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#####################################################################################
#
# liering.gi Serena Cicalo' and Willem de Graaf
#
#
# The package LieRing is free software; you can redistribute it and/or modify it under the
# terms of the GNU General Public License as published by the Free Software Foundation;
# either version 2 of the License, or (at your option) any later version.
# A Lie ring element is represented as a list [ index, cft, index, cft, ... ]
############################################################################
##
#M ObjByExtRep( <fam>, <list> )
#M ExtRepOfObj( <obj> )
##
InstallMethod( ObjByExtRep,
"for family of LR elements, and list",
true, [ IsLRElementFamily, IsList ], 0,
function( fam, list )
return Objectify( fam!.packedLRElementDefaultType,
[ Immutable(list) ] );
end );
InstallMethod( ExtRepOfObj,
"for an LR element",
true, [ IsLRElement ], 0,
function( obj )
return obj![1];
end );
###########################################################################
##
#M PrintObj( <m> ) . . . . . . . . . . . . . . . . for an LR element
##
InstallMethod( PrintObj,
"for LR element",
true, [IsLRElement], 0,
function( x )
local lst, n, i;
# This function prints an element of a Lie ring.
lst:= x![1];
n:= Length( lst );
if n = 0 then Print("0"); fi;
for i in [1,3..n-1] do
if lst[i+1] = -1 then
Print("-");
elif lst[i+1]<>1 then
if i<>1 then Print("+"); fi;
Print(lst[i+1],"*");
elif i <> 1 then
Print("+");
fi;
Print("v_",lst[i]);
od;
end );
#############################################################################
##
#M ZeroOp( <m> ) . . . . . . . . . . . . . . . for a LR element
#M \<( <m1>, <m2> ) . . . . . . . . . . . . . . for two LR elements
#M \=( <m1>, <m2> ) . . . . . . . . . . . . . . for two LR elements
#M \+( <m1>, <m2> ) . . . . . . . . . . . . . . for two LR elements
#M \-( <m> ) . . . . . . . . . . . . . . for a LR element
#M \in( <U>, <u> ) . . . . . . . . . . . . . . for LR, and element
##
InstallMethod( ZeroOp,
"for LR element",
true, [ IsLRElement ], 0,
function( x )
return ObjByExtRep( FamilyObj( x ), [ ] );
end );
InstallMethod( \<,
"for two LR elements",
IsIdenticalObj, [ IsLRElement ,IsLRElement ], 0,
function( x, y )
return x![1]< y![1];
end );
InstallMethod( \=,
"for two LR elements",
IsIdenticalObj, [ IsLRElement ,IsLRElement ], 0,
function( x, y )
return x![1] = y![1];
end );
InstallMethod( \+,
"for two LR elements",
IsIdenticalObj, [ IsLRElement ,IsLRElement ], 0,
function( x, y )
local tor, sum, i;
tor:= FamilyObj( x )!.torsion;
sum:= ZIPPED_SUM_LISTS( x![1], y![1], 0, [ \<, \+ ] );
for i in [2,4..Length(sum)] do
if tor[ sum[i-1] ] <> 0 then
sum[i]:= sum[i] mod tor[sum[i-1]];
if sum[i] = 0 then
Unbind( sum[i-1] );
Unbind( sum[i] );
fi;
fi;
od;
sum:= Filtered( sum, x -> IsBound(x) );
return ObjByExtRep( FamilyObj(x), sum );
end );
InstallMethod( AdditiveInverseSameMutability,
"for LR element",
true, [ IsLRElement ], 0,
function( x )
local ex, i;
