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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, and Willem de Graaf.
##
## 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 generic methods for algebras and algebras-with-one.
##
#############################################################################
##
#M Representative( <A> ) . . . . . . . . one element of a left operator ring
##
InstallMethod( Representative,
"for left operator ring with known generators",
[ IsLeftOperatorRing and HasGeneratorsOfLeftOperatorRing ],
RepresentativeFromGenerators( GeneratorsOfLeftOperatorRing ) );
#############################################################################
##
#M Representative( <A> ) . . . one element of a left operator ring-with-one
##
InstallMethod( Representative,
"for left operator ring-with-one with known generators",
[ IsLeftOperatorRingWithOne and HasGeneratorsOfLeftOperatorRingWithOne ],
RepresentativeFromGenerators( GeneratorsOfLeftOperatorRingWithOne ) );
#############################################################################
##
#M FLMLORByGenerators( <R>, <gens> ) . . . . <R>-FLMLOR generated by <gens>
#M FLMLORByGenerators( <R>, <gens>, <zero> )
##
InstallMethod( FLMLORByGenerators,
"for ring and collection",
[ IsRing, IsCollection ],
function( R, gens )
local A;
A:= Objectify( NewType( FamilyObj( gens ),
IsFLMLOR and IsAttributeStoringRep ),
rec() );
SetLeftActingDomain( A, R );
SetGeneratorsOfLeftOperatorRing( A, AsList( gens ) );
CheckForHandlingByNiceBasis( R, gens, A, false );
return A;
end );
InstallOtherMethod( FLMLORByGenerators,
"for ring, homogeneous list, and ring element",
[ IsRing, IsHomogeneousList, IsRingElement ],
function( R, gens, zero )
local A;
A:= Objectify( NewType( CollectionsFamily( FamilyObj( zero ) ),
IsFLMLOR and IsAttributeStoringRep ),
rec() );
SetLeftActingDomain( A, R );
SetGeneratorsOfLeftOperatorRing( A, gens );
SetZero( A, zero );
if IsEmpty( gens ) then
SetDimension( A, 0 );
SetGeneratorsOfLeftModule( A, gens );
fi;
CheckForHandlingByNiceBasis( R, gens, A, zero );
return A;
end );
#############################################################################
##
#M FLMLORWithOneByGenerators( <R>, <gens> ) unit. <R>-FLMLOR gen. by <gens>
#M FLMLORWithOneByGenerators( <R>, <gens>, <zero> )
##
InstallMethod( FLMLORWithOneByGenerators,
"for ring and collection",
[ IsRing, IsCollection ],
function( R, gens )
local A;
A:= Objectify( NewType( FamilyObj( gens ),
IsFLMLORWithOne and IsAttributeStoringRep ),
rec() );
SetLeftActingDomain( A, R );
SetGeneratorsOfLeftOperatorRingWithOne( A, AsList( gens ) );
CheckForHandlingByNiceBasis( R, gens, A, false );
return A;
end );
InstallOtherMethod( FLMLORWithOneByGenerators,
"for ring, homogeneous list, and ring element",
[ IsRing, IsHomogeneousList, IsRingElement ],
function( R, gens, zero )
local A;
A:= Objectify( NewType( CollectionsFamily( FamilyObj( zero ) ),
IsFLMLORWithOne and IsAttributeStoringRep ),
rec() );
SetLeftActingDomain( A, R );
SetGeneratorsOfLeftOperatorRingWithOne( A, AsList( gens ) );
SetZero( A, zero );
CheckForHandlingByNiceBasis( R, gens, A, zero );
return A;
end );
#############################################################################
##
#F Algebra( <F>, <gens> )
#F Algebra( <F>, <gens>, <zero> )
#F Algebra( <F>, <gens>, "basis" )
#F Algebra( <F>, <gens>, <zero>, "basis" )
##
InstallGlobalFunction( FLMLOR, function( arg )
local A;
# ring and list of generators
if Length( arg ) = 2 and IsRing( arg[1] )
and IsList( arg[2] ) and 0 < Length( arg[2] ) then
A:= FLMLORByGenerators( arg[1], arg[2] );
# ring, list of generators plus zero
elif Length( arg ) = 3 and IsRing( arg[1] )
and IsList( arg[2] ) then
if arg[3] = "basis" then
A:= FLMLORByGenerators( arg[1], arg[2] );
UseBasis( A, arg[2] );
else
A:= FLMLORByGenerators( arg[1], arg[2], arg[3] );
fi;
# ring, list of generators plus zero
elif Length( arg ) = 4 and IsRing( arg[1] )
and IsList( arg[2] )
and arg[4] = "basis" then
A:= FLMLORByGenerators( arg[1], arg[2], arg[3] );
UseBasis( A, arg[2] );
# no argument given, error
else
Error( "usage: FLMLOR( <F>, <gens> ), ",
"FLMLOR( <F>, <gens>, <zero> )" );
fi;
