping and IsAlgebraGeneralMapping and IsInducedCoMultMap and IsAlgebraHomomorphism), rec( origMap:= f, canMap:= CanonicalMapping( U ), tendeg:= d ) ); fam!.coMaps[d]:= map; return map; end ); InstallMethod( PrintObj, "for a comultmap", true, [ IsCoMultMap ], 0, function( map ) Print("<Comultiplication of ",Source(map),", degree ",map!.tendeg,">"); 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, "for comult map, and elm", true, [ IsGenericCoMultMap, IsQEAElement ], 0, function( map, x ) local rew_K, ex, U, B, R, posR, sim, noposR, rank, i, j, im, u, qp, T; rew_K:= function( a, b, delta, s, qpar ) local res, i; res:= a^delta; for i in [1..s] do res:= res*( qpar^(-i+1)*a-qpar^(i-1)*b )/( qpar^i-qpar^-i ); od; return res; end; ex:= ExtRepOfObj( x ); U:= Source( map ); T:= Range( map ); R:= RootSystem( U ); B:= BilinearFormMatNF( R ); posR:= PositiveRootsInConvexOrder( R ); sim:= SimpleSystemNF( R ); noposR:= map!.noPosR; rank:= map!.rank; im:= Zero( T ); for i in [1,3..Length(ex)-1] do u:= ex[i+1]*One( T ); for j in [1,3..Length(ex[i])-1] do if IsList( ex[i][j] ) then #it is a K...; more difficult. qp:= _q^( sim[ ex[i][j][1]-noposR ]*( B*sim[ ex[i][j][1]-noposR ] )/2); u:= u*rew_K( map!.images[ ex[i][j][1] ], map!.images[ ex[i][j][1]+map!.rank ], ex[i][j][2], ex[i][j+1], qp ); elif ex[i][j] <= map!.noPosR then #it is an F... qp:= _q^( posR[ ex[i][j] ]*( B*posR[ ex[i][j] ] )/2 ); u:= u*( map!.images[ ex[i][j] ]^ex[i][j+1] )/ GaussianFactorial( ex[i][j+1], qp ); else #it is an E... qp:= _q^( posR[ ex[i][j]-noposR-rank ]*( B*posR[ ex[i][j]-noposR-rank ] )/2 ); u:= u*( map!.images[ ex[i][j]+rank ]^ex[i][j+1] )/ GaussianFactorial( ex[i][j+1], qp ); fi; od; im:= im+u; od; return im; end ); InstallMethod( ImageElm, "for induced comult map, and elm", true, [ IsInducedCoMultMap, IsQEAElement ], 0, function( map, x ) local U, T, qp, U0, f0, im, ex, canmap, i, j, y, tn; U:= Source( map ); T:= Range( map ); qp:= QuantumParameter( U ); U0:= QuantizedUEA( RootSystem( U ) ); f0:= map!.origMap; im:= Zero( T ); ex:= ExtRepOfObj( x ); canmap:= map!.canMap; for i in [1,3..Length(ex)-1] do y:= ObjByExtRep( ElementsFamily(FamilyObj(U0)), [ ShallowCopy(ex[i]), QuantumParameter(U0)^0 ] ); tn:= ShallowCopy( ExtRepOfObj( Image( f0, y ) ) ); for j in [2,4..Length(tn)] do tn[j]:= Value( tn[j], qp )*ex[i+1]; if IsZero( tn[j] ) then Unbind( tn[j] ); Unbind(tn[j-1]); else tn[j-1]:= List( tn[j-1], x -> Image( canmap, x ) ); fi; od; tn:= Filtered( tn, x -> IsBound(x) ); im:= im + ObjByExtRep( ElementsFamily(FamilyObj(T)), tn ); od; return im; end ); ############################################################################## ## #M TensorProductOfAlgebraModules( <list> ) ## ## constructs the tensor product of left modules over quantized uea. ## InstallMethod( TensorProductOfAlgebraModules, "for a list of quantum algebra modules", true, [ IsDenseList ], 100, 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, A, VT, dtab, delta, s, rank, g, i, Tprod, wlist, wts, vecs, ei, wt, j, pos, left_quantum_action_induced; # The following is a function that takes an element from the qea, and an # element from the tensor product. It returns the result of letting # x act on tn. left_quantum_action_generic:= function( x, tn ) # When everyting is defined over Q(q). `x' is the algebra elt, tn is an element # of the tensor product of modules. local R, noPosR, rank, B, deg, delta, res, ex, etn, i, vec, len, j, rule, k, vec1, u, v, ind, cf, ka, tv, pos, d; if tn = 0*tn then return tn; fi; R:= FamilyObj(x)!.rootSystem; noPosR:= Length( PositiveRoots(R) ); rank:= Length( CartanMatrix(R) ); B:= BilinearFormMatNF( R ); deg:= FamilyObj( tn )!.degree; delta:= FamilyObj( tn )!.deltaTab; 