ex:= ShallowCopy(x![1]);
for i in [2,4..Length(ex)] do
ex[i]:= -ex[i];
od;
return ObjByExtRep( FamilyObj(x), ex );
end );
InstallMethod( AdditiveInverseMutable,
"for LR element",
true, [ IsLRElement ], 0,
function( x )
local ex, i;
ex:= ShallowCopy(x![1]);
for i in [2,4..Length(ex)] do
ex[i]:= -ex[i];
od;
return ObjByExtRep( FamilyObj(x), ex );
end );
#############################################################################
##
#M \*( <scal>, <m> ) . . . . . . . . .for a scalar and a LR element
#M \*( <m>, <scal> ) . . . . . . . . .for a scalar and a LR element
##
InstallMethod( \*,
"for scalar and LR element",
true, [ IS_INT, IsLRElement ], 0,
function( scal, x )
local ex, i, tor;
if IsZero( scal ) then return Zero(x); fi;
ex:= ShallowCopy( x![1] );
tor:= FamilyObj(x)!.torsion;
for i in [2,4..Length(ex)] do
ex[i]:= scal*ex[i];
if tor[ ex[i-1] ] <> 0 then
ex[i]:= ex[i] mod tor[ ex[i-1] ];
if ex[i] = 0 then
Unbind( ex[i-1] );
Unbind( ex[i] );
fi;
fi;
od;
ex:= Filtered( ex, x -> IsBound( x ) );
return ObjByExtRep( FamilyObj(x), ex );
end);
InstallMethod( \*,
"for LR element and scalar",
true, [ IsLRElement, IsScalar ], 0,
function( x, scal )
local ex, i, tor;
if IsZero( scal ) then return Zero(x); fi;
ex:= ShallowCopy( x![1] );
tor:= FamilyObj(x)!.torsion;
for i in [2,4..Length(ex)] do
ex[i]:= scal*ex[i];
if tor[ ex[i-1] ] <> 0 then
ex[i]:= ex[i] mod tor[ ex[i-1] ];
if ex[i] = 0 then
Unbind( ex[i-1] );
Unbind( ex[i] );
fi;
fi;
od;
ex:= Filtered( ex, x -> IsBound( x ) );
return ObjByExtRep( FamilyObj(x), ex );
end);
InstallMethod( \in,
"for LR element and Lie Ring",
true, [ IsObject, IsLieRing ], 0,
function( x, L )
local B, cc;
if not IsLRElement(x) then return false; fi;
if IsStandardBasisOfLieRing( Basis(L) ) then
return IsIdenticalObj( ElementsFamily( FamilyObj(L) ), FamilyObj(x) );
else
B:= Basis(L);
cc:= SolutionIntMat( B!.mat, Coefficients( B!.ambient, x ) );
if cc <> fail then return true; else return false; fi;
fi;
end );
InstallMethod( \=,
"for two Lie Rings",
true, [ IsLieRing, IsLieRing ], 0,
function( K, L )
if not ForAll( GeneratorsOfAlgebra(K), x -> x in L ) then
return false;
else
return ForAll( GeneratorsOfAlgebra(L), x -> x in K );
fi;
end );
InstallMethod( PrintObj,
"for a Lie ring",
true, [IsLieRing], 0,
function( L )
Print("<Lie ring with ",Length(GeneratorsOfAlgebra(L))," generators>");
end );
InstallMethod( ViewObj,
"for a Lie ring",
true, [IsLieRing], 0,
function( L )
Print("<Lie ring with ",Length(GeneratorsOfAlgebra(L))," generators>");
end );
#############################################################################
##
#M \*( <x>, <y> ) . . . . . . . . . . . . . . for two LR elements
##
##
InstallMethod( \*,
"for two Lie Ring elements",
IsIdenticalObj, [ IsLRElement, IsLRElement ], 0,
function( x, y )
local ex, ey, i, j, k, tor, T, res, cc, cf, prod;
ex:= ExtRepOfObj(x);
ey:= ExtRepOfObj(y);
tor:= FamilyObj(x)!.torsion;
T:= FamilyObj( x )!.multTab;
res:= [ ]; # list with holes, i-th position has cft of ith basis element in result.