# Return the result.
return A;
end );
#############################################################################
##
#F Subalgebra( <A>, <gens> ) . . . . . subalgebra of <A> generated by <gens>
#F Subalgebra( <A>, <gens>, "basis" )
##
InstallGlobalFunction( SubFLMLOR, function( arg )
local S;
if Length( arg ) <= 1
or not IsFLMLOR( arg[1] )
or not IsHomogeneousList( arg[2] ) then
Error( "first argument must be a FLMLOR,\n",
"second argument must be a list of generators" );
elif IsEmpty( arg[2] ) then
return SubFLMLORNC( arg[1], arg[2] );
elif IsIdenticalObj( FamilyObj( arg[1] ),
FamilyObj( arg[2] ) )
and ForAll( arg[2], v -> v in arg[1] ) then
S:= FLMLORByGenerators( LeftActingDomain( arg[1] ), arg[2] );
SetParent( S, arg[1] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( S, arg[2] );
fi;
return S;
fi;
Error( "usage: SubFLMLOR( <V>, <gens> [, \"basis\"] )" );
end );
#############################################################################
##
#F SubalgebraNC( <A>, <gens>, "basis" )
#F SubalgebraNC( <A>, <gens> )
##
InstallGlobalFunction( SubFLMLORNC, function( arg )
local S;
if IsEmpty( arg[2] ) then
S:= Objectify( NewType( FamilyObj( arg[1] ),
IsFLMLOR
and IsTrivial
and IsTwoSidedIdealInParent
and IsAttributeStoringRep ),
rec() );
SetDimension(S, 0);
SetLeftActingDomain( S, LeftActingDomain( arg[1] ) );
SetGeneratorsOfLeftModule( S, AsList( arg[2] ) );
else
S:= FLMLORByGenerators( LeftActingDomain( arg[1] ), arg[2] );
fi;
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( S, arg[2] );
fi;
SetParent( S, arg[1] );
return S;
end );
#############################################################################
##
#F AlgebraWithOne( <F>, <gens> )
#F AlgebraWithOne( <F>, <gens>, <zero> )
#F AlgebraWithOne( <F>, <gens>, "basis" )
#F AlgebraWithOne( <F>, <gens>, <zero>, "basis" )
##
InstallGlobalFunction( FLMLORWithOne, function( arg )
local A;
# ring and list of generators
if Length( arg ) = 2 and IsRing( arg[1] )
and IsList( arg[2] ) and 0 < Length( arg[2] ) then
A:= FLMLORWithOneByGenerators( arg[1], arg[2] );
# ring, list of generators plus zero
elif Length( arg ) = 3 and IsRing( arg[1] )
and IsList( arg[2] ) then
if arg[3] = "basis" then
A:= FLMLORWithOneByGenerators( arg[1], arg[2] );
UseBasis( A, arg[2] );
else
A:= FLMLORWithOneByGenerators( arg[1], arg[2], arg[3] );
fi;
# ring, list of generators plus zero
elif Length( arg ) = 4 and IsRing( arg[1] )
and IsList( arg[2] )
and arg[4] = "basis" then
A:= FLMLORWithOneByGenerators( arg[1], arg[2], arg[3] );
UseBasis( A, arg[2] );
# no argument given, error
else
Error( "usage: FLMLORWithOne( <F>, <gens> ), ",
"FLMLORWithOne( <F>, <gens>, <zero> )" );
fi;
# Return the result.
return A;
end );
#############################################################################
##
#F SubalgebraWithOne( <A>, <gens> ) subalg.-with-one of <A> gen. by <gens>
##
InstallGlobalFunction( SubFLMLORWithOne, function( arg )
local S;
if Length( arg ) <= 1 or not IsFLMLOR( arg[1] )
or not IsHomogeneousList( arg[2] ) then
Error( "first argument must be a FLMLOR,\n",
"second argument must be a list of generators" );
elif IsEmpty( arg[2] ) then
return SubFLMLORWithOneNC( arg[2], arg[2] );
elif IsIdenticalObj( FamilyObj( arg[1] ),
FamilyObj( arg[2] ) )
and ForAll( arg[2], v -> v in arg[1] )
and ( IsFLMLORWithOne( arg[1] ) or One( arg[1] ) <> fail ) then
S:= FLMLORWithOneByGenerators( LeftActingDomain( arg[1] ), arg[2] );
SetParent( S, arg[1] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( S, arg[2] );
fi;
return S;
fi;
Error( "usage: SubFLMLORWithOne( <V>, <gens> [, \"basis\"] )" );
end );
#############################################################################
##
#F SubalgebraWithOneNC( <A>, <gens> )
##
InstallGlobalFunction( SubFLMLORWithOneNC, function( arg )
local S, gens;
if IsEmpty( arg[2] ) then
# Note that `S' is in general not trivial,
# and if we call `Objectify' here then `S' does not get a special
# representation (e.g., as a matrix algebra).
# So the argument that special methods would catch this case
# does not hold!
gens:= [ One( arg[1] ) ];
S:= FLMLORWithOneByGenerators( LeftActingDomain( arg[1] ), gens );
UseBasis( S, gens );
else
S:= FLMLORWithOneByGenerators( LeftActingDomain( arg[1] ), arg[2] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( S, arg[2] );
fi;
fi;
SetParent( S, arg[1] );
return S;
end );
#############################################################################
##
#M LieAlgebraByDomain( <A> )
##
## The Lie algebra of the associative algebra <A>
##
InstallMethod( LieAlgebraByDomain,
"for an algebra",
[ IsAlgebra ],
function( A )
local T, n, zero, nullvec, S, i, j, k, m, cfs, cij, cji;
if not IsAssociative( A ) then TryNextMethod(); fi;