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..1] do if not IsList( ex[i][j] ) then # It is an E or an F... rule:= delta[ ex[i][j] ]; for k in [1..ex[i][j+1]] do # we apply rule to all tensors in vec, and collect # everything in vec1 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]) ); Add( vec1, vec[u+1]*rule[v+1] ); od; od; vec:= vec1; od; # take into account that F_1^(k) = F_i^k/[k]! if ex[i][j] <= noPosR then ind:= LongestWeylWord(R)[ ex[i][j] ]; else ind:= LongestWeylWord(R)[ ex[i][j]-noPosR-rank ]; fi; cf:= GaussianFactorial( ex[i][j+1], _q^(B[ind][ind]/2) ); for k in [2,4..Length(vec)] do vec[k]:= vec[k]/cf; od; else # it is a `K'; we take the image under d... (I know nothing better). ka:= ObjByExtRep( FamilyObj(x), [ [ ex[i][j], ex[i][j+1] ], ex[i+1]^0] ); rule:= ExtRepOfObj( Image( d, ka ) ); # we apply rule to all tensors in vec, and collect # everything in vec1 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]) ); Add( vec1, vec[u+1]*rule[v+1] ); od; od; vec:= vec1; fi; # intermediate normalization... tv:= ObjByExtRep( FamilyObj(tn), vec ); tv![2]:= false; tv:= ConvertToNormalFormMonomialElement(tv); vec:= ShallowCopy( tv![1] ); od; vec:= ShallowCopy( vec ); for k in [2,4..Length(vec)] do vec[k]:= ex[i+1]*vec[k]; od; # add vec to `res' for k in [1,3..Length(vec)-1] do pos:= Position( res, vec[k] ); if pos = fail then Add( res, vec[k] ); Add( res, vec[k+1] ); else res[pos+1]:= res[pos+1]+vec[k+1]; if res[pos+1] = 0*res[pos+1] then Unbind( res[pos] ); Unbind( res[pos+1] ); res:= Filtered( res, x -> IsBound(x) ); fi; fi; od; od; tv:= ObjByExtRep( FamilyObj(tn), res ); tv![2]:= false; tv:= ConvertToNormalFormMonomialElement(tv); return tv; end; left_quantum_action_induced:= function( x, tn ) # When the quantum parameter is not `q'. `x' is the algebra elt, tn is an element # of the tensor product of modules. local res, ex, etn, i, vec, len, j, rule, k, vec1, u, v, ind, cf, tv, pos, d; if tn = 0*tn then 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..1] do rule:= ExtRepOfObj( Image( d, ObjByExtRep( FamilyObj(x), [ [ ex[i][j], ex[i][j+1] ], ex[i+1]^0] ) ) ); # we apply rule to all tensors in vec, and collect # everything in vec1 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]) ); Add( vec1, vec[u+1]*rule[v+1] ); od; od; vec:= vec1; # intermediate normalization... tv:= ObjByExtRep( FamilyObj(tn), vec ); tv![2]:= false; tv:= ConvertToNormalFormMonomialElement(tv); vec:= ShallowCopy( tv![1] ); od; vec:= ShallowCopy( vec ); for k in [2,4..Length(vec)] do vec[k]:= ex[i+1]*vec[k]; od; # add vec to `res' for k in [1,3..Length(vec)-1] do pos:= Position( res, vec[k] ); if pos = fail then Add( res, vec[k] ); Add( res, vec[k+1] ); else res[pos+1]:= res[pos+1]+vec[k+1]; if res[pos+1] = 0*res[pos+1] then Unbind( res[pos] ); Unbind( res[pos+1] ); res:= Filtered( res, x -> IsBound(x) ); fi; fi; od; od; 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 TryNextMethod(); fi; A:= LeftActingAlgebra( list[1] ); if not ForAll( [2..Length(list)], x -> LeftActingAlgebra(list[x])=A) then Error( "All modules must have the same acting algebra"); fi; if not IsQuantumUEA(A) then TryNextMethod(); fi; VT:= TensorProduct( list ); ElementsFamily( FamilyObj(VT) )!.degree:= Length(list); dtab:= ComultiplicationMap( A, Length(list) ); ElementsFamily( FamilyObj( VT ) )!.deltaMap:= dtab; if IsGenericCoMultMap( dtab ) then delta:= [ ]; s:= dtab!.noPosR; rank:= dtab!.rank; g:= GeneratorsOfAlgebra( A ); for i in [1..s] do delta[i]:= ExtRepOfObj( Image( dtab, g[i] ) ); delta[s+rank+i]:= ExtRepOfObj( Image( dtab, g[s+2*rank+i] ) ); od; ElementsFamily( FamilyObj( VT ) )!.deltaTab:= delta; Tprod:= LeftAlgebraModule( A, left_quantum_action_generic, VT ); else Tprod:= LeftAlgebraModule( A, left_quantum_action_induced, VT ); fi; # Set the attribute `WeightsAndVectors', if each element # in the list has this attribute set. if ForAll( list, HasWeightsAndVectors ) then rank:= Length( CartanMatrix( RootSystem( LeftActingAlgebra(list[1]) ) ) ); wlist:= List( list, x -> WeightsAndVectors(x) ); wts:= [ ]; vecs:= [ ]; for i in Basis(Tprod) do ei:= ExtRepOfObj( ExtRepOfObj( i ) )[1]; wt:= [1..rank]*0; for j in [1..Length(ei)] do pos:= PositionProperty( wlist[j][2], x -> ei[j] in x ); wt:= wt + wlist[j][1][pos]; od; pos:= Position( wts, wt ); if pos = fail then Add( wts, wt ); Add( vecs, [ i ] ); else Add( vecs[pos], i ); fi; od; SetWeightsAndVectors( Tprod, [ wts, vecs ] ); fi; return Tprod; end ); InstallMethod( UseTwistedHopfStructure, "for a quea, and an (anti-)automorphism", true, [ IsQuantumUEA, IsAlgebraHomomorphism, IsAlgebraHomomorphism ], 0, function( U, f, finv ) if (not IsQUEAAutomorphism( f )) and (not IsQUEAAntiAutomorphism( f )) then Error("<f> must be an (anti-)automorphism"); fi; if not IsIdenticalObj( U, Source(f) ) then Error("<f> is a homomorphism of the wrong algebra"); fi; SetHopfStructureTwist( U, [ f, finv ] ); end ); InstallMethod( AntipodeMap, "for a qea", true, [ IsQuantumUEA ], 0, function( U ) local U0, g, R, sim, posR, s, rank, imgs, i, pos, S, antiaut; # 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(E) = -EK^-1, S^-1(F) = -KF, S^-1(K) = 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 # f*S*f^-1, and if it is twisted by an antiautomorphism f, then # the antipode is f*S^-1*f^-1. # We first construct the antipode of the corresponding generic # quea: if IsGenericQUEA( U ) then U0:= U; else U0:= QuantizedUEA( RootSystem( U ) ); fi; if HasHopfStructureTwist( U ) then if IsQUEAAutomorphism( HopfStructureTwist( U )[1] ) then antiaut:= false; else antiaut:= true; fi; else antiaut:= false; fi; g:= GeneratorsOfAlgebra( U0 ); R:= RootSystem( U0 ); sim:= SimpleSystemNF( R ); posR:= PositiveRootsInConvexOrder( R ); s:= Length( posR ); rank:= Length( sim ); imgs:= [ ]; if not antiaut then # We construct S: for i in [1..rank] do pos:= Position( posR, sim[i] ); 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 i in [1..rank] do pos:= Position( posR, sim[i] ); imgs[i]:= -g[s+2*i-1]*g[pos]; imgs[3*rank+i]:= -g[s+2*rank+pos]*g[s+2*i]; 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; fi; if HasHopfStructureTwist( U ) then return HopfStructureTwist( U )[1]*S*HopfStructureTwist( U )[2]; else return S; fi; end ); 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 j in [1,3..Length(eu[i])-1] do if eu[i][j+1] > 0 then a:= 0*a; break; fi; od; res:= eu[i+1]*a+res; fi; od; return res; end; if HasHopfStructureTwist( U ) then return function( u ) return e( Image( HopfStructureTwist( U )[2], u ) ); 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. InstallMethod( ObjByExtRep, "make a dual element", true, [ IsDualElementFamily, IsList ], 0, function( fam, list ) local m; m:= Objectify( fam!.packedType, rec( val:=Immutable(list) ) ); SetSource( m, fam!.source ); SetRange( m, fam!.range ); return m; end ); InstallMethod( ExtRepOfObj, "for a dual element", true, [ IsDualElement ], 0, function( d ) return d!.val; end ); InstallMethod( PrintObj, "for dual elements", true, [ IsDualElement ], 0, function( df ) local ed, i; ed:= ExtRepOfObj( df ); if ed = [ ] then Print( "<zero function>" ); fi; for i in [1,3..Length(ed)-1] do if i > 1 then Print(" + "); fi; Print("(",ed[i],")*F@",ed[i+1]); od; end ); ############################################################################java ## ## Spaces of elements of a dual space are handled by nice bases; ## we use sparse vectors for