Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/hap/lib/NonabelianTensor/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 19.6.2025 mit Größe 8 kB image not shown  

Quellcode-Bibliothek tensor.gi   Sprache: unbekannt

 



#####################################################
#####################################################
InstallOtherMethod(NonabelianTensorProduct,
"Nonabelian tensor product of two crossed modules",
[IsHapCrossedModule,IsHapCrossedModule],
function(MG,NG)
local M, N, G, F, MhomG, NhomG, gensM, gensN, gensMf, gensNf, gensF, 
      gensF1, gensF2, MhomMfp, NhomNfp, MfhomF, NfhomF, Mfp, Nfp,
      MhomF, NhomF, FhomG, relsF,r,m,x,y,z,zz,v,w, FhomT, T,actM, actN,
      TN, TM, ShomTT, TT, tensor, delta, gensTT, gens, TThomG, action,
      S, ThomS, X, Y ;;

if not (IsHapCrossedModule(MG) and IsHapCrossedModule(NG)) 
then Print("Arguments must be crossed modules"); return fail; fi;
if not Target(MG!.map)=Target(NG!.map) then
Print("Arguments must have common codomain"); return fail; fi;
if not 
(IsNilpotent(Source(CatOneGroupByCrossedModule(MG)!.sourceMap))
and
IsNilpotent(Source(CatOneGroupByCrossedModule(MG)!.sourceMap)) )
then
Print("WARNING: The largest nilpotent quotient of the tensor product will be returned.\n\n");
fi;


MhomG:=MG!.map;
NhomG:=NG!.map;
actM:=MG!.action;
actN:=NG!.action;
M:=Source(MhomG);
N:=Source(NhomG);
G:=Target(MhomG);

MhomMfp:=IsomorphismFpGroup(M);#
NhomNfp:=IsomorphismFpGroup(N);#

Mfp:=Range(MhomMfp);
Nfp:=Range(NhomNfp);

gensM:=List(GeneratorsOfGroup(Mfp),x->PreImagesRepresentative(MhomMfp,x));#
gensN:=List(GeneratorsOfGroup(Nfp),x->PreImagesRepresentative(NhomNfp,x));#
gensMf:=FreeGeneratorsOfFpGroup(Mfp);
gensNf:=FreeGeneratorsOfFpGroup(Nfp);
F:=FreeGroup(Length(gensM)+Length(gensN));
gensF:=GeneratorsOfGroup(F);
gensF1:=gensF{[1..Length(gensM)]};
gensF2:=gensF{Length(gensM)+[1..Length(gensN)]};
FhomG:=GroupHomomorphismByImagesNC(F,G,gensF,
Concatenation(List(gensM,x->Image(MhomG,x)),
List(gensN,x->Image(NhomG,x))));

MfhomF:=GroupHomomorphismByImagesNC(Group(gensMf),F,gensMf,gensF1);
NfhomF:=GroupHomomorphismByImagesNC(Group(gensNf),F,gensNf,gensF2);
MhomF:=GroupHomomorphismByImagesNC(M,F,gensM,gensF1);
NhomF:=GroupHomomorphismByImagesNC(N,F,gensN,gensF2);



relsF:=[];
for r in RelatorsOfFpGroup(Mfp) do
Add(relsF,Image(MfhomF,r));
od;
for r in RelatorsOfFpGroup(Nfp) do
Add(relsF,Image(NfhomF,r));
od;

X:=Kernel(MhomG);
Y:=MinimalGeneratingSet(X);
X:=RightTransversal(M,X);
X:=Concatenation(X,Y);

for z in X do
zz:=Image(MhomG,z);
for x in gensM do
for y in gensN do
v:=Comm(Image(MhomF,x),Image(NhomF,y))^Image(MhomF,z) ;
w:=Comm(Image(NhomF,actN(zz,y)),Image(MhomF,actM(zz,x)) );
Append(relsF,[v*w]);
od;od;od;

X:=Kernel(NhomG);
Y:=MinimalGeneratingSet(X);
X:=RightTransversal(N,X);
X:=Concatenation(X,Y);

for z in X do
zz:=Image(NhomG,z);
for x in gensM do
for y in gensN do
v:=Comm(Image(MhomF,x),Image(NhomF,y))^Image(NhomF,z) ;
w:=Comm(Image(NhomF,actN(zz,y)),Image(MhomF,actM(zz,x)) );
Append(relsF,[v*w]);
od;od;od;