# We construct a structure constants table for the Lie algebra
# corresponding to <A>. If the structure constants of <A> are given by
# d_{ij}^k, then the structure constants of the Lie algebra will be given
# by d_{ij}^k - d_{ji}^k.
T:= StructureConstantsTable( Basis( A ) );
n:= Dimension( A );
zero:= Zero( LeftActingDomain( A ) );
nullvec:= List( [1..n], x -> zero );
S:= EmptySCTable( n, zero, "antisymmetric" );
for i in [1..n] do
for j in [i+1..n] do
cfs:= ShallowCopy( nullvec );
cij:= T[i][j]; cji:= T[j][i];
for m in [1..Length(cij[1])] do
k:= cij[1][m];
cfs[k]:= cfs[k] + cij[2][m];
od;
for m in [1..Length(cji[1])] do
k:= cji[1][m];
cfs[k]:= cfs[k] - cji[2][m];
od;
cij:= [ ];
for m in [1..n] do
if cfs[m] <> zero then
Add( cij, cfs[m] );
Add( cij, m );
fi;
od;
SetEntrySCTable( S, i, j, cij );
od;
od;
return LieAlgebraByStructureConstants( LeftActingDomain( A ), S );
end );
#############################################################################
##
#F LieAlgebra( <A> )
#F LieAlgebra( <F>, <gens> )
#F LieAlgebra( <F>, <gens>, <zero> )
#F LieAlgebra( <F>, <gens>, "basis" )
#F LieAlgebra( <F>, <gens>, <zero>, "basis" )
##
InstallGlobalFunction( LieAlgebra, function( arg )
#T check that the families have the same characteristic?
#T `CharacteristicFamily' ?
local A,gens;
# In the case of one domain argument,
# construct the isomorphic Lie algebra.
if Length( arg ) = 1 and IsDomain( arg[1] ) then
A:= LieAlgebraByDomain( arg[1] );
# division ring and list of generators
elif Length( arg ) >= 2 and IsList( arg[2] ) then
gens:= List( arg[2], x -> LieObject( x ) );
if Length( arg ) = 2 and IsDivisionRing( arg[1] )
and 0 < Length( arg[2] ) then
A:= AlgebraByGenerators( arg[1], gens );
# division ring, list of generators plus zero
elif Length( arg ) = 3 and IsDivisionRing( arg[1] ) then
if arg[3] = "basis" then
A:= AlgebraByGenerators( arg[1], gens );
UseBasis( A, gens );
else
A:= AlgebraByGenerators( arg[1], gens, arg[3] );
fi;
# division ring, list of generators plus zero
elif Length( arg ) = 4 and IsDivisionRing( arg[1] )
and arg[4] = "basis" then
A:= AlgebraByGenerators( arg[1], gens, arg[3] );
UseBasis( A, gens );
else
Error( "usage: LieAlgebra( <F>, <gens> ), ",
"LieAlgebra( <F>, <gens>, <zero> ), LieAlgebra( <D> )");
fi;
# no argument given, error
else
Error( "usage: LieAlgebra( <F>, <gens> ), ",
"LieAlgebra( <F>, <gens>, <zero> ), LieAlgebra( <D> )");
fi;
# Return the result.
return A;
end );
#############################################################################
##
#F EmptySCTable( <dim>, <zero> )
#F EmptySCTable( <dim>, <zero>, \"symmetric\" )
#F EmptySCTable( <dim>, <zero>, \"antisymmetric\" )
##
InstallGlobalFunction( EmptySCTable, function( arg )
local dim, T, entry, i;
if 2 <= Length( arg )
and IsInt( arg[1] ) and IsZero( arg[2] )
and ( Length( arg ) = 2 or IsString( arg[3] ) ) then
dim:= arg[1];
T:= [];
entry:= Immutable( [ [], [] ] );
for i in [ 1 .. dim ] do
T[i]:= List( [ 1 .. dim ], x -> entry );
od;
# Store the symmetry flag.
if Length( arg ) = 3 then
if arg[3] = "symmetric" then
Add( T, 1 );
elif arg[3] = "antisymmetric" then
Add( T, -1 );
else
Error("third argument must be \"symmetric\" or \"antisymmetric\"");
fi;
else
Add( T, 0 );
fi;
# Store the zero coefficient.
Add( T, arg[2] );
else
Error( "usage: EmptySCTable( <dim>, <zero> [,\"symmetric\"] )" );
fi;
return T;
end );
#############################################################################
##
#F SetEntrySCTable( <T>, <i>, <j>, <list> )
##
InstallGlobalFunction( SetEntrySCTable, function( T, i, j, list )
local range, zero, Fam, entry, k, val, pos;
# Check that `i' and `j' are admissible.
range:= [ 1 .. Length( T ) - 2 ];
if not i in range then
Error( "<i> must lie in ", range );
elif not j in range then
Error( "<j> must lie in ", range );
fi;
# Check `list', and construct the table entry.
zero:= Last(T);
Fam:= FamilyObj( zero );
entry:= [ [], [] ];
for k in [ 1, 3 .. Length( list ) -1 ] do
val:= list[k];
pos:= list[k+1];
# Check that `pos' is inside the table,
# and that its entry is assigned only once.
if not pos in range then
Error( "list entry ", list[k+1], " must lie in ", range );
elif pos in entry[1] then
Error( "position ", pos, " must occur at most once in <list>" );
fi;
# Check that the coefficients either fit to the zero element
# or are rationals (with suitable denominators).
if FamilyObj( val ) = Fam then
if val <> zero then
Add( entry[1], pos );
Add( entry[2], val );
fi;
elif IsRat( val ) then
if val <> 0 then
Add( entry[1], pos );
Add( entry[2], val * One( zero ) );
fi;
else
Error( "list entry ", list[k], " does not fit to zero element" );
fi;
od;