for i in [1,3..Length(ex)-1] do
for j in [1,3..Length(ey)-1] do
cc:= T[ex[i]][ey[j]];
cf:= ex[i+1]*ey[j+1];
for k in [1..Length(cc[1])] do
if not IsBound( res[cc[1][k]] ) then
res[cc[1][k]]:= cc[2][k]*cf;
else
res[cc[1][k]]:= res[cc[1][k]] + cc[2][k]*cf;
fi;
od;
od;
od;
prod:= [ ];
for k in [1..Length(tor)] do
if IsBound(res[k]) then
cf:=res[k];
if tor[k] <> 0 then
cf:= cf mod tor[k];
fi;
if cf <> 0 then
Add( prod, k ); Add( prod, cf );
fi;
fi;
od;
return ObjByExtRep( FamilyObj(x), prod );
end);
InstallMethod( ViewObj,
"for a Lie ring",
true, [IsLieRing], 100,
function( L )
Print("<Lie ring with ",Length(GeneratorsOfLeftOperatorRing(L))," generators>");
end );
#########################################################################
##
#M LieRingByStructureConstants( <tors>, <T> )
##
InstallMethod( LieRingByStructureConstants,
"for structure constants table, and torsion list",
true, [ IsList, IsList ], 0,
function( tors, T )
local fam, bas, L, B;
fam:= NewFamily( "LREltFam", IsLRElement );
fam!.packedLRElementDefaultType:= NewType( fam, IsPackedElementDefaultRep );
fam!.torsion:= tors;
fam!.multTab:= T;
bas:= List( [1..Length(tors)], x -> ObjByExtRep( fam, [ x, 1 ] ) );
L:= Objectify( NewType( CollectionsFamily( FamilyObj( bas[1] ) ),
IsLieRing
and IsAttributeStoringRep ),
rec() );
B:= Objectify( NewType( FamilyObj( L ),
IsFiniteBasisDefault and
IsBasisOfLieRing and
IsAttributeStoringRep ),
rec() );
SetLeftActingDomain( L, Integers );
SetGeneratorsOfLeftOperatorRing( L, bas );
SetGeneratorsOfAdditiveGroup( L, bas );
SetUnderlyingLeftModule( B, L );
SetBasisVectors( B, bas );
SetTorsion( B, tors );
SetStructureConstantsTable( B, T );
SetIsStandardBasisOfLieRing( B, true );
SetBasis( L, B );
SetParent( L, L );
return L;
end );
InstallAccessToGenerators( IsLieRing,
"structure constants Lie ring",
Basis );
InstallMethod( Coefficients,
"for basis of a Lie ring, and Lie ring element",
true, [ IsBasisOfLieRing,
IsLRElement ], 0,
function( B, v )
local cf, cc, k, heads, tb, cfs, y, ey, pos, a, b, r, t;
if IsStandardBasisOfLieRing(B) then
cf:= 0*FamilyObj( v )!.torsion;
cc:= v![1];
for k in [1,3..Length(cc)-1] do
cf[cc[k]]:= cc[k+1];
od;
return cf;
else
cc:= SolutionIntMat( B!.mat, Coefficients( B!.ambient, v ) );
cc:= cc{[1..Length(B)]};
t:= Torsion( B );
for k in [1..Length(t)] do
if t[k] <> 0 then cc[k]:= cc[k] mod t[k]; fi;
od;
return cc;
fi;
end );
LRPrivateFunctions.sub_lie_ring:= function( arg )
local B, n, bas, heads, foundbasis, newelms, x, c, cf, i, pos, a, b,
r, q, eqns, row, A, S, tors, Q, Qi, nb, K, BK, L, vv, is_bas,
start, mat, t, v;
L:= arg[1];
vv:= arg[2];
if Length(arg)=3 then
is_bas:= true;
else
is_bas:= false;
fi;
if Length(vv)=0 then vv:= [ Zero(L) ]; fi;
B:= Basis( Parent(L) );
n:= Length( BasisVectors( B ) );
bas:= TriangulizedIntegerMat( List( vv, x -> Coefficients( B, x ) ) );
bas:= Filtered( bas, x -> x <> 0*x );
if bas = [ ] then
K:= Objectify( NewType( FamilyObj( L ),
IsLieRing
and IsAttributeStoringRep ),
rec() );
BK:= Objectify( NewType( FamilyObj( K ),
IsFiniteBasisDefault and
IsBasisOfLieRing and
IsAttributeStoringRep ),
rec() );
# BK has a record component with useful information:
#
# in this case it is empty
BK!.mat:= [ ];
BK!.ambient:= B;
SetLeftActingDomain( K, Integers );
SetGeneratorsOfLeftOperatorRing( K, [ ] );
SetUnderlyingLeftModule( BK, K );
SetBasisVectors( BK, [ ] );
SetTorsion( BK, [ ] );
SetIsStandardBasisOfLieRing( BK, false );
SetBasis( K, BK );
SetParent( K, Parent(L) );
return K;
fi;
heads:= List( bas, PositionNonZero );
if not is_bas then
foundbasis:= false;
while not foundbasis do
newelms:= [ ];
for x in vv do
for c in bas do
cf:= Coefficients( B, x*LinearCombination(B,c) );
for i in [1..Length(heads)] do
pos:= heads[i];
a:= cf[ pos ]; b:= bas[i][pos];
r:= a mod b;
q:= (a-r)/b;
cf:= cf - q*bas[i];
od;
if cf <> 0*cf then
Add( newelms, cf );
fi;
od;
od;
if newelms = [ ] then
foundbasis:= true;
else
Append( bas, newelms );
bas:= TriangulizedIntegerMat( bas );
bas:= Filtered( bas, x -> x <> 0*x );
heads:= List( bas, PositionNonZero );
fi;
od;
fi;