T:=F/relsF;
FhomT:=GroupHomomorphismByImagesNC(F,T,gensF,GeneratorsOfGroup(T));;

ThomS:=IsomorphismSimplifiedFpGroup(T);
S:=Image(ThomS);

ShomTT:=NqEpimorphismNilpotentQuotient(S);

TT:=Image(ShomTT);

TM:=List(gensF1,x->Image(ShomTT,Image(ThomS,Image(FhomT,x))));;
TM:=Group(TM);
TM:=NormalClosure(TT,TM);
TN:=List(gensF2,x->Image(ShomTT,Image(ThomS,Image(FhomT,x))));;
TN:=Group(TN);
TN:=NormalClosure(TT,TN);
tensor:= Intersection(TM,TN); 

gensTT:=GeneratorsOfGroup(TT);
gens:=List(gensTT,x->PreImagesRepresentative(ShomTT,x));
gens:=List(gens,x->PreImagesRepresentative(ThomS,x));
gens:=List(gens,x->PreImagesRepresentative(FhomT,x));
gens:=List(gens,x->Image(FhomG,x));
TThomG:=GroupHomomorphismByImagesNC(TT,G,gensTT,gens);

delta:=GroupHomomorphismByFunction(tensor,G,x->Image(TThomG,x));
########################
action:=function(g,t)
local gg;
gg:=PreImagesRepresentative(TThomG,g);
return gg^-1*t*gg;
end;
########################


return Objectify(HapCrossedModule,
       rec(map:=delta,action:=action));
end);
#####################################################
#####################################################




#####################################################
#####################################################
InstallOtherMethod(NonabelianExteriorProduct,
"Nonabelian exterior product of two crossed modules",
[IsHapCrossedModule,IsHapCrossedModule],
function(MG,NG)
local M, N, G, F, MhomG, NhomG, gensM, gensN, gensMf, gensNf, gensF,
      gensF1, gensF2, MhomMfp, NhomNfp, MfhomF, NfhomF, Mfp, Nfp,
      MhomF, NhomF, FhomG, relsF,r,m,x,y,z,zz,v,w, FhomT, T,actM, actN,
      TN, TM, ShomTT, TT, tensor, delta, gensTT, gens, TThomG, action,
      S, ThomS, X, Y, imM, imN, MN,PreMN,KerM, KerN ;;

if not (IsHapCrossedModule(MG) and IsHapCrossedModule(NG))
then Print("Arguments must be crossed modules"); return fail; fi;
if not Target(MG!.map)=Target(NG!.map) then
Print("Arguments must have common codomain"); return fail; fi;



MhomG:=MG!.map;
NhomG:=NG!.map;
imM:=Image(MhomG);
imN:=Image(NhomG);
KerM:=Kernel(MhomG);
KerN:=Kernel(NhomG);
KerM:=MinimalGeneratingSet(KerM);
KerN:=MinimalGeneratingSet(KerN);
MN:=Intersection(imM,imN);
PreMN:=List(MN,x->[PreImagesRepresentative(MhomG,x),PreImagesRepresentative(NhomG,x)]);
actM:=MG!.action;
actN:=NG!.action;
M:=Source(MhomG);
N:=Source(NhomG);
G:=Target(MhomG);
#gensM:=MinimalGeneratingSet(M);
#gensN:=MinimalGeneratingSet(N);
#MhomMfp:=IsomorphismFpGroupByGenerators(M,gensM);
#NhomNfp:=IsomorphismFpGroupByGenerators(N,gensN);

MhomMfp:=IsomorphismFpGroup(M);#
NhomNfp:=IsomorphismFpGroup(N);#

Mfp:=Range(MhomMfp);
Nfp:=Range(NhomNfp);

gensM:=List(GeneratorsOfGroup(Mfp),x->PreImagesRepresentative(MhomMfp,x));#
gensN:=List(GeneratorsOfGroup(Nfp),x->PreImagesRepresentative(NhomNfp,x));#
gensMf:=FreeGeneratorsOfFpGroup(Mfp);
gensNf:=FreeGeneratorsOfFpGroup(Nfp);
F:=FreeGroup(Length(gensM)+Length(gensN));
gensF:=GeneratorsOfGroup(F);
gensF1:=gensF{[1..Length(gensM)]};
gensF2:=gensF{Length(gensM)+[1..Length(gensN)]};
FhomG:=GroupHomomorphismByImagesNC(F,G,gensF,
Concatenation(List(gensM,x->Image(MhomG,x)),
List(gensN,x->Image(NhomG,x))));

MfhomF:=GroupHomomorphismByImagesNC(Group(gensMf),F,gensMf,gensF1);
NfhomF:=GroupHomomorphismByImagesNC(Group(gensNf),F,gensNf,gensF2);
MhomF:=GroupHomomorphismByImagesNC(M,F,gensM,gensF1);
NhomF:=GroupHomomorphismByImagesNC(N,F,gensN,gensF2);


relsF:=[];
for r in RelatorsOfFpGroup(Mfp) do
Add(relsF,Image(MfhomF,r));
od;
for r in RelatorsOfFpGroup(Nfp) do
Add(relsF,Image(NfhomF,r));
od;

X:=Kernel(MhomG);
Y:=MinimalGeneratingSet(X);
X:=RightTransversal(M,X);
X:=Concatenation(X,Y);

for z in X do
zz:=Image(MhomG,z);
for x in gensM do
for y in gensN do
v:=Comm(Image(MhomF,x),Image(NhomF,y))^Image(MhomF,z) ;
w:=Comm(Image(NhomF,actN(zz,y)),Image(MhomF,actM(zz,x)) );
Append(relsF,[v*w]);
od;od;od;

X:=Kernel(NhomG);
Y:=MinimalGeneratingSet(X);
X:=RightTransversal(N,X);
X:=Concatenation(X,Y);

for z in X do
zz:=Image(NhomG,z);
for x in gensM do
for y in gensN do
v:=Comm(Image(MhomF,x),Image(NhomF,y))^Image(NhomF,z) ;
w:=Comm(Image(NhomF,actN(zz,y)),Image(MhomF,actM(zz,x)) );
Append(relsF,[v*w]);
od;od;od;

for x in PreMN do
v:=Comm(Image(MhomF,x[1]),Image(NhomF,x[2]));
Add(relsF,v);
for y in KerN do
v:=Comm(Image(MhomF,x[1]),Image(NhomF,y));
Add(relsF,v);
od;
for y in KerM do
v:=Comm(Image(MhomF,y),Image(NhomF,x[2]));
Add(relsF,v);
od;


od;


T:=F/relsF;
FhomT:=GroupHomomorphismByImagesNC(F,T,gensF,GeneratorsOfGroup(T));;

ThomS:=IsomorphismSimplifiedFpGroup(T);
S:=Image(ThomS);

ShomTT:=NqEpimorphismNilpotentQuotient(S);
TT:=Image(ShomTT);


TM:=List(gensF1,x->Image(ShomTT,Image(ThomS,Image(FhomT,x))));;
TM:=Group(TM);
TM:=NormalClosure(TT,TM);
TN:=List(gensF2,x->Image(ShomTT,Image(ThomS,Image(FhomT,x))));;
TN:=Group(TN);
TN:=NormalClosure(TT,TN);
tensor:= Intersection(TM,TN);

gensTT:=GeneratorsOfGroup(TT);
gens:=List(gensTT,x->PreImagesRepresentative(ShomTT,x));
gens:=List(gens,x->PreImagesRepresentative(ThomS,x));
gens:=List(gens,x->PreImagesRepresentative(FhomT,x));
gens:=List(gens,x->Image(FhomG,x));
TThomG:=GroupHomomorphismByImagesNC(TT,G,gensTT,gens);

delta:=GroupHomomorphismByFunction(tensor,G,x->Image(TThomG,x));
########################
action:=function(g,t)
local gg;
gg:=PreImagesRepresentative(TThomG,g);
return gg^-1*t*gg;
end;
########################


return Objectify(HapCrossedModule,
       rec(map:=delta,action:=action));
end);
#####################################################
#####################################################

[ 0.28Quellennavigators  Projekt   ]