# Set the table entry.
SortParallel( entry[1], entry[2] );
T[i][j]:= Immutable( entry );
# Add the value `T[j][i]' in the case of (anti-)symmetric tables.
if T[ Length(T) - 1 ] = 1 then
T[j][i]:= T[i][j];
elif T[ Length(T) - 1 ] = -1 then
T[j][i]:= Immutable( [ entry[1], -entry[2] ] );
fi;
end );
#############################################################################
##
#F ReducedSCTable( <T>, <one> )
##
InstallGlobalFunction( ReducedSCTable, function( T, one )
local new, n, i, j, entry;
new:= [];
n:= Length( T ) - 2;
# Reduce the entries.
for i in [ 1 .. n ] do
new[i]:= [];
for j in [ 1 .. n ] do
entry:= T[i][j];
entry:= [ Immutable( entry[1] ), entry[2] * one ];
MakeImmutable( entry );
new[i][j]:= entry;
od;
od;
# Store zero coefficient and symmetry flag.
new[ n+1 ]:= T[ n+1 ];
new[ n+2 ]:= T[ n+2 ] * one;
# Return the immutable new table.
MakeImmutable( new );
return new;
end );
#############################################################################
##
#F GapInputSCTable( <T>, <varnam> )
##
InstallGlobalFunction( GapInputSCTable, function( T, varnam )
local dim, str, lower, i, j, entry, k;
# Initialize, and set the ranges for the loops.
dim:= Length( T ) - 2;
str:= Concatenation( varnam, ":= EmptySCTable( ",
String( dim ), ", ", String( Last(T) ) );
lower:= [ 1 .. dim ];
if T[ dim+1 ] = 1 then
Append( str, ", \"symmetric\"" );
elif T[ dim+1 ] = -1 then
Append( str, ", \"antisymmetric\"" );
else
lower:= ListWithIdenticalEntries( dim, 1 );
fi;
Append( str, " );\n" );
# Fill up the table.
for i in [ 1 .. dim ] do
for j in [ lower[i] .. dim ] do
entry:= T[i][j];
if not IsEmpty( entry[1] ) then
Append( str, "SetEntrySCTable( " );
Append( str, varnam );
Append( str, ", " );
Append( str, String(i) );
Append( str, ", " );
Append( str, String(j) );
Append( str, ", [" );
for k in [ 1 .. Length( entry[1] )-1 ] do
Append( str, String( entry[2][k] ) );
Add( str, ',' );
Append( str, String( entry[1][k] ) );
Add( str, ',' );
od;
k:= Length( entry[1] );
Append( str, String( entry[2][k] ) );
Add( str, ',' );
Append( str, String( entry[1][k] ) );
Append( str, "] );\n" );
fi;
od;
od;
ConvertToStringRep( str );
return str;
end );
#############################################################################
##
#F IdentityFromSCTable( <T> )
##
InstallGlobalFunction( IdentityFromSCTable, function( T )
local n, # dimension of the underlying algebra
equ, # equation system to solve
zero, # zero of the field
zerovec, # zero vector
vec, # right hand side of the equation system
one, # identity of the field
i, j, k, # loop over rows of `equ'
row, # one row of the equation system
Tpos, #
Tval, #
p, #
sol,
sum;
n:= Length( T ) - 2;
zero:= T[ n+2 ];
# If the table belongs to a trivial algebra,
# the identity is equal to the zero.
if n = 0 then
return EmptyRowVector( FamilyObj( zero ) );
fi;
# Set up the equation system,
# in row $i$ and column $(k-1)*n + j$ we have $c_{ijk}$.
equ:= [];
zerovec:= ListWithIdenticalEntries( n^2, zero );
vec:= ShallowCopy( zerovec );
one:= One( zero );
for i in [ 1 .. n ] do
row:= ShallowCopy( zerovec );
for j in [ 1 .. n ] do
Tpos:= T[i][j][1];
Tval:= T[i][j][2];
p:= (j-1)*n;
for k in [ 1 .. Length( Tpos ) ] do
row[ p + Tpos[k] ]:= Tval[k];
od;
od;
Add( equ, row );
vec[ (i-1)*n + i ]:= one;
od;
sol:= SolutionMat( equ, vec );
# If we have a candidate and if the algebra is not known
# to be commutative then check whether the candidate
# acts trivially also from the right.
if sol <> fail and T[ n+1 ] <> 1 then
for j in [ 1 .. n ] do
for k in [ 1 .. n ] do
sum:= zero;
for i in [ 1 .. n ] do
Tpos:= T[j][i];
p:= Position( Tpos[1], k );
#T cheaper !!!