# in order to get the normal basis with have to find
# the linear relations among the basis elements....
eqns:= ShallowCopy( bas );
for i in [1..n] do
row:= List( [1..n], x -> 0 );
row[ i ]:= Torsion(B)[i];
Add( eqns, row );
od;
A:= NullspaceIntMat( eqns );
A:= List( A, x -> x{[1..Length(bas)]} );
A:= Filtered( A, x -> x <> 0*x );
if Length(A) > 0 then
S:= SmithNormalFormIntegerMatTransforms( A );
start:= 1;
while start <= Length( S.normal ) and S.normal[start][start] = 1 do
start:= start+1;
od;
tors:= List( [start..Length(S.normal)], x -> S.normal[x][x] );
for i in [Length(S.normal)+1..Length(bas)] do Add( tors, 0 ); od;
Q:= S.coltrans;
Qi:= Q^-1;
nb:= List( Qi{[start..Length(Qi)]}, x -> LinearCombination( bas, x ) );
nb:= List( nb, x -> LinearCombination( B, x ) );
else
tors:= List( bas, x -> 0 );
nb:= bas;
nb:= List( nb, x -> LinearCombination( B, x ) );
fi;
K:= Objectify( NewType( FamilyObj( L ),
IsLieRing
and IsAttributeStoringRep ),
rec() );
BK:= Objectify( NewType( FamilyObj( K ),
IsFiniteBasisDefault and
IsBasisOfLieRing and
IsAttributeStoringRep ),
rec() );
# BK has a record component with useful information:
#
# .mat is the matrix of the basis vectors with respect to the
# "ambient" basis, concatenated with some rows that record the
# torsion of the "ambient" basis elements.
# .ambient is the basis of the "ambient" Lie ring.
mat:= List( nb, x -> Coefficients( B, x ) );
for i in [1..Length(Torsion(B))] do
t:= Torsion(B)[i];
if t <> 0 then
v:= List( B, x -> 0 );
v[i]:= t;
Add( mat, v );
fi;
od;
BK!.mat:= mat;
BK!.ambient:= B;
SetLeftActingDomain( K, Integers );
SetGeneratorsOfLeftOperatorRing( K, nb );
SetUnderlyingLeftModule( BK, K );
SetBasisVectors( BK, nb );
SetTorsion( BK, tors );
SetIsStandardBasisOfLieRing( BK, false );
SetBasis( K, BK );
SetParent( K, Parent(L) );
return K;
end;
InstallMethod( SubLieRing,
"for Lie ring and list of its elements",
true, [ IsLieRing,
IsList ], 0,
function( L, vv )
return LRPrivateFunctions.sub_lie_ring( L, vv );
end );
InstallOtherMethod( SubLieRing,
"for Lie ring and list of its elements and string",
true, [ IsLieRing,
IsLRElementCollection, IsString ], 0,
function( L, vv, str )
if str <> "basis" then
Error("The third argument has to be the string \"basis\".");
fi;
return LRPrivateFunctions.sub_lie_ring( L, vv, str );
end );
LRPrivateFunctions.ideal_lie_ring:= function( arg )
local B, n, bas, heads, foundbasis, newelms, x, c, cf, i, pos, a, b,
r, q, eqns, row, A, S, tors, Q, Qi, nb, K, BK, L, vv, is_bas,
start, ss, mat, t, v;
L:= arg[1];