that. ## InstallHandlingByNiceBasis detect(,,V ) return IsDualElementCollection( V ); end, NiceFreeLeftModuleInfo := ReturnFalse, NiceVector: ) local ev, nums, cfs, B, i, vec; ev: ExtRepOfObj(v ); tensor:=Filtered ,x->IsBound(x ; B:= FamilyObj ] (ev)1] Add( numsreslen:=reslen+i] (, []; od; SortParallel ,cfs ; vec: ] for############################################################################# function qt2) java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 Add res, i1*j+ ; return ObjByExtRep( FamilyObj( v )!.niceVectorFam, vec ); end, UglyVector := function( V, vec local ev,vv ,B ,java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds f # We do the inverse of `NiceVector'. ) vv:= [ ]## ; gth(ev)1 do Add ,i]; (vv, [ev[]])java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for od; SortParallel( vv, cfs ); m:= [ ] lse for i) Add( m, fam!.: java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 ObjByExtRep(ElementsFamily FamilyObj(V)), m ); end ) ); InstallMethod( DualSpace, "for a vector space", true, [ IsLeftModule ], 0, function( V ) local fam, type, niceVF, B, vecs, one, v; fam:= NewFamily( "DualEltsFam", IsDualElement ); type:= NewType( fam, IsAttributeStoringRep ); fam!.packedType:= type; fam!.source:= V; fam!.range:= LeftActingDomain( V ); SetFamilySource( fam, ElementsFamily( FamilyObj( V ) ) ); niceVF:= NewFamily( "NiceVectorFam", IsSparseRowSpaceElement ); niceVF!.zeroCoefficient:= Zero( LeftActingDomain(V) ); niceVF!.sparseRowSpaceElementDefaultType:= NewType( niceVF, IsPackedElementDefaultRep ); fam!.niceVectorFam:= niceVF; B:= Basis(g]= [j; fam!.basisV:= B; vecs:= [ ]; one:= One( LeftActingDomain( V ) ); Addsensi][]:= ; fam!.basisVdual.ensorPowers[]=T; java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 2 ; +java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18 for elements", IsIdenticalObj, [ IsDualElement,IsDualElement #W tensor.gi QuaGroup Willem de Graafreturn !d; locale1, e2,m1, c1, m2, ,i pos,e; sim:SimpleSystemNF(R ; :ExtRepOfObj(d1) # Catch trivial case: if s java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 m1:= e1{ java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds f c1:= e1{[1,fijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds fo c2:=e2{[13.for 1.( sim )]do for i in [ k1< fail then posifk1 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for leng if pos > tt:= ShallowCopytensors[k] Add( m1, m2[i] ); Add( c1, c2[i] ); else if m1[pos ; [pos]=( tt ; if Unbind( c1[pos] cfts Unbind(rel[pos]; [1] Unbind( m1pos (,1) m1:Unbind cfts[k] ); c1: fi InsertElmList ,, m2[] ); InsertElmList( c1 pos, c2[i] ); fi; ; : []java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length :Tjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 2 ,[i ) Add , :GaussianFactorial(el[+] ## return ObjByExtRep IsIdenticalObj,tens:=tens+[k+1*; endlocal res,e1 e2, i j,java.lang.StringIndexOutOfBoundsException: Index 1 out of bo java.lang.StringIndexOutOfBoundsException: Range [12, 2) out of bounds for lengt "for a scalar then IsDualElement ], function( a, d ) local ed, i; ifIsZero(a)then Zero d ) ; ed:= ShallowCopy for in[1,3..Length()-] do edifork .Length( od return ObjByExtRep( FamilyObj( d ), ed ); endre![2: true; InstallMethod( \*, "for " ajava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for leng IsDualElement function( d, a ) local ed, i; if IsZero( a qfamodjava.lang.StringIndexOutOfBoundsException: Index 16 out of bound ExtRepOfObj(d ) fi; ed[i]: od; java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for leng end Addrel java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds fo "for#Apply to... for onet:tens)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds f Addjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for l ,i () ed[ return return ObjByExtRepQGPrivateFunctions.make_generic_comulmapQ end ) InstallMethod( AdditiveInverseMutable, "for a dual element", java.lang.StringIndexOutOfBoundsException: Index 33 out of bou [ ,0 (d) localed ; s: Length(PositiveRootsR); posR=PositiveRootsInConvexOrder(R); [i]=-[i]; od return[+i : [s * 1 ; end ); InstallMethod( ZeroOp# in that case we can make the table directly, : ; java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 function=java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds ( ,[; end )ten: 1.],- g[] ; ajava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for len function( =* then return d = Zero(d); end ); InstallMethod(\, "for# ###W tensor.gi QuaGroup Willem de Graaf [ java.lang.StringIndexOutOfBoundsException: Range [24, 12) out of bounds for len function ) java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length endc ,tt1 len; InstallMethod(\,java.lang.StringIndexOutOfBoundsException: Index 19 out of bound "for dual elements", true, [ IsDualElement, IsDualElement ], 0, function( d1, d2 ) ExtRepOfObj()=ExtRepOfObj(d2; end ); InstallMethod( ImageElmAdd(fgens,tens) od; ,[IsDualElement,IsObject ] 0, function( d, x ) localB, , ed, , pos; B:= FamilyObj(d)!.basisV; cf: Coefficients( B, x ) if cf = fail then return fail; fi; res :=ExtRepOfObj(d; for i in [1,3..Length(ed)-1] do pos:= Position( (cfts,eu[+] ; edipos : fgens[] java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for le return res; end ); InstallMethod # we make the `standard' table. " a left module over a quantized "true IsAlgebraModule andjava.lang.StringIndexOutOfBoundsException: Index 33 out of bou function( M ) local U, Mstr, apode, action, gens, R, posR, s, sim, java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 U=LeftActingAlgebra(M)java.lang.StringIndexOutOfBoundsException: Index 31 out of if notSortParallel(tensors,cfts) Mstr=DualSpace(M) apode:= AntipodeMap( U ); not(, ;Add( , ) fi; # with the action given by apode... (ufjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length localQuantumParameterU^ x:= Image( for [. Appendten (1d,x>1 ; FamilyObj(java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bound kin[.FamilyObj(t1!degree do return LeftAlgebraModule( U, action, Mstr ); fi; # (nthe genericcase) we construct the action ofthe # generators, and then construct the module by DIYModule; this gives # a module where the action can be calculated much faster. List( 1.FamilyObj R:=RootSystem( U)java.lang.StringIndexOutOfBoundsException: Index 24 out of bound posR=PositiveRootsInConvexOrder(R); s:= sim=SimpleSystemNF( ) java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for leng g:ifIsBound( qfam!tensorPowers for i in [1..rank] do g dt:= QGPrivateFunctions.DeltaTable( U, d, fgens ); g[rank+fi *rank+]=gens[ s* java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds g : (fam,; od; : java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for len twistingand notnice vv: (, >ExtRepOfObj)2 ) of M imgs java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length for1Length) : fori in [.rank]do : ,v- uv) qfam for # calculate the image g[k]^ff[j]## enerators of <>( Image## s elements are the co-products of the F-elements, then there are r T, tensorfam , B , famten=ObjByExtRep(tfamten ; od; Add od; DIYModule ,java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bound ) ; deltatab "for a quantized uea", true, [ IsQuantumUEA function function( U local F, V; F=LeftActingDomainU) V:= F return LeftAlgebraModule( U, function( u=Position convR []-sim[]) CounitMap() )*v;end V ; end )
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2026-11-01