if p <> fail then
sum:= sum + sol[i] * Tpos[2][p];
fi;
od;
if ( j = k and sum <> one ) or ( j <> k and sum <> zero ) then
return fail;
fi;
od;
od;
fi;
# Return the result.
return sol;
end );
#############################################################################
##
#F QuotientFromSCTable( <T>, <num>, <den> )
##
## We solve the equation system $<num> = x <den>$.
## If no solution exists, `fail' is returned.
##
## In terms of the basis $B$ with vectors $b_1, \ldots, b_n$ this means
## for $<num> = \sum_{i=1}^n a_i b_i$,
## $<den> = \sum_{i=1}^n c_i b_i$,
## $x = \sum_{i=1}^n x_i b_i$ that
## $a_k = \sum_{i,j} c_i x_j c_{ijk}$ for all $k$.
## Here $c_{ijk}$ denotes the structure constants w.r.t. $B$.
## This means $a = x M$ with $M_{ik} = \sum_{j=1}^n c_{ijk} c_j$.
##
InstallGlobalFunction( QuotientFromSCTable, function( T, x, c )
local M, # matrix of the equation system
n, # dimension of the algebra
zero, # zero vector
i, j, # loop variables
row, # one row of `M'
entry, #
val; #
M:= [];
n:= Length( c );
# If the algebra is zero dimensional,
# the zero is also the identity and thus also its inverse.
if n = 0 then
return c;
fi;
zero:= ListWithIdenticalEntries( n, Last(T) );
for i in [ 1 .. n ] do
row:= ShallowCopy( zero );
for j in [ 1 .. n ] do
entry:= T[i][j];
val:= c[j];
row{ entry[1] }:= row{ entry[1] } + val * entry[2];
#T better!
od;
Add( M, row );
od;
# Return the quotient, or `fail'.
return SolutionMat( M, x );
end );
#############################################################################
##
#F TestJacobi( <T> )
##
## We check whether for all $1 \leq m \leq n$ the equality
## $\sum_{l=1}^n c_{jkl} c_{ilm} + c_{kil} c_{jlm} + c_{ijl} c_{klm} = 0$
## holds.
##
InstallGlobalFunction( TestJacobi, function( T )
local zero, # the zero of the field
n, # dimension of the algebra
i, j, k, m, # loop variables
cij, cki, cjk, # structure constant vectors
sum,
t;
zero:= Last(T);
n:= Length( T ) - 2;
for i in [ 1 .. n ] do
for j in [ i+1 .. n ] do
cij:= T[i][j];
for k in [ j+1 .. n ] do
cki:= T[k][i];
cjk:= T[j][k];
for m in [ 1 .. n ] do
sum:= zero;
for t in [ 1 .. Length( cjk[1] ) ] do
sum:= sum + cjk[2][t] * SCTableEntry( T, i, cjk[1][t], m );
od;
for t in [ 1 .. Length( cki[1] ) ] do
sum:= sum + cki[2][t] * SCTableEntry( T, j, cki[1][t], m );
od;
for t in [ 1 .. Length( cij[1] ) ] do
sum:= sum + cij[2][t] * SCTableEntry( T, k, cij[1][t], m );
od;
if sum <> zero then
return [ i, j, k ];
fi;
od;
od;
od;
od;
return true;
end );
#############################################################################
##
#M MultiplicativeNeutralElement( <A> )
##
## is the multiplicative neutral element of <A> if this exists,
## otherwise is `fail'.
##
## Let $(b_1, b_2, \ldots, b_n)$ be a basis of $A$, and $e$ the result of
## `MultiplicativeNeutralElement( <A> )'.
## Then $e = \sum_{i=1}^n a_i b_i$, and for $1 \leq k \leq n$ we have
## $e \cdot b_j = b_j$, or equivalently
## $\sum_{i=1}^n a_i b_i \cdot b_j = b_j$.
## Define the structure constants by
## $b_i \cdot b_j = \sum_{k=1}^n c_{ijk} b_k$.
##
## Then $\sum_{i=1}^n a_i c_{ijk} = \delta_{jk}$ for $1 \leq k \leq n$.
##
## This yields $n^2$ linear equations for the $n$ indeterminates $a_i$,
## and a solution is a left identity.
## For this we have to test whether it is also a right identity.
##
InstallMethod( MultiplicativeNeutralElement,
[ IsFLMLOR and IsFiniteDimensional ],
function( A )
local B, # basis of `A'
one; # result
B:= Basis( A );
one:= IdentityFromSCTable( StructureConstantsTable( B ) );
if one <> fail then
one:= LinearCombination( B, one );
fi;
return one;
end );
#############################################################################
##
#M IsAssociative( <A> )