vv:= arg[2];
if Length(arg)=3 then
is_bas:= true;
else
is_bas:= false;
fi;
if Length(vv)=0 then vv:= [ Zero(L) ]; fi;
B:= Basis( Parent(L) );
n:= Length( BasisVectors( B ) );
bas:= TriangulizedIntegerMat( List( vv, x -> Coefficients( B, x ) ) );
bas:= Filtered( bas, x -> x <> 0*x );
if bas = [ ] then
K:= Objectify( NewType( FamilyObj( L ),
IsLieRing
and IsAttributeStoringRep ),
rec() );
BK:= Objectify( NewType( FamilyObj( K ),
IsFiniteBasisDefault and
IsBasisOfLieRing and
IsAttributeStoringRep ),
rec() );
# BK has three record components with useful information:
#
# in this case all are empty
BK!.mat:= [ ];
BK!.ambient:= [ ];
SetLeftActingDomain( K, Integers );
SetGeneratorsOfLeftOperatorRing( K, [ ] );
SetGeneratorsOfIdeal( K, vv );
SetUnderlyingLeftModule( BK, K );
SetBasisVectors( BK, [ ] );
SetTorsion( BK, [ ] );
SetIsStandardBasisOfLieRing( BK, false );
SetBasis( K, BK );
SetParent( K, Parent(L) );
return K;
fi;
heads:= List( bas, PositionNonZero );
if not is_bas then
foundbasis:= false;
while not foundbasis do
newelms:= [ ];
for x in Basis(L) do
for c in bas do
cf:= Coefficients( B, x*LinearCombination(B,c) );
for i in [1..Length(heads)] do
pos:= heads[i];
a:= cf[ pos ]; b:= bas[i][pos];
r:= a mod b;
q:= (a-r)/b;
cf:= cf - q*bas[i];
od;
if cf <> 0*cf then
Add( newelms, cf );
fi;
od;
od;
if newelms = [ ] then
foundbasis:= true;
else
Append( bas, newelms );
bas:= TriangulizedIntegerMat( bas );
bas:= Filtered( bas, x -> x <> 0*x );
heads:= List( bas, PositionNonZero );
fi;
od;
fi;
# in order to get the normal basis with have to write the find
# the linear relations among the basis elements....
eqns:= ShallowCopy( bas );
for i in [1..n] do
row:= List( [1..n], x -> 0 );
row[ i ]:= Torsion(B)[i];
Add( eqns, row );
od;
A:= NullspaceIntMat( eqns );
A:= List( A, x -> x{[1..Length(bas)]} );
A:= Filtered( A, x -> x <> 0*x );
if A = [ ] then
Q:= IdentityMat( Length(bas) );
Qi:= Q;
S:= [ ];
else
ss:= SmithNormalFormIntegerMatTransforms( A );
S:= ss.normal;
Q:= ss.coltrans;
Qi:= Q^-1;
fi;
start:= 1;
while start <= Length( S ) and S[start][start] = 1 do
start:= start+1;
od;
tors:= List( [start..Length(S)], x -> S[x][x] );
for i in [Length(S)+1..Length(bas)] do Add( tors, 0 ); od;
nb:= List( Qi{[start..Length(Qi)]}, x -> LinearCombination( bas, x ) );
nb:= List( nb, x -> LinearCombination( B, x ) );
K:= Objectify( NewType( FamilyObj( L ),
IsLieRing
and IsAttributeStoringRep ),
rec() );
BK:= Objectify( NewType( FamilyObj( K ),
IsFiniteBasisDefault and
IsBasisOfLieRing and
IsAttributeStoringRep ),
rec() );
