|
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains methods for algebra(-with-one) general mappings.
##
## There are two default representations of such general mappings,
## one by generators and images (see the file `vspchom.gi'),
## the other as (linear) operation homomorphism
##
## 1. methods for algebra general mappings given by images
## 2. methods for operation algebra homomorphisms
## 3. methods for natural homomorphisms from algebras
## 4. methods for isomorphisms to matrix algebras
## 5. methods for isomorphisms to f.p. algebras
##
#############################################################################
##
## 1. methods for algebra general mappings given by images
##
#############################################################################
##
#R IsAlgebraGeneralMappingByImagesDefaultRep
##
## is a default representation of algebra general mappings between two
## algebras $A$ and $B$ where $F$ is equal to the left acting
## domain of $A$ and of $B$.
##
## Algebra generators of $A$ and $B$ images are stored in the attribute
## `MappingGeneratorsImages'.
##
## The general mapping is defined as the closure of the relation that joins
## the $i$-th generator of $A$ and the $i$-th generator of $B$
## w.r.t. linearity and multiplication.
##
## It is handled using the attribute `AsLinearGeneralMappingByImages'.
##
DeclareRepresentation( "IsAlgebraGeneralMappingByImagesDefaultRep",
IsAlgebraGeneralMapping and IsAdditiveElementWithInverse
and IsAttributeStoringRep, [] );
DeclareRepresentation( "IsPolynomialRingDefaultGeneratorMapping",
IsAlgebraGeneralMappingByImagesDefaultRep,[]);
#############################################################################
##
#M AlgebraGeneralMappingByImages( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( AlgebraGeneralMappingByImages,
"for two FLMLORs and two homogeneous lists",
[ IsFLMLOR, IsFLMLOR, IsHomogeneousList, IsHomogeneousList ],
function( S, R, gens, imgs )
local map, # general mapping from <S> to <R>, result
filter,
i,basic;
# Handle the case that `gens' is empty.
# We can form a left module general mapping directly.
if IsEmpty( gens ) then
map:= LeftModuleGeneralMappingByImages( S, R, gens, imgs );
SetIsAlgebraGeneralMapping( map, true );
return map;
fi;
# Check the arguments.
if Length( gens ) <> Length( imgs ) then
Error( "<gens> and <imgs> must have the same length" );
elif not IsSubset( S, gens ) then
Error( "<gens> must lie in <S>" );
elif not IsSubset( R, imgs ) then
Error( "<imgs> must lie in <R>" );
elif LeftActingDomain( S ) <> LeftActingDomain( R ) then
Error( "<S> and <R> must have same left acting domain" );
fi;
# type setting
filter:=IsSPGeneralMapping
and IsAlgebraGeneralMapping
and IsAlgebraGeneralMappingByImagesDefaultRep;
#special case: test whether polynomial ring is mapped via 1 and free
#generators
if IsPolynomialRing(S) then
basic:=ForAll(imgs,x->ForAll(imgs,y->x*y=y*x));
for i in [1..Length(gens)] do
if IsOne(gens[i]) then
if not IsOne(imgs[i]) then basic:=false;fi;
elif not gens[i] in IndeterminatesOfPolynomialRing(S) then
basic:=false;
fi;
od;
if basic=true then
filter:=filter and IsPolynomialRingDefaultGeneratorMapping;
fi;
fi;
# Make the general mapping.
map:= Objectify( TypeOfDefaultGeneralMapping( S, R,filter),
rec(
# generators := gens,
# genimages := imgs
) );
SetMappingGeneratorsImages(map,[Immutable(gens),Immutable(imgs)]);
# return the general mapping
return map;
end );
#############################################################################
##
#M AlgebraHomomorphismByImagesNC( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( AlgebraHomomorphismByImagesNC,
"for two FLMLORs and two homogeneous lists",
[ IsFLMLOR, IsFLMLOR, IsHomogeneousList, IsHomogeneousList ],
function( S, R, gens, imgs )
local map; # homomorphism from <source> to <range>, result
map:= AlgebraGeneralMappingByImages( S, R, gens, imgs );
SetIsSingleValued( map, true );
SetIsTotal( map, true );
return map;
end );
#############################################################################
##
#M AlgebraWithOneGeneralMappingByImages( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( AlgebraWithOneGeneralMappingByImages,
"for two FLMLORs and two homogeneous lists",
[ IsFLMLOR, IsFLMLOR, IsHomogeneousList, IsHomogeneousList ],
function( S, R, gens, imgs )
local map; # homomorphism from <source> to <range>, result
gens:= Concatenation( gens, [ One( S ) ] );
imgs:= Concatenation( imgs, [ One( R ) ] );
map:= AlgebraGeneralMappingByImages( S, R, gens, imgs );
SetRespectsOne( map, true );
return map;
end );
#############################################################################
##
#M AlgebraWithOneHomomorphismByImagesNC( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( AlgebraWithOneHomomorphismByImagesNC,
"for two FLMLORs and two homogeneous lists",
true,
[ IsFLMLOR, IsFLMLOR, IsHomogeneousList, IsHomogeneousList ], 0,
function( S, R, gens, imgs )
local map; # homomorphism from <source> to <range>, result
gens:= Concatenation( gens, [ One( S ) ] );
imgs:= Concatenation( imgs, [ One( R ) ] );
map:= AlgebraHomomorphismByImagesNC( S, R, gens, imgs );
SetRespectsOne( map, true );
return map;
end );
#############################################################################
##
#F AlgebraHomomorphismByImages( <S>, <R>, <gens>, <imgs> )
##
InstallGlobalFunction( AlgebraHomomorphismByImages,
function( S, R, gens, imgs )
local hom;
hom:= AlgebraGeneralMappingByImages( S, R, gens, imgs );
if IsMapping( hom ) then
return AlgebraHomomorphismByImagesNC( S, R, gens, imgs );
else
return fail;
fi;
end );
#############################################################################
##
#F AlgebraWithOneHomomorphismByImages( <S>, <R>, <gens>, <imgs> )
##
InstallGlobalFunction( AlgebraWithOneHomomorphismByImages,
function( S, R, gens, imgs )
local hom;
hom:= AlgebraWithOneGeneralMappingByImages( S, R, gens, imgs );
if IsMapping( hom ) then
return AlgebraWithOneHomomorphismByImagesNC( S, R, gens, imgs );
else
return fail;
fi;
end );
#############################################################################
##
#M AlgebraHomomorphismByFunction( <A>, <B>, <f> )
#M AlgebraWithOneHomomorphismByFunction( <A>, <B>, <f> )
##
InstallMethod( AlgebraHomomorphismByFunction,
"for two algebras and a function",
true,
[ IsAlgebra, IsAlgebra, IsFunction ],
function( A, B, f )
return Objectify( TypeOfDefaultGeneralMapping( A, B,
IsSPMappingByFunctionRep and IsAlgebraHomomorphism ), rec(fun:=f) );
end);
InstallMethod(AlgebraWithOneHomomorphismByFunction,
"for two algebras and a function",
true,
[ IsAlgebraWithOne, IsAlgebraWithOne, IsFunction ],
function( A, B, f )
return Objectify( TypeOfDefaultGeneralMapping( A, B,
IsSPMappingByFunctionRep and IsAlgebraWithOneHomomorphism ),
rec(fun:=f) );
end);
#############################################################################
##
#M ViewObj( <map> ) . . . . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( ViewObj, "for an algebra g.m.b.i", true,
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
mapi:=MappingGeneratorsImages(map);
View(mapi[1]);
Print(" -> ");
View(mapi[2]);
end );
#############################################################################
##
#M PrintObj( <map> ) . . . . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( PrintObj, "for an algebra-with-one hom. b.i", true,
[ IsMapping and RespectsOne
and IsAlgebraGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
mapi:=MappingGeneratorsImages(map);
Print( "AlgebraWithOneHomomorphismByImages( ",
Source( map ), ", ", Range( map ), ", ",
mapi[1], ", ", mapi[2], " )" );
end );
InstallMethod( PrintObj, "for an algebra hom. b.i.", true,
[ IsMapping
and IsAlgebraGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
mapi:=MappingGeneratorsImages(map);
Print( "AlgebraHomomorphismByImages( ",
Source( map ), ", ", Range( map ), ", ",
mapi[1], ", ", mapi[2], " )" );
end );
InstallMethod( PrintObj, "for an algebra-with-one g.m.b.i", true,
[ IsGeneralMapping and RespectsOne
and IsAlgebraGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
mapi:=MappingGeneratorsImages(map);
Print( "AlgebraWithOneGeneralMappingByImages( ",
Source( map ), ", ", Range( map ), ", ",
mapi[1], ", ", mapi[2], " )" );
end );
InstallMethod( PrintObj, "for an algebra g.m.b.i", true,
[ IsGeneralMapping
and IsAlgebraGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
mapi:=MappingGeneratorsImages(map);
Print( "AlgebraGeneralMappingByImages( ",
Source( map ), ", ", Range( map ), ", ",
mapi[1], ", ", mapi[2], " )" );
end );
#############################################################################
##
#M AsLeftModuleGeneralMappingByImages( <alg_gen_map> )
##
## If necessary then we compute a basis of the preimage,
## and images of its basis vectors.
##
## Note that we must prescribe also the products of basis vectors and
## their images if <alg_gen_map> is not known to be a mapping.
##
InstallMethod( AsLeftModuleGeneralMappingByImages,
"for an algebra general mapping by images",
[ IsAlgebraGeneralMapping
and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( alg_gen_map )
local origgenerators, # list of algebra generators of the preimage
origgenimages, # list of images of `origgenerators'
generators, # list of left module generators of the preimage
genimages, # list of images of `generators'
A, # source of the general mapping
left, # is it necessary to multiply also from the left?
# (not if `A' is associative or a Lie algebra)
maxdim, # upper bound on the dimension
MB, # mutable basis of the preimage
dim, # dimension of the actual left module
len, # number of algebra generators
i, j, # loop variables
gen, # loop over generators
prod, #
result; #
A:=MappingGeneratorsImages(alg_gen_map);
origgenerators := A[1];
origgenimages := A[2];
A:= Source( alg_gen_map );
if IsBasis( origgenerators ) then
generators := origgenerators;
genimages := origgenimages;
else
generators := ShallowCopy( origgenerators );
genimages := ShallowCopy( origgenimages );
left:= not ( ( HasIsAssociative( A ) and IsAssociative( A ) )
or ( HasIsLieAlgebra( A ) and IsLieAlgebra( A ) ) );
if HasDimension( A ) then
maxdim:= Dimension( A );
else
maxdim:= infinity;
fi;
# $A_1$
MB:= MutableBasis( LeftActingDomain( A ), generators,
Zero( A ) );
dim:= 0;
len:= Length( origgenerators );
while dim < NrBasisVectors( MB ) and NrBasisVectors( MB ) < maxdim do
# `MB' is a mutable basis of $A_i$.
dim:= NrBasisVectors( MB );
# Compute $\bigcup_{g \in S} ( A_i g \cup A_i g )$.
for i in [ 1 .. len ] do
gen:= origgenerators[i];
for j in [ 1 .. Length( generators ) ] do
prod:= generators[j] * gen;
if not IsContainedInSpan( MB, prod ) then
Add( generators, prod );
Add( genimages, genimages[j] * origgenimages[i] );
CloseMutableBasis( MB, prod );
fi;
od;
od;
if left then
# Compute $\bigcup_{g \in S} ( A_i g \cup g A_i )$.
for i in [ 1 .. len ] do
gen:= origgenerators[i];
for j in [ 1 .. Length( generators ) ] do
prod:= gen * generators[j];
if not IsContainedInSpan( MB, prod ) then
Add( generators, prod );
Add( genimages, origgenimages[i] * genimages[j] );
CloseMutableBasis( MB, prod );
fi;
od;
od;
fi;
od;
fi;
# If it is not known whether alg_gen_map is single valued, we need to
# perform some extra work.
if not (HasIsSingleValued( alg_gen_map ) and IsSingleValued( alg_gen_map )) then
# TODO: This code below is far from optimal. Indeed, it would suffice to
# loop over a basis; and we don't need to record all generator / image
# pairs we obtain below, but rather only those that are not linearly
# dependent on the already known pairs.
len := Length( generators );
generators:= ShallowCopy( generators );
genimages:= ShallowCopy( genimages );
for i in [ 1 .. len ] do
for j in [ 1 .. len ] do
Add( generators, generators[i] * generators[j] );
Add( genimages, genimages[i] * genimages[j] );
od;
od;
fi;
# Construct the left module (general) mapping.
result := LeftModuleGeneralMappingByImages( A, Range( alg_gen_map ),
generators, genimages );
# Transfer properties of alg_gen_map to result (in particular whether this is
# a homomorphism).
if HasIsSingleValued( alg_gen_map ) then
SetIsSingleValued( result, IsSingleValued( alg_gen_map ) );
fi;
if HasIsTotal( alg_gen_map ) then
SetIsTotal( result, IsTotal( alg_gen_map ) );
fi;
return result;
end );
#############################################################################
##
#M ImagesSource( <map> ) . . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( ImagesSource,
"for an algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map )
local asmap;
if HasAsLeftModuleGeneralMappingByImages( map ) then
asmap:= AsLeftModuleGeneralMappingByImages( map );
if IsLinearGeneralMappingByImagesDefaultRep( asmap )
and IsBound( asmap!.basisimage ) then
return SubFLMLORNC( Range( map ),
asmap!.basisimage, "basis" );
fi;
fi;
return SubFLMLORNC( Range( map ), MappingGeneratorsImages(map)[2] );
end );
#############################################################################
##
#M PreImagesRange( <map> ) . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( PreImagesRange,
"for an algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map )
local asmap;
if HasAsLeftModuleGeneralMappingByImages( map ) then
asmap:= AsLeftModuleGeneralMappingByImages( map );
if IsLinearGeneralMappingByImagesDefaultRep( asmap )
and IsBound( asmap!.basispreimage ) then
return SubFLMLORNC( Source( map ),
asmap!.basispreimage, "basis" );
fi;
fi;
return SubFLMLORNC( Source( map ), MappingGeneratorsImages(map)[1]);
end );
#############################################################################
##
#M CoKernelOfAdditiveGeneralMapping( <map> ) . . . . . for algebra g.m.b.i.
##
InstallMethod( CoKernelOfAdditiveGeneralMapping,
"for algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map )
local asmap, genimages, coker;
asmap:= AsLeftModuleGeneralMappingByImages( map );
if not IsBound( asmap!.corelations ) then
MakeImagesInfoLinearGeneralMappingByImages( asmap );
fi;
genimages:= MappingGeneratorsImages(asmap)[2];
coker:= SubFLMLORNC( Range( map ),
List( asmap!.corelations,
r -> LinearCombination( genimages, r ) ) );
SetCoKernelOfAdditiveGeneralMapping( asmap, coker );
return coker;
end );
#############################################################################
##
#M IsSingleValued( <map> ) . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( IsSingleValued,
"for algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function(map)
local S;
S:=Source(map);
# rewriting to left modules is not feasible for infinite dimensional
# domains
if not IsFiniteDimensional(S) then
TryNextMethod();
fi;
return IsSingleValued( AsLeftModuleGeneralMappingByImages( map ) );
end);
#############################################################################
##
#M IsSingleValued( <map> ) . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallTrueMethod( IsSingleValued, IsGeneralMapping and
IsPolynomialRingDefaultGeneratorMapping );
#############################################################################
##
#M KernelOfAdditiveGeneralMapping( <map> ) . . . . . . for algebra g.m.b.i.
##
InstallMethod( KernelOfAdditiveGeneralMapping,
"for algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map )
local asmap, generators, ker;
asmap:= AsLeftModuleGeneralMappingByImages( map );
if not IsBound( asmap!.relations ) then
MakePreImagesInfoLinearGeneralMappingByImages( asmap );
fi;
generators:= MappingGeneratorsImages(asmap)[1];
ker:= SubFLMLORNC( Source( map ),
List( asmap!.relations,
r -> LinearCombination( generators, r ) ) );
SetKernelOfAdditiveGeneralMapping( asmap, ker );
if HasIsTotal( map ) and IsTotal( map ) then
SetIsTwoSidedIdealInParent( ker, true );
fi;
return ker;
end );
#############################################################################
##
#M IsInjective( <map> ) . . . . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( IsInjective,
"for algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
map -> IsInjective( AsLeftModuleGeneralMappingByImages( map ) ) );
#############################################################################
##
#M ImagesRepresentative( <map>, <elm> ) . . . . . . . for algebra g.m.b.i.
##
InstallMethod( ImagesRepresentative,
"for algebra g.m.b.i., and element",
FamSourceEqFamElm,
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep,
IsObject ],
function( map, elm )
return ImagesRepresentative( AsLeftModuleGeneralMappingByImages( map ),
elm );
end );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> ) . . . . . . for algebra g.m.b.i.
##
InstallMethod( PreImagesRepresentative,
"for algebra g.m.b.i., and element",
FamRangeEqFamElm,
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep,
IsObject ],
function( map, elm )
return PreImagesRepresentative( AsLeftModuleGeneralMappingByImages(map),
elm );
end );
#############################################################################
##
#M \*( <c>, <map> ) . . . . . . . . . . . . for scalar and algebra g.m.b.i.
##
InstallMethod( \*,
"for scalar and algebra g.m.b.i.",
[ IsMultiplicativeElement,
IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( scalar, map )
return scalar * AsLeftModuleGeneralMappingByImages( map );
end );
#############################################################################
##
#M AdditiveInverseOp( <map> ) . . . . . . . . . . . . for algebra g.m.b.i.
##
InstallMethod( AdditiveInverseOp,
"for algebra g.m.b.i.",
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
map -> AdditiveInverse( AsLeftModuleGeneralMappingByImages( map ) ) );
#############################################################################
##
#M CompositionMapping2( <map2>, map1> ) for lin. mapping & algebra g.m.b.i.
##
InstallMethod( CompositionMapping2,
"for left module hom. and algebra g.m.b.i.",
FamSource1EqFamRange2,
[ IsLeftModuleHomomorphism,
IsAlgebraGeneralMapping
and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map2, map1 )
# Composition of two algebra homomorphisms is handled by another method.
if HasRespectsMultiplication( map2 )
and HasRespectsMultiplication( map2 ) then
TryNextMethod();
fi;
return CompositionMapping( map2,
AsLeftModuleGeneralMappingByImages( map1 ) );
end );
#############################################################################
##
#M CompositionMapping2( <map2>, map1> ) for algebra hom. & algebra g.m.b.i.
##
InstallMethod( CompositionMapping2,
"for algebra hom. and algebra g.m.b.i.",
FamSource1EqFamRange2,
[ IsAlgebraHomomorphism,
IsAlgebraGeneralMapping
and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map2, map1 )
local comp, # composition of <map2> and <map1>, result
gens,
genimages,
mapi1,mapi2;
mapi1:=MappingGeneratorsImages(map1);
# Compute images for the generators of `map1'.
gens:= mapi1[1];
if IsAlgebraGeneralMappingByImagesDefaultRep( map2 ) then
mapi2:= MappingGeneratorsImages( map2 );
if mapi1[2] = mapi2[1] then
genimages:= mapi2[2];
else
genimages:= List( mapi1[2],
v -> ImagesRepresentative( map2, v ) );
fi;
else
genimages:= List( mapi1[2],
v -> ImagesRepresentative( map2, v ) );
fi;
# Construct the linear general mapping.
comp:= AlgebraGeneralMappingByImages(
Source( map1 ), Range( map2 ), gens, genimages );
# Maintain info.
if HasRespectsOne( map1 ) and HasRespectsOne( map2 )
and RespectsOne( map1 ) and RespectsOne( map2 ) then
SetRespectsOne( comp, true );
fi;
if HasAsLeftModuleGeneralMappingByImages( map1 )
and HasAsLeftModuleGeneralMappingByImages( map2 ) then
SetAsLeftModuleGeneralMappingByImages( comp,
CompositionMapping( AsLeftModuleGeneralMappingByImages( map2 ),
AsLeftModuleGeneralMappingByImages( map1 ) ) );
fi;
# Return the composition.
return comp;
end );
#############################################################################
##
#M \+( <map1>, map2> ) . . . . . . . . . . . . . . . . for algebra g.m.b.i.
##
## The sum of an algebra general mapping and a left module general mapping
## is in general only a left module general mapping.
## So we delegate to the methods for left module general mappings.
##
InstallOtherMethod( \+,
"for an algebra g.m.b.i. and general mapping",
IsIdenticalObj,
[ IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep,
IsGeneralMapping ],
function( map1, map2 )
return AsLeftModuleGeneralMappingByImages( map1 ) + map2;
end );
InstallOtherMethod( \+,
"for general mapping and algebra g.m.b.i.",
IsIdenticalObj,
[ IsGeneralMapping,
IsGeneralMapping and IsAlgebraGeneralMappingByImagesDefaultRep ],
function( map1, map2 )
return map1 + AsLeftModuleGeneralMappingByImages( map2 );
end );
#############################################################################
##
## 2. methods for operation algebra homomorphisms
##
#############################################################################
##
#R IsOperationAlgebraHomomorphismDefaultRep
##
## is a default representation of operation homomorphisms to matrix FLMLORs.
## It assumes that a basis of the operation domain is known.
## (For operation homomorphisms from f.~p. algebras to matrix algebras,
## see `IsAlgebraHomomorphismFromFpRep'.)
##
## Defining components are
##
## `basis'
## basis of the domain on that the source acts
##
## `operation'
## the function via that the source acts
##
## Images can be computed by the action, w.r.t. the basis `basisImage'.
## Preimages can be computed using the components
##
## `basisImage'
## basis of the image
##
## `preimagesBasisImage'
## list of preimages of the basis vectors of `basisImage'.
##
## These components are computed as soon as they are needed.
##
## Note that we cannot use the attribute `AsLinearGeneralMappingByImages'
## because the source may be infinite dimensional, i.e., we cannot write
## down the left module general mapping.
##
DeclareRepresentation( "IsOperationAlgebraHomomorphismDefaultRep",
IsAlgebraHomomorphism and IsAdditiveElementWithInverse
and IsAttributeStoringRep,
[ "basis", "operation",
"basisImage", "preimagesBasisImage" ] );
#############################################################################
##
#M ViewObj( <ophom> ) . . . . . . . . for an operation algebra homomorphism
##
InstallMethod( ViewObj,
"for an operation algebra homomorphism",
[ IsOperationAlgebraHomomorphismDefaultRep ],
function( ophom )
Print( "<op. hom. ", Source( ophom ), " -> matrices of dim. ",
Length( BasisVectors( ophom!.basis ) ), ">" );
end );
#############################################################################
##
#M PrintObj( <ophom> ) . . . . . . . . for an operation algebra homomorphism
##
InstallMethod( PrintObj,
"for an operation algebra homomorphism",
[ IsOperationAlgebraHomomorphismDefaultRep ],
function( ophom )
if ophom!.operation = OnRight then
Print( "OperationAlgebraHomomorphism( ",
Source( ophom ), ", ", ophom!.basis, " )" );
else
Print( "OperationAlgebraHomomorphism( ",
Source( ophom ), ", ", ophom!.basis, ", ",
ophom!.operation, " )" );
fi;
end );
#############################################################################
##
#F InducedLinearAction( <basis>, <elm>, <opr> )
##
InstallGlobalFunction( InducedLinearAction, function( basis, elm, opr )
return List( BasisVectors( basis ),
x -> Coefficients( basis, opr( x, elm ) ) );
end );
#############################################################################
##
#M MakePreImagesInfoOperationAlgebraHomomorphism( <ophom> )
##
InstallMethod( MakePreImagesInfoOperationAlgebraHomomorphism,
"for an operation algebra homomorphism",
[ IsOperationAlgebraHomomorphismDefaultRep ],
function( ophom )
local A, # source of the general mapping
F, # left acting domain
origgenerators, # list of algebra generators of the preimage
origgenimages, # list of images of `origgenerators'
I, # image of the mapping
genimages, # list of left module generators of the image
preimages, # list of preimages of `genimages'
maxdim, # upper bound on the dimension
MB, # mutable basis of the image
dim, # dimension of the actual left module
len, # number of algebra generators
i, j, # loop variables
gen, # loop over generators
prod; #
A:= Source( ophom );
F:= LeftActingDomain( A );
dim:= Length( BasisVectors( ophom!.basis ) );
if IsRingWithOne( A ) then
origgenerators:= GeneratorsOfAlgebraWithOne( A );
origgenimages:= List( origgenerators,
a -> InducedLinearAction( ophom!.basis, a, ophom!.operation ) );
if IsEmpty( origgenimages ) then
I:= FLMLORWithOneByGenerators( F, origgenimages,
Immutable( NullMat( F, dim, dim ) ) );
else
I:= FLMLORWithOneByGenerators( F, origgenimages );
fi;
else
origgenerators:= GeneratorsOfAlgebra( A );
origgenimages:= List( origgenerators,
a -> InducedLinearAction( ophom!.basis, a, ophom!.operation ) );
if IsEmpty( origgenimages ) then
I:= FLMLORByGenerators( F, origgenimages,
Immutable( NullMat( F, dim, dim ) ) );
else
I:= FLMLORByGenerators( F, origgenimages );
fi;
fi;
preimages := [ One( A ) ];
genimages := [ InducedLinearAction( ophom!.basis, One( A ),
ophom!.operation ) ];
maxdim:= dim^2;
# $A_1$
MB:= MutableBasis( F, genimages, Zero( I ) );
dim:= 0;
len:= Length( origgenimages );
while dim < NrBasisVectors( MB ) and NrBasisVectors( MB ) < maxdim do
# `MB' is a mutable basis of $A_i$.
dim:= NrBasisVectors( MB );
# Compute $\bigcup_{g \in S} ( A_i g \cup A_i g )$.
for i in [ 1 .. len ] do
gen:= origgenimages[i];
for j in [ 1 .. Length( genimages ) ] do
prod:= genimages[j] * gen;
if not IsContainedInSpan( MB, prod ) then
Add( genimages, prod );
Add( preimages, preimages[j] * origgenerators[i] );
CloseMutableBasis( MB, prod );
fi;
od;
od;
od;
# Set the desired components.
ophom!.basisImage:= BasisNC( I, genimages );
ophom!.preimagesBasisImage:= Immutable( preimages );
end );
#############################################################################
##
#M ImagesRepresentative( <ophom>, <elm> ) . . . . . . . . for op. alg. hom.
##
InstallMethod( ImagesRepresentative,
"for an operation algebra homomorphism, and an element",
FamSourceEqFamElm,
[ IsOperationAlgebraHomomorphismDefaultRep, IsRingElement ],
function( ophom, elm )
return InducedLinearAction( ophom!.basis, elm, ophom!.operation );
end );
#############################################################################
##
#M PreImagesRepresentative( <ophom>, <mat> )
##
BindGlobal( "PreImagesRepresentativeOperationAlgebraHomomorphism", function( ophom, mat )
if not IsBound( ophom!.basisImage ) then
MakePreImagesInfoOperationAlgebraHomomorphism( ophom );
fi;
mat:= Coefficients( ophom!.basisImage, mat );
if mat <> fail then
mat:= LinearCombination( ophom!.preimagesBasisImage, mat );
fi;
return mat;
end );
InstallMethod( PreImagesRepresentative,
"for an operation algebra homomorphism, and an element",
FamRangeEqFamElm,
[ IsOperationAlgebraHomomorphismDefaultRep, IsMatrix ],
PreImagesRepresentativeOperationAlgebraHomomorphism );
#############################################################################
##
#R IsAlgebraHomomorphismFromFpRep
##
## is a representation of operation homomorphisms from f.~p. FLMLORs
## to matrix FLMLORs.
## Contrary to `IsOperationAlgebraHomomorphismDefaultRep', no basis of the
## source is needed, computing images is done via `MappedExpression'.
##
## Defining components are
##
## `Agenerators'
## generators of the f.~p. algebra
##
## `Agenimages'
## images of `Agenerators'
##
## Preimages can be computed using the components
##
## `basisImage'
## basis of the image
##
## `preimagesBasisImage'
## list of preimages of the basis vectors of `basisImage'.
##
## (This works analogously to `IsOperationAlgebraHomomorphismDefaultRep'.)
## These components are computed as soon as they are needed.
##
## Note that also here, we cannot use the attribute
## `AsLinearGeneralMappingByImages'.
##
DeclareRepresentation( "IsAlgebraHomomorphismFromFpRep",
IsAlgebraHomomorphism and IsAdditiveElementWithInverse
and IsAttributeStoringRep,
[ "Agenerators", "Agenimages",
"basisImage", "preimagesBasisImage" ] );
#############################################################################
##
#M ViewObj( <ophom> ) . . . . . . . . for an algebra homomorphism from f.p.
##
InstallMethod( ViewObj,
"for an alg. hom. from f. p. algebra",
[ IsAlgebraHomomorphismFromFpRep ],
function( ophom )
Print( "<op. hom. ", Source( ophom ), " -> matrices of dim. ",
Length( ophom!.Agenimages[1] ), ">" );
end );
#############################################################################
##
#M PrintObj( <hom> ) . . . . . . . . . for an algebra homomorphism from f.p.
##
InstallMethod( PrintObj,
"for an alg. hom. from f. p. algebra",
[ IsAlgebraHomomorphismFromFpRep ],
function( hom )
Print( "AlgebraHomomorphismByImages( ",
Source( hom ), ", ", Range( hom ), ", ",
hom!.Agenerators, ", ", hom!.Agenimages, " )" );
end );
#T this does not admit to recover the homomorphism from the printed data;
#T in fact we have no function to construct such a homomorphism ...
#############################################################################
##
#M MakePreImagesInfoOperationAlgebraHomomorphism( <ophom> )
##
InstallMethod( MakePreImagesInfoOperationAlgebraHomomorphism,
"for an alg. hom. from f. p. algebra",
[ IsAlgebraHomomorphismFromFpRep ],
function( ophom )
local A, # source of the general mapping
F, # left acting domain
I, # image of the homomorphism
origgenerators, # list of algebra generators of the preimage
origgenimages, # list of images of `origgenerators'
genimages, # list of left module generators of the image
preimages, # list of preimages of `genimages'
maxdim, # upper bound on the dimension
MB, # mutable basis of the image
dim, # dimension of the actual left module
len, # number of algebra generators
i, j, # loop variables
gen, # loop over generators
prod; #
A:= Source( ophom );
F:= LeftActingDomain( A );
I:= ImagesSource( ophom );
origgenerators := ophom!.Agenerators;
origgenimages := ophom!.Agenimages;
dim:= Length( origgenimages[1] );
if dim = 0 then
ophom!.basisImage:= BasisNC( I, [] );
ophom!.preimagesBasisImage:= Immutable( [] );
return;
fi;
maxdim:= dim^2;
preimages := [ One( A ) ];
genimages := [ One( origgenimages[1] ) ];
# $A_1$
MB:= MutableBasis( F, genimages, Zero( I ) );
dim:= 0;
len:= Length( origgenimages );
while dim < NrBasisVectors( MB ) and NrBasisVectors( MB ) < maxdim do
# `MB' is a mutable basis of $A_i$.
dim:= NrBasisVectors( MB );
# Compute $\bigcup_{g \in S} ( A_i g \cup A_i g )$.
for i in [ 1 .. len ] do
gen:= origgenimages[i];
for j in [ 1 .. Length( genimages ) ] do
prod:= genimages[j] * gen;
if not IsContainedInSpan( MB, prod ) then
Add( genimages, prod );
Add( preimages, preimages[j] * origgenerators[i] );
CloseMutableBasis( MB, prod );
fi;
od;
od;
od;
# Set the desired components.
ophom!.basisImage:= BasisNC( I, genimages );
ophom!.preimagesBasisImage:= Immutable( preimages );
end );
#############################################################################
##
#M ImagesRepresentative( <ophom>, <elm> ) . . . . . . . . for op. alg. hom.
##
InstallMethod( ImagesRepresentative,
"for an alg. hom. from f. p. algebra, and an element",
FamSourceEqFamElm,
[ IsAlgebraHomomorphismFromFpRep, IsRingElement ],
function( ophom, elm )
return MappedExpression( elm, ophom!.Agenerators, ophom!.Agenimages );
end );
#############################################################################
##
#M PreImagesRepresentative( <ophom>, <mat> )
##
InstallMethod( PreImagesRepresentative,
"for an alg. hom. from f. p. algebra, and an element",
FamRangeEqFamElm,
[ IsAlgebraHomomorphismFromFpRep, IsMatrix ],
PreImagesRepresentativeOperationAlgebraHomomorphism );
#############################################################################
##
#M OperationAlgebraHomomorphism( <A>, <basis>, <opr> )
##
InstallMethod( OperationAlgebraHomomorphism,
"for a FLMLOR, a basis, and a function",
[ IsFLMLOR, IsBasis, IsFunction ],
function( A, basis, opr )
local ophom, image;
# Make the general mapping.
ophom:= Objectify( NewType( GeneralMappingsFamily(
ElementsFamily( FamilyObj( A ) ),
CollectionsFamily( FamilyObj(
LeftActingDomain( A ) ) ) ),
IsSPGeneralMapping
and IsAlgebraHomomorphism
and IsOperationAlgebraHomomorphismDefaultRep ),
rec(
operation := opr,
basis := basis
) );
SetSource( ophom, A );
# Handle the case that the basis is empty.
if IsEmpty( basis ) then
image := NullAlgebra( LeftActingDomain( A ) );
ophom!.basisImage := Basis( image );
ophom!.preimagesBasisImage := Immutable( [ Zero( A ) ] );
SetRange( ophom, image );
SetKernelOfAdditiveGeneralMapping( ophom, A );
SetIsSurjective( ophom, true );
fi;
# Return the operation homomorphism.
return ophom;
end );
#############################################################################
##
#M OperationAlgebraHomomorphism( <A>, <C> )
##
## Add the default argument `OnRight'.
##
InstallOtherMethod( OperationAlgebraHomomorphism,
"for a FLMLOR and a collection (add `OnRight' argument)",
[ IsFLMLOR, IsCollection ],
function( A, C )
return OperationAlgebraHomomorphism( A, C, OnRight );
end );
#############################################################################
##
#M OperationAlgebraHomomorphism( <A>, <V>, <opr> )
##
## For a finite dimensional free left module <V> with known generators,
## we assume that a basis can be computed.
##
InstallOtherMethod( OperationAlgebraHomomorphism,
"for a FLMLOR, a free left module with known generators, and a function",
[ IsFLMLOR,
IsFreeLeftModule and IsFiniteDimensional and HasGeneratorsOfLeftModule,
IsFunction ],
function( A, V, opr )
return OperationAlgebraHomomorphism( A, Basis( V ), opr );
end );
#############################################################################
##
#M Range( <ophom> ) . . . . . . . . . . for operation algebra homomorphism
##
## An operation algebra homomorphism that does not know its range cannot be
## forced to be surjective; so we may choose a full matrix FLMLOR.
##
InstallMethod( Range,
"for operation algebra homomorphism (set full matrix FLMLOR)",
[ IsOperationAlgebraHomomorphismDefaultRep ],
ophom -> FullMatrixFLMLOR( LeftActingDomain( Source( ophom ) ),
Length( BasisVectors( ophom!.basis ) ) ) );
#############################################################################
##
#M KernelOfAdditiveGeneralMapping( <ophom> ) . . for operation algebra hom.
##
## For a finite dimensional acting algebra, we compute a basis of the kernel
## by solving a linear equation system.
##
InstallMethod( KernelOfAdditiveGeneralMapping,
"for operation algebra hom. with fin. dim. source",
[ IsMapping and IsOperationAlgebraHomomorphismDefaultRep ],
function( ophom )
local A, # source of the homomorphism
BA, # basis of `A'
BV, # basis of the module
opr, # operation of `A' on the vectors
nullsp; # coefficients vectors of a basis of the kernel
A:= Source( ophom );
if IsTrivial( A ) then
return A;
elif not IsFiniteDimensional( A ) then
TryNextMethod();
fi;
BA:= Basis( A );
BV:= ophom!.basis;
opr:= ophom!.operation;
nullsp:= NullspaceMat( List( BA,
a -> Concatenation( List( BV,
v -> Coefficients( BV, opr( v, a ) ) ) ) ) );
nullsp:= SubFLMLORNC( A,
List( nullsp, v -> LinearCombination( BA, v ) ), "basis" );
SetIsTwoSidedIdealInParent( nullsp, true );
return nullsp;
end );
#############################################################################
##
#M RepresentativeLinearOperation( <A>, <v>, <w>, <opr> )
##
## Let <A> be a finite dimensional algebra over the ring $R$,
## <v> and <w> either elements in <A> or tuples of elements in <A>,
## and <opr> equal to `OnRight' or `OnTuples', respectively.
## We compute an element of <A> that maps <v> to <w>.
##
## We compute the coefficients $a_i$ in the equation system
## $\sum_{i=1}^n a_i <opr>( <v>, b_i ) = <w>$,
## where $(b_1, b_2, \ldots, b_n)$ is a basis of <A>.
##
## For a tuple $(v_1, \ldots, v_k)$ of vectors we simply replace $v b_i$ by
## the concatenation of the $v_j b_i$ for all $j$, and replace $w$ by the
## concatenation $(w_1, \ldots, w_k)$, and solve this system.
##
## (There are also methods for matrix algebras acting on row vectors via
## `OnRight' or `OnTuples'.)
##
InstallMethod( RepresentativeLinearOperation,
"for a FLMLOR, two elements in it, and `OnRight'",
IsCollsElmsElmsX,
[ IsFLMLOR, IsVector, IsVector, IsFunction ],
function( A, v, w, opr )
local B, vectors, a;
if not ( v in A and w in A and opr = OnRight ) then
TryNextMethod();
fi;
if IsTrivial( A ) then
if IsZero( w ) then
return Zero( A );
else
return fail;
fi;
fi;
B:= Basis( A );
vectors:= BasisVectors( B );
# Compute the matrix of the equation system,
# the coefficient vector $a$, \ldots
a:= SolutionMat( List( vectors, x -> Coefficients( B, v * x ) ),
Coefficients( B, w ) );
if a = fail then
return fail;
fi;
# \ldots and the representative.
return LinearCombination( B, a );
end );
InstallOtherMethod( RepresentativeLinearOperation,
"for a FLMLOR, two tuples of elements in it, and `OnTuples'",
IsFamFamFamX,
[ IsFLMLOR, IsHomogeneousList, IsHomogeneousList, IsFunction ],
function( A, vs, ws, opr )
local B, vectors, a;
if not ( Length( vs ) = Length( ws )
and IsSubset( A, vs ) and IsSubset( A, ws )
and opr = OnTuples ) then
TryNextMethod();
fi;
if IsTrivial( A ) then
if ForAll( ws, IsZero ) then
return Zero( A );
else
return fail;
fi;
fi;
B:= Basis( A );
vectors:= BasisVectors( B );
# Compute the matrix of the equation system,
# the coefficient vector $a$, \ldots
a:= SolutionMat( List( vectors,
x -> Concatenation( List( vs,
v -> Coefficients( B, v * x ) ) ) ),
Concatenation( List( ws,
w -> Coefficients( B, w ) ) ) );
if a = fail then
return fail;
fi;
# \ldots and the representative.
return LinearCombination( B, a );
end );
#############################################################################
##
## 3. methods for natural homomorphisms from algebras
##
#M NaturalHomomorphismByIdeal( <A>, <I> ) . . . . . map onto factor algebra
##
## <#GAPDoc Label="NaturalHomomorphismByIdeal_algebras">
## <ManSection>
## <Meth Name="NaturalHomomorphismByIdeal" Arg='A, I'
## Label="for an algebra and an ideal"/>
##
## <Description>
## For an algebra <A>A</A> and an ideal <A>I</A> in <A>A</A>,
## the return value of <Ref Oper="NaturalHomomorphismByIdeal"/>
## is a homomorphism of algebras, in particular the range of this mapping
## is also an algebra.
## <P/>
## <Example><![CDATA[
## gap> L:= FullMatrixLieAlgebra( Rationals, 3 );;
## gap> C:= LieCentre( L );
## <two-sided ideal in <Lie algebra of dimension 9 over Rationals>,
## (dimension 1)>
## gap> hom:= NaturalHomomorphismByIdeal( L, C );
## <linear mapping by matrix, <Lie algebra of dimension
## 9 over Rationals> -> <Lie algebra of dimension 8 over Rationals>>
## gap> ImagesSource( hom );
## <Lie algebra of dimension 8 over Rationals>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
#############################################################################
##
#M NaturalHomomorphismByIdeal( <A>, <triv> ) . . . . . . onto trivial FLMLOR
##
## Return the identity mapping.
##
InstallMethod( NaturalHomomorphismByIdeal,
"for FLMLOR and trivial FLMLOR",
IsIdenticalObj,
[ IsFLMLOR, IsFLMLOR and IsTrivial ], SUM_FLAGS,
function( A, I )
return IdentityMapping( A );
end );
#############################################################################
##
#M NaturalHomomorphismByIdeal( <A>, <I> ) . . . . for two fin. dim. FLMLORs
##
## We return a left module m.b.m. from <A> onto the factor `<A>/<I>'.
## The image is a s.c. algebra if <I> is nontrivial,
## and otherwise the identity mapping of <A>.
##
InstallMethod( NaturalHomomorphismByIdeal,
"for two finite dimensional FLMLORs",
IsIdenticalObj,
[ IsFLMLOR, IsFLMLOR ],
function( A, I )
local F, # left acting domain of `A'
zero, # zero of `F'
Ivectors, # basis vectors of a basis of `I'
mb, # mutable basis of `I'
compl, # basis vectors of a complement of `I' in `A'
gen, # loop over a basis of `A'
B, # basis of `A', through `I'
k, # length of `Ivectors'
n, # length of `compl'
T, # s.c. table of a basis of the image
i, j, # loop variables
coeff, # coefficients of a product
pos, # relevant positions
img, # image of the homomorphism
canbas, # canonical basis of the image
Bimgs, # images of the vectors of `B' under the hom.
nathom; # the homomorphism, result
# Check that the FLMLORs are finite dimensional.
if not IsFiniteDimensional( A ) or not IsFiniteDimensional( I ) then
TryNextMethod();
fi;
# If `A' is equal to `I', return a zero mapping.
if not IsIdeal( A, I ) then
Error( "<I> must be an ideal in <A>" );
elif Dimension( A ) = Dimension( I ) then
return ZeroMapping( A, NullAlgebra( LeftActingDomain( A ) ) );
fi;
# If `I' is trivial, return the identity mapping.
if IsTrivial( I ) then
return IdentityMapping( A );
fi;
# If the left acting domains are different, adjust them.
F:= LeftActingDomain( A );
if F <> LeftActingDomain( I ) then
F:= Intersection2( A, LeftActingDomain( I ) );
A:= AsFLMLOR( F, A );
I:= AsFLMLOR( F, I );
fi;
# Compute a basis of `A' through a basis of `I'.
Ivectors:= BasisVectors( Basis( I ) );
mb:= MutableBasis( F, Ivectors );
compl:= [];
for gen in BasisVectors( Basis( A ) ) do
if not IsContainedInSpan( mb, gen ) then
Add( compl, gen );
CloseMutableBasis( mb, gen );
fi;
od;
B:= BasisNC( A, Concatenation( Ivectors, compl ) );
# Compute the structure constants of the quotient algebra.
zero:= Zero( F );
k:= Length( Ivectors );
n:= Length( compl );
if HasIsCommutative( A ) and IsCommutative( A ) then
T:= EmptySCTable( n, Zero( F ), "symmetric" );
elif HasIsAnticommutative( A ) and IsAnticommutative( A ) then
T:= EmptySCTable( n, Zero( F ), "antisymmetric" );
else
T:= EmptySCTable( n, Zero( F ) );
fi;
for i in [ 1 .. n ] do
for j in [ 1 .. n ] do
coeff:= Coefficients( B, compl[i] * compl[j] ){ [ k+1 .. k+n ] };
pos:= Filtered( [ 1 .. n ], i -> coeff[i] <> zero );
if not IsEmpty( pos ) then
T[i][j]:= Immutable( [ pos, coeff{ pos } ] );
fi;
od;
od;
#T use (anti)symm. here!!!
# Compute the linear mapping by images.
img:= AlgebraByStructureConstants( F, T );
canbas:= CanonicalBasis( img );
zero:= zero * [ 1 .. n ];
Bimgs:= Concatenation( List( [ 1 .. k ], v -> zero ),
Immutable( IdentityMat( n, F ) ) );
nathom:= LeftModuleHomomorphismByMatrix( B, Bimgs, canbas );
#T take a special representation for nat. hom.s,
#T (just compute coefficients, and then choose a subset ...)
SetIsAlgebraWithOneHomomorphism( nathom, true );
SetIsInjective( nathom, false );
SetIsSurjective( nathom, true );
# Enter the preimages info.
nathom!.basisimage:= canbas;
nathom!.preimagesbasisimage:= Immutable( compl );
#T relations are not needed if the kernel is known ?
SetKernelOfAdditiveGeneralMapping( nathom, I );
# Run the implications for the factor.
UseFactorRelation( A, I, img );
return nathom;
end );
#############################################################################
##
## 4. methods for isomorphisms to matrix algebras
##
#############################################################################
##
#M IsomorphismMatrixFLMLOR( <A> ) . . . . . . for a fin. dim. assoc. FLMLOR
##
## A FLMLOR with a multiplicative neutral element acts faithfully on itself
## via right multiplication.
## So we get for an $n$ dimensional algebra a representation with matrices
## of dimension $n \times n$.
##
InstallMethod( IsomorphismMatrixFLMLOR,
"for a finite dimensional associative FLMLOR with identity",
[ IsFLMLOR ],
function( A )
local B, # basis of `A'
F, # left acting domain of `A'
I, # image of the isomorphism
map, # isomorphism, result
gens, # algebra generators of `A'
imgs, # images of `gens' under the action from the right
dim; # dimension of `A'
if IsSubalgebraFpAlgebra( A ) # avoid to call `IsFiniteDimensional'
# in this case
or not IsFiniteDimensional( A )
or not IsAssociative( A )
or MultiplicativeNeutralElement( A ) = fail then
TryNextMethod();
fi;
B:= Basis( A );
F:= LeftActingDomain( A );
if IsEmpty( B ) then
# Handle the case that `A' is trivial.
I:= NullAlgebra( F );
map:= LeftModuleHomomorphismByImagesNC( A, I, B, Basis( I ) );
SetRespectsMultiplication( map, true );
else
if IsRingWithOne( A ) then
gens:= GeneratorsOfAlgebraWithOne( A );
else
gens:= GeneratorsOfAlgebra( A );
fi;
imgs:= List( gens, a -> InducedLinearAction( B, a, OnRight ) );
if IsEmpty( imgs ) then
dim:= Dimension( A );
imgs[1]:= Immutable( NullMat( F, dim, dim ) );
fi;
I:= FLMLORByGenerators( F, imgs );
UseIsomorphismRelation( A, I );
# Make an operation algebra homomorphism.
map:= Objectify( NewType( GeneralMappingsFamily(
ElementsFamily( FamilyObj( A ) ),
ElementsFamily( FamilyObj( imgs ) ) ),
IsSPGeneralMapping
and IsAlgebraHomomorphism
and IsOperationAlgebraHomomorphismDefaultRep ),
rec(
operation := OnRight,
basis := B
) );
SetSource( map, A );
SetRange( map, I );
fi;
SetIsSurjective( map, true );
SetIsInjective( map, true );
return map;
end );
#############################################################################
##
## 5. methods for isomorphisms to f.p. algebras
##
#############################################################################
##
#M IsomorphismFpFLMLOR( <A> ) . . . . . . . . for a fin. dim. assoc. FLMLOR
##
## Construct the free (associative) algebra $F$ on generators of <A>,
## and factor out the two-sided ideal $I$ spanned by the structure relators
## w.r.t. a basis of <A>.
## Then clearly the kernel of the homomorphism from $F$ to <A> contains $I$,
## on the other hand any expression in the kernel can be reduced to a sum
## of generators modulo the structure relators of <A>, and this must be
## trivial since the images of generators were assumed to be linearly
## independent.
##
## We write down all relations to reduce words of length two to
## linear combinations of the generators.
## So it makes no difference whether the f.p. algebra is constructed
## from a free algebra or from a free associative algebra.
## But if <A> knows to be associative then we take a free associative
## algebra.
##
InstallMethod( IsomorphismFpFLMLOR,
"for a finite dimensional FLMLOR-with-one",
[ IsFLMLORWithOne ],
function( A )
local Agens, # list of algebra generators of `A'
F, # free (associative) algebra
Fgens, # list of images of `Agens'
generators, # list of left module generators of the preimage
genimages, # list of images of `generators'
left, # is it necessary to multiply also from the left?
maxdim, # upper bound on the dimension
MB, # mutable basis of the preimage
dim, # dimension of the actual left module
len, # number of algebra generators
i, j, # loop variables
gen, # loop over generators
prod, #
rels, # relators list
rel, # one relator
coeff, # coefficients of product of basis vectors
k, # loop over `coeff'
B, # basis of `A'
Fp, # f.p. algebra
Fam, # elements family of the family of `Fp'
map; # the isomorphism, result
if not IsFiniteDimensional( A ) then
TryNextMethod();
fi;
Agens:= GeneratorsOfAlgebraWithOne( A );
if HasIsAssociative( A ) and IsAssociative( A ) then
F:= FreeAssociativeAlgebraWithOne( LeftActingDomain( A ),
Length( Agens ) );
left:= false;
else
F:= FreeAlgebraWithOne( LeftActingDomain( A ), Length( Agens ) );
left:= true;
fi;
Fgens:= GeneratorsOfAlgebraWithOne( F );
generators := ShallowCopy( Agens );
genimages := ShallowCopy( Fgens );
if HasDimension( A ) then
maxdim:= Dimension( A );
else
maxdim:= infinity;
fi;
# $A_1$
MB:= MutableBasis( LeftActingDomain( A ), generators,
Zero( A ) );
dim:= 0;
len:= Length( Agens );
while dim < NrBasisVectors( MB ) and NrBasisVectors( MB ) < maxdim do
# `MB' is a mutable basis of $A_i$.
dim:= NrBasisVectors( MB );
# Compute $\bigcup_{g \in S} ( A_i g \cup A_i g )$.
for i in [ 1 .. len ] do
gen:= Agens[i];
for j in [ 1 .. Length( generators ) ] do
prod:= generators[j] * gen;
if not IsContainedInSpan( MB, prod ) then
Add( generators, prod );
Add( genimages, genimages[j] * Fgens[i] );
CloseMutableBasis( MB, prod );
fi;
od;
od;
if left then
# Compute $\bigcup_{g \in S} ( A_i g \cup g A_i )$.
for i in [ 1 .. len ] do
gen:= Agens[i];
for j in [ 1 .. Length( generators ) ] do
prod:= gen * generators[j];
if not IsContainedInSpan( MB, prod ) then
Add( generators, prod );
Add( genimages, Fgens[i] * genimages[j] );
CloseMutableBasis( MB, prod );
fi;
od;
od;
fi;
od;
B:= BasisNC( A, generators );
dim:= Length( generators );
# Construct the relators given by the multiplication table.
rels:= [];
for i in [ 1 .. dim ] do
for j in [ 1 .. dim ] do
coeff:= Coefficients( B, generators[i] * generators[j] );
rel:= genimages[i] * genimages[j];
for k in [ 1 .. dim ] do
rel:= rel - coeff[k] * genimages[k];
od;
if not IsZero( rel ) then
Add( rels, rel );
fi;
od;
od;
# Remove duplicate relators.
rels:= Set( rels );
# Construct the f.p. algebra.
Fp:= FactorFreeAlgebraByRelators( F, rels );
Fam:= ElementsFamily( FamilyObj( Fp ) );
# Set useful information.
UseIsomorphismRelation( A, Fp );
# Map the elements of the free algebra into the f.p. algebra.
Fgens:= List( Fgens, a -> ElementOfFpAlgebra( Fam, a ) );
genimages:= List( genimages, a -> ElementOfFpAlgebra( Fam, a ) );
# Set the info to compute with a basis of the f.p. algebra.
SetNiceAlgebraMonomorphism( Fp,
Objectify( NewType( GeneralMappingsFamily(
ElementsFamily( FamilyObj( Fp ) ),
ElementsFamily( FamilyObj( A ) ) ),
IsSPGeneralMapping
and IsAlgebraHomomorphism
and IsAlgebraHomomorphismFromFpRep ),
rec( Agenerators := Fgens,
Agenimages := Agens,
basisImage := B,
preimagesBasisImage := genimages ) ) );
# We know left module generators of the f.p. algebra,
# and we know the isomorphic nice free left module.
# (Note that in general, `NiceAlgebraMonomorphism' is valid also for
# subalgebras.)
SetGeneratorsOfLeftModule( Fp, genimages );
SetNiceFreeLeftModule( Fp, UnderlyingLeftModule( B ) );
# Construct the isomorphism.
map:= AlgebraWithOneHomomorphismByImagesNC( A, Fp, B, genimages );
SetIsSurjective( map, true );
SetIsInjective( map, true );
# Return the isomorphism.
return map;
end );
#T special representation to improve computing preimages?
#T (the element in the nice module is first computed, then decomposed
#T and composed again; one can avoid the last two steps)
#############################################################################
##
#M IsomorphismFpFLMLOR( <A> ) . . . . . . . . . . . . . . . for f.p. FLMLOR
##
## Return the identity mapping.
##
InstallMethod( IsomorphismFpFLMLOR,
"for f.p. FLMLOR (return the identity mapping)",
[ IsSubalgebraFpAlgebra ], SUM_FLAGS,
IdentityMapping );
#############################################################################
##
#M IsomorphismSCFLMLOR( <A> ) . . . . . . . . . . . . . . . . for a FLMLOR
##
InstallMethod( IsomorphismSCFLMLOR,
"for a finite dimensional FLMLOR (delegate to the method for a basis)",
[ IsFLMLOR ],
A -> IsomorphismSCFLMLOR( Basis( A ) ) );
#############################################################################
##
#M IsomorphismSCFLMLOR( <B> ) . . . . . . . . . . . for a basis of a FLMLOR
##
InstallMethod( IsomorphismSCFLMLOR,
"for a basis (of a finite dimensional FLMLOR)",
[ IsBasis ],
function( B )
local A, # underlying FLMLOR of `B'
T, # structure constants table w.r.t. `B'
I, # s.c. FLMLOR, image of the isomorphism
map; # isomorphism from `A' to `I', result
A:= UnderlyingLeftModule( B );
if not IsFLMLOR( A ) then
Error( "<A> must be a FLMLOR" );
fi;
# Construct the image.
T:= StructureConstantsTable( B );
I:= AlgebraByStructureConstants( LeftActingDomain( A ), T );
UseIsomorphismRelation( A, I );
# Construct the isomorphism.
map:= LeftModuleHomomorphismByImagesNC( A, I, B, CanonicalBasis( I ) );
SetIsBijective( map, true );
SetIsAlgebraHomomorphism( map, true );
if IsFLMLORWithOne( A ) then
SetRespectsOne( map, true );
fi;
# Return the result.
return map;
end );
#############################################################################
##
#M IsomorphismSCFLMLOR( <A> ) . . . . . . . . . . . . . . . for s.c. FLMLOR
##
## Return the identity mapping.
##
InstallMethod( IsomorphismSCFLMLOR,
"for s.c. FLMLOR (return the identity mapping)",
[ IsFLMLOR and IsSCAlgebraObjCollection ], SUM_FLAGS,
IdentityMapping );
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