##
## We check whether the vectors of a basis satisfy the associativity law.
## (Bilinearity of the multiplication is of course assumed.)
##
## If $b_i \cdot b_j = \sum_{l=1}^n c_{ijl} b_l$ then we have
## $b_i \cdot ( b_j \cdot b_k ) = ( b_i \cdot b_j ) \cdot b_k$
## if and only if
## $\sum_{l=1}^n c_{jkl} c_{ilm} = \sum_{l=1}^n c_{ijl} c_{lkm}$ for all
## $1 \leq m \leq n$.
##
## We check this equality for all $1 \leq i, j, k \leq n$.
##
InstallMethod( IsAssociative,
"generic method for a (finite dimensional) FLMLOR",
[ IsFLMLOR ],
function( A )
local T, # structure constants table w.r.t. a basis of `A'
zero,
range,
i, j, k, l, m,
Ti,
Tj,
cijpos,
cijval,
cjkpos,
cjkval,
sum,
x,
pos;
if not IsFiniteDimensional( A ) then
TryNextMethod();
fi;
T:= StructureConstantsTable( Basis( A ) );
zero:= Zero( LeftActingDomain( A ) );
range:= [ 1 .. Length( T[1] ) ];
for i in range do
Ti:= T[i];
for j in range do
cijpos:= Ti[j][1];
cijval:= Ti[j][2];
Tj:= T[j];
for k in range do
cjkpos:= Tj[k][1];
cjkval:= Tj[k][2];
for m in range do
sum:= zero;
for l in [ 1 .. Length( cjkpos ) ] do
x:= Ti[ cjkpos[l] ];
pos:= Position( x[1], m );
if pos <> fail then
sum:= sum + cjkval[l] * x[2][ pos ];
fi;
od;
for l in [ 1 .. Length( cijpos ) ] do
x:= T[ cijpos[l] ][k];
pos:= Position( x[1], m );
if pos <> fail then
sum:= sum - cijval[l] * x[2][ pos ];
fi;
od;
if sum <> zero then
# $i, j, k$ fail
Info( InfoAlgebra, 2,
"IsAssociative fails for i = ", i, ", j = ", j,
", k = ", k );
return false;
fi;
od;
od;
od;
od;
return true;
end );
#############################################################################
##
#M IsAnticommutative( <A> ) . . . . . . . . . . . . .for a fin.-dim. FLMLOR
##
## is `true' if the multiplication in <A> is anticommutative,
## and `false' otherwise.
##
InstallMethod( IsAnticommutative,
"generic method for a (finite dimensional) FLMLOR",
[ IsFLMLOR ],
function( A )
local n, # dimension of `A'
T, # table of structure constants for `A'
i, j; # loop over rows and columns ot `T'
if not IsFiniteDimensional( A ) then
TryNextMethod();
fi;
n:= Dimension( A );
T:= StructureConstantsTable( Basis( A ) );
for i in [ 2 .. n ] do
for j in [ 1 .. i-1 ] do
if T[i][j][1] <> T[j][i][1]
or ( not IsEmpty( T[i][j][1] )
and PositionNonZero( T[i][j][2] + T[j][i][2] )
<= Length( T[i][j][2] ) ) then
return false;
fi;
od;
od;
if Characteristic( A ) <> 2 then
# The values on the diagonal must be zero.
for i in [ 1 .. n ] do
if not IsEmpty( T[i][i][1] ) then
return false;
fi;
od;
fi;
return true;
end );
#############################################################################
##
#M IsCommutative( <A> ) . . . . . . . . . . . for finite dimensional FLMLOR
##
## Check whether every basis vector commutes with every basis vector.
##
InstallMethod( IsCommutative,
"generic method for a finite dimensional FLMLOR",
[ IsFLMLOR ],
IsCommutativeFromGenerators( GeneratorsOfVectorSpace ) );
#T use structure constants!
#############################################################################
##
#M IsCommutative( <A> ) . . . . . . . . . . . . . for an associative FLMLOR
##
## If <A> is associative then we can restrict the check to a smaller
## equation system than that for arbitrary algebras, since we have to check
## $x a = a x$ only for algebra generators $a$ and $x$, not for all vectors
## of a basis.
##
InstallMethod( IsCommutative,
"for an associative FLMLOR",
[ IsFLMLOR and IsAssociative ],
IsCommutativeFromGenerators( GeneratorsOfAlgebra ) );
InstallMethod( IsCommutative,
"for an associative FLMLOR-with-one",
[ IsFLMLORWithOne and IsAssociative ],
IsCommutativeFromGenerators( GeneratorsOfAlgebraWithOne ) );
#############################################################################
##
#M IsZeroSquaredRing( <A> ) . . . . . . . . for a finite dimensional FLMLOR
##
InstallMethod( IsZeroSquaredRing,
"for a finite dimensional FLMLOR",
[ IsFLMLOR ],
function( A )
if not IsAnticommutative( A ) then
# Every zero squared ring is anticommutative.
return false;
elif ForAny( BasisVectors( Basis( A ) ),
x -> not IsZero( x*x ) ) then
# If not all basis vectors are zero squared then we return `false'.
return false;
elif IsCommutative( LeftActingDomain( A ) ) then
# If otherwise the left acting domain is commutative then we return
# `true' because we know that <A> is anticommutative and the basis
# vectors are zero squared.
return true;
else
# Otherwise we give up.
TryNextMethod();
fi;
end );
#############################################################################
##
#M IsJacobianRing( <A> )
##
InstallMethod( IsJacobianRing,
"for a (finite dimensional) FLMLOR",
[ IsFLMLOR ],
function( A )
local n, # dimension of `A'
T, # table of structure constants for `A'
i; # loop over the diagonal of `T'
if not IsFiniteDimensional( A ) then
TryNextMethod();
fi;
# In characteristic 2 we have to make sure that $a \* a = 0$.
#T really?
T:= StructureConstantsTable( Basis( A ) );
if Characteristic( A ) = 2 then
n:= Dimension( A );
for i in [ 1 .. n ] do
if not IsEmpty( T[i][i][1] ) then
return false;
fi;
od;
fi;
# Check the Jacobi identity $[a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0$.
return TestJacobi( T ) = true;
end );
#############################################################################
##
#M Intersection2( <A1>, <A2> ) . . . . . . . . . intersection of two FLMLORs
##
InstallMethod( Intersection2,
"generic method for two FLMLORs",
IsIdenticalObj,
[ IsFLMLOR, IsFLMLOR ],
Intersection2Spaces( AsFLMLOR, SubFLMLORNC, FLMLOR ) );
#############################################################################
##
#M Intersection2( <A1>, <A2> ) . . . . intersection of two FLMLORs-with-one
##
InstallMethod( Intersection2,
"generic method for two FLMLORs-with-one",
IsIdenticalObj,
[ IsFLMLORWithOne, IsFLMLORWithOne ],
Intersection2Spaces( AsFLMLORWithOne, SubFLMLORWithOneNC,
FLMLORWithOne ) );
#############################################################################
##
#M \/( <A>, <I> ) . . . . . . . . . . . . factor of an algebra by an ideal
#M \/( <A>, <relators> ) . . . . . . . . . factor of an algebra by an ideal
##
## is the factor algebra of the finite dimensional algebra <A> modulo
## the ideal <I> or the ideal spanned by the collection <relators>.
##
InstallOtherMethod( \/,
"for FLMLOR and collection",
IsIdenticalObj,
[ IsFLMLOR, IsCollection ],
function( A, relators )
if IsFLMLOR( relators ) then
TryNextMethod();
else
return A / TwoSidedIdealByGenerators( A, relators );
fi;
end );
InstallOtherMethod( \/,
"for FLMLOR and empty list",
[ IsFLMLOR, IsList and IsEmpty ],
function( A, empty )
# `NaturalHomomorphismByIdeal( A, TrivialSubFLMLOR( A ) )' is the
# identity mapping on `A', and `ImagesSource' of it yields `A'.
return A;
end );
InstallOtherMethod( \/,
"generic method for two FLMLORs",
IsIdenticalObj,
[ IsFLMLOR, IsFLMLOR ],
function( A, I )
return ImagesSource( NaturalHomomorphismByIdeal( A, I ) );
end );
#############################################################################
##
#M TrivialSubadditiveMagmaWithZero( <A> ) . . . . . . . . . . for a FLMLOR
##
InstallMethod( TrivialSubadditiveMagmaWithZero,
"for a FLMLOR",
[ IsFLMLOR ],
A -> SubFLMLORNC( A, [] ) );
#############################################################################
##
#M AsFLMLOR( <R>, <D> ) . . view a collection as a FLMLOR over the ring <R>
##
InstallMethod( AsFLMLOR,
"for a ring and a collection",
[ IsRing, IsCollection ],
function( F, D )
local A, L;
D:= AsSSortedList( D );
L:= ShallowCopy( D );
A:= TrivialSubFLMLOR( AsFLMLOR( F, D ) );
SubtractSet( L, AsSSortedList( A ) );
while 0 < Length(L) do
A:= ClosureLeftOperatorRing( A, L[1] );
#T call explicit function that maintains an elements list?
SubtractSet( L, AsSSortedList( A ) );
od;
if Length( AsList( A ) ) <> Length( D ) then
return fail;
fi;
A:= FLMLOR( F, GeneratorsOfLeftOperatorRing( A ), Zero( D[1] ) );
SetAsSSortedList( A, D );
SetSize( A, Length( D ) );
SetIsFinite( A, true );
#T ?
# Return the FLMLOR.
return A;
end );
#############################################################################
##
#M AsFLMLOR( <F>, <V> ) . . view a left module as FLMLOR over the field <F>
##
## is an algebra over <F> that is equal (as set) to <V>.
## For that, perhaps the field of <A> has to be changed before
## getting the correct list of generators.
##
InstallMethod( AsFLMLOR,
"for a division ring and a free left module",
[ IsDivisionRing, IsFreeLeftModule ],
function( F, V )
local L, A;
if LeftActingDomain( V ) = F then
A:= FLMLOR( F, GeneratorsOfLeftModule( V ) );
if A <> V then
return fail;
fi;
if HasBasis( V ) then
SetBasis( A, Basis( V ) );
fi;
elif IsTrivial( V ) then
# We need the zero.
A:= FLMLOR( F, [], Zero( V ) );
elif IsSubset( LeftActingDomain( V ), F ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( F, LeftActingDomain(V) ) ) );
L:= Concatenation( List( L, x -> List( GeneratorsOfLeftModule( V ),
y -> x * y ) ) );
A:= FLMLOR( F, L );
if A <> V then
return fail;
fi;
elif IsSubset( F, LeftActingDomain( V ) ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( LeftActingDomain(V), F ) ) );
if ForAny( L, x -> ForAny( GeneratorsOfLeftModule( V ),
y -> not x * y in V ) ) then
return fail;
fi;
A:= FLMLOR( F, GeneratorsOfLeftModule( V ) );
if A <> V then
return fail;
fi;
else
V:= AsFLMLOR( Intersection( F, LeftActingDomain( V ) ), V );
return AsFLMLOR( F, V );
fi;
UseIsomorphismRelation( V, A );
UseSubsetRelation( V, A );
return A;
end );
#############################################################################
##
#M AsFLMLOR( <F>, <A> ) . . . view an algebra as algebra over the field <F>
##
## is an algebra over <F> that is equal (as set) to <D>.
## For that, perhaps the field of <A> has to be changed before
## getting the correct list of generators.
##
InstallMethod( AsFLMLOR,
"for a division ring and an algebra",
[ IsDivisionRing, IsFLMLOR ],
function( F, D )
local L, A;
if LeftActingDomain( D ) = F then
return D;
elif IsTrivial( D ) then
# We need the zero.
A:= FLMLOR( F, [], Zero( D ) );
elif IsSubset( LeftActingDomain( D ), F ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( F, LeftActingDomain(D) ) ) );
L:= Concatenation( List( L, x -> List( GeneratorsOfAlgebra( D ),
y -> x * y ) ) );
A:= FLMLOR( F, L );
elif IsSubset( F, LeftActingDomain( D ) ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( LeftActingDomain(D), F ) ) );
if ForAny( L, x -> ForAny( GeneratorsOfAlgebra( D ),
y -> not x * y in D ) ) then
return fail;
fi;
A:= FLMLOR( F, GeneratorsOfAlgebra( D ) );
else
D:= AsFLMLOR( Intersection( F, LeftActingDomain( D ) ), D );
return AsFLMLOR( F, D );
fi;
UseIsomorphismRelation( D, A );
UseSubsetRelation( D, A );
return A;
end );
#############################################################################
##
#M AsFLMLORWithOne( <R>, <D> ) . . . . . . view a coll. as a FLMLOR-with-one
##
InstallMethod( AsFLMLORWithOne,
"for a ring and a collection",
[ IsRing, IsCollection ],
function( F, D )
return AsFLMLORWithOne( AsFLMLOR( F, D ) );
end );
#############################################################################
##
#M AsFLMLORWithOne( <F>, <V> ) . . view a left module as an algebra-with-one
##
InstallMethod( AsFLMLORWithOne,
"for a division ring and a free left module",
[ IsDivisionRing, IsFreeLeftModule ],
function( F, V )
local L, A;
# Check that `V' contains the identity.
if One( V ) = fail then
return fail;
elif LeftActingDomain( V ) = F then
A:= FLMLORWithOne( F, GeneratorsOfLeftModule( V ) );
if A <> V then
return fail;
fi;
# Left module generators and basis are maintained.
if HasGeneratorsOfLeftModule( V ) then
SetGeneratorsOfLeftModule( A, GeneratorsOfLeftModule( V ) );
fi;
if HasBasis( V ) then
SetBasis( A, Basis( V ) );
fi;
elif IsSubset( LeftActingDomain( V ), F ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( F, LeftActingDomain(V) ) ) );
L:= Concatenation( List( L, x -> List( GeneratorsOfLeftModule( V ),
y -> x * y ) ) );
A:= FLMLORWithOne( F, L );
if A <> V then
return fail;
fi;
elif IsSubset( F, LeftActingDomain( V ) ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( LeftActingDomain(V), F ) ) );
if ForAny( L, x -> ForAny( GeneratorsOfLeftModule( V ),
y -> not x * y in V ) ) then
return fail;
fi;
A:= FLMLORWithOne( F, GeneratorsOfLeftModule( V ) );
if A <> V then
return fail;
fi;
else
# Note that we need not use the isomorphism and subset relations
# (see below) because this is the task of the calls to
# `AsAlgebraWithOne'.
A:= AsAlgebraWithOne( Intersection( F, LeftActingDomain( V ) ), V );
return AsAlgebraWithOne( F, A );
fi;
UseIsomorphismRelation( V, A );
UseSubsetRelation( V, A );
return A;
end );
#############################################################################
##
#M AsFLMLORWithOne( <F>, <D> ) . . . view an algebra as an algebra-with-one
##
InstallMethod( AsFLMLORWithOne,
"for a division ring and an algebra",
[ IsDivisionRing, IsFLMLOR ],
function( F, D )
local L, A;
# Check that `D' contains the identity.
if One( D ) = fail then
return fail;
elif LeftActingDomain( D ) = F then
A:= FLMLORWithOne( F, GeneratorsOfLeftOperatorRing( D ) );
# Left module generators and basis are maintained.
if HasGeneratorsOfLeftModule( D ) then
SetGeneratorsOfLeftModule( A, GeneratorsOfLeftModule( D ) );
fi;
if HasBasis( D ) then
SetBasis( A, Basis( D ) );
fi;
elif IsSubset( LeftActingDomain( D ), F ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( F, LeftActingDomain(D) ) ) );
L:= Concatenation( List( L, x -> List( GeneratorsOfAlgebra( D ),
y -> x * y ) ) );
A:= FLMLORWithOne( F, L );
elif IsSubset( F, LeftActingDomain( D ) ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( LeftActingDomain(D), F ) ) );
if ForAny( L, x -> ForAny( GeneratorsOfAlgebra( D ),
y -> not x * y in D ) ) then
return fail;
fi;
A:= FLMLORWithOne( F, GeneratorsOfLeftOperatorRing( D ) );
else
# Note that we need not use the isomorphism and subset relations
# (see below) because this is the task of the calls to
# `AsAlgebraWithOne'.
D:= AsAlgebraWithOne( Intersection( F, LeftActingDomain( D ) ), D );
return AsAlgebraWithOne( F, D );
fi;
UseIsomorphismRelation( D, A );
UseSubsetRelation( D, A );
return A;
end );
#############################################################################
##
#M AsFLMLORWithOne( <F>, <D> ) . . view an alg.-with-one as an alg.-with-one
##
InstallMethod( AsFLMLORWithOne,
"for a