# BK has a record component with useful information:
#
# .mat is the matrix of the basis vectors with respect to the
# "ambient" basis, concatenated with some rows that record the
# torsion of the "ambient" basis elements.
# .ambient is the basis of the "ambient" Lie ring.
mat:= List( nb, x -> Coefficients( B, x ) );
for i in [1..Length(Torsion(B))] do
t:= Torsion(B)[i];
if t <> 0 then
v:= List( B, x -> 0 );
v[i]:= t;
Add( mat, v );
fi;
od;
BK!.mat:= mat;
BK!.ambient:= B;
SetLeftActingDomain( K, Integers );
SetGeneratorsOfLeftOperatorRing( K, nb );
SetGeneratorsOfIdeal( K, vv );
SetUnderlyingLeftModule( BK, K );
SetBasisVectors( BK, nb );
SetTorsion( BK, tors );
SetIsStandardBasisOfLieRing( BK, false );
SetBasis( K, BK );
SetParent( K, Parent(L) );
return K;
end;
InstallMethod( LieRingIdeal,
"for Lie ring and list of its elements",
true, [ IsLieRing,
IsList ], 0,
function( L, vv )
return LRPrivateFunctions.ideal_lie_ring( L, vv );
end );
InstallOtherMethod( LieRingIdeal,
"for Lie ring and list of its elements and string",
true, [ IsLieRing,
IsLRElementCollection, IsString ], 0,
function( L, vv, str )
if str <> "basis" then
Error("The third argument has to be the string \"basis\".");
fi;
return LRPrivateFunctions.ideal_lie_ring( L, vv, str );
end );
InstallOtherMethod( NaturalHomomorphismByIdeal,
"for two Lie rings",
IsIdenticalObj,
[ IsLieRing, IsLieRing ],
function( K, I )
local vv, S, start, Q, Qi, tors, i, j, k, M, bas, cc, entry,
imgs, ss, f, mat, e, T;
vv:= BasisVectors( Basis(I) );
mat:= [ ];
tors:= Torsion( Basis(K) );
for i in [1..Length(tors)] do
if tors[i] <> 0 then
e:= List( tors, x -> 0 );
e[i]:= tors[i];
Add( mat, e );
fi;
od;
Append( mat, List( vv, x -> Coefficients( Basis(K), x ) ) );
ss:= SmithNormalFormIntegerMatTransforms( mat );
S:= ss.normal;
S:= Filtered( S, x -> x <> 0*x );
start:= 1;
while start <= Length(S) and S[start][start] = 1 do
start:= start+1;
od;
Q:= ss.coltrans;
Qi:= Q^-1;
tors:= List( [1..Length(S)], x -> S[x][x] );
for i in [Length(S)+1..Length(Q)] do
Add( tors, 0 );
od;
bas:= List( Qi{[start..Length(Qi)]}, x ->
LinearCombination( Basis(K), x ) );
T:= EmptySCTable( Length(bas), 0, "antisymmetric" );
for i in [1..Length(bas)] do
for j in [i+1..Length(bas)] do
cc:= Coefficients( Basis(K), bas[i]*bas[j])*Q;
for k in [1..Length(tors)] do
if tors[k] <> 0 then
cc[k]:= cc[k] mod tors[k];
fi;
od;
entry:= [ ];
for k in [start..Length(cc)] do
if cc[k] <> 0 then
Add( entry, cc[k] );
Add( entry, k-start+1 );
fi;
od;
SetEntrySCTable( T, i, j, entry );
od;
od;
M:= LieRingByStructureConstants( tors{[start..Length(tors)]}, T );
imgs:= [ ];
for i in [1..Length(Basis(K))] do
cc:= Q[i];
for k in [1..Length(tors)] do
if tors[k] <> 0 then
cc[k]:= cc[k] mod tors[k];
fi;
od;
Add( imgs, LinearCombination( Basis(M), cc{[start..Length(cc)]} ) );
od;
# a part from vspchom.gi, to get around the requirement of free left module...
f:= Objectify( TypeOfDefaultGeneralMapping( K, M,
IsSPGeneralMapping
and IsLeftModuleGeneralMapping
and IsLinearGeneralMappingByImagesDefaultRep ),
rec() );
SetMappingGeneratorsImages(f,[BasisVectors(Basis(K)),imgs]);
SetIsSingleValued( f, true );
SetIsTotal( f, true );
# f:= LeftModuleHomomorphismByImagesNC( K, M, BasisVectors(Basis(K)),
# imgs );
f!.basisimage:= Basis( M );
f!.preimagesbasisimage:= bas;
f!.basispreimage:= Basis(K);
f!.imagesbasispreimage:= imgs;
return f;
end );
InstallMethod( LieLowerCentralSeries,
"for a Lie ring",
true, [ IsLieRing ], 0,
function( L )
local lc, b, done, a, x, y, S;
lc:= [ L ];
b:= BasisVectors( Basis(L) );
done:= false;
while not done do
a:= [ ];
for x in BasisVectors( Basis(L) ) do
for y in b do
Add( a, x*y );
od;
od;
S:= SubLieRing( L, a );
if S = lc[ Length(lc) ] then
done:= true;
elif GeneratorsOfAlgebra(S) = [ ] then
done:= true;
Add( lc, S );
else
Add( lc, S );
fi;
b:= BasisVectors( Basis(S) );
od;
return lc;
end );
InstallMethod( LieCentre,
"for a Lie ring",
true, [ IsLieRing ], 0,
function( L )
local n, T, d, eqns, i, j, k, c, eq, sol, bas;
n:= Length( Basis(L) );
T:= StructureConstantsTable( Basis(L) );
d:= Torsion( Basis(L) );
eqns:= NullMat( n+n^2, n^2 );
for i in [1..n] do
for j in [1..n] do
c:= T[i][j];
for k in [1..Length(c[1])] do
eqns[j][(i-1)*n+c[1][k]]:= c[2][k];
od;
eqns[n+(i-1)*n+j][(i-1)*n+j]:= d[j];
od;
od;
sol:= NullspaceIntMat( eqns );
bas:= List( sol, x -> LinearCombination( Basis(L), x{[1..n]} ) );
return SubLieRing( L, bas, "basis" );
end );
InstallMethod( TensorWithField,
"for a Lie ring and a field",
true, [ IsLieRing, IsField ], 0,
function( L, F )
local p, inds, k, l, i, j, r, ckl, cf, T, dim, t;
p:= Characteristic(F);
inds:= [ ]; # some basis elements reduce to zero,
# if bas elt i goes to zero we put a zero on inds[i],
# otherwise we put the index of the new bas elt in the Lie algebra
k:= 1;
t:= Torsion( Basis(L) );
for i in [1..Length(t)] do
if (t[i]>0 and p=0) or (p>0 and t[i] mod p <> 0) then
# goes to zero...
inds[i]:= 0;
else
inds[i]:= k; k:= k+1;
fi;
od;
dim:= k-1;
T:= EmptySCTable( dim, Zero( F ), "antisymmetric" );
for i in [1..Length(t)] do
if inds[i] <> 0 then
k:= inds[i];
for j in [i+1..Length(t)] do
if inds[j] <> 0 then
l:= inds[j];
ckl:= [ ];
cf:= Coefficients( Basis(L), Basis(L)[i]*Basis(L)[j] );
for r in [1..Length(cf)] do
if cf[r] <> 0 and inds[r] <> 0 then
Append( ckl, [ cf[r]*One(F), inds[r] ] );
fi;
od;
SetEntrySCTable( T, k, l, ckl );
fi;
od;
fi;
od;
return LieAlgebraByStructureConstants( F, T );
end );
InstallMethod( SmallNEngelLieRing,
"for two integers",
true, [ IsInt, IsInt ], 0,
function( n, k )
# i.e., n-engel, with k gens..
local s, t, S, T, u;
if not [n,k] in [ [4,3], [3,2], [3,3], [3,4], [4,2] ] then
s:= "The ";
Append( s, String(n) );
Append( s, "-Engel Lie ring with " );
Append( s, String(k) );
Append( s, " generators is not contained in the database" );
Error(s);
fi;
if [n,k] = [4,3] then
t:= LRPrivateFunctions.n_engelLR.t43;
S:= LRPrivateFunctions.n_engelLR.S43;
elif [n,k] = [3,2] then
t:= LRPrivateFunctions.n_engelLR.t32;
S:= LRPrivateFunctions.n_engelLR.S32;
elif [n,k] = [3,3] then
t:= LRPrivateFunctions.n_engelLR.t33;
S:= LRPrivateFunctions.n_engelLR.S33;
elif [n,k] = [3,4] then
t:= LRPrivateFunctions.n_engelLR.t34;
S:= LRPrivateFunctions.n_engelLR.S34;
elif [n,k] = [4,2] then
t:= LRPrivateFunctions.n_engelLR.t42;
S:= LRPrivateFunctions.n_engelLR.S42;
fi;
T:= EmptySCTable( Length(t), 0, "antisymmetric" );
for u in S do SetEntrySCTable( T, u[1], u[2], u[3] ); od;
return LieRingByStructureConstants( t, T );
end );
[ Dauer der Verarbeitung: 0.30 Sekunden
(vorverarbeitet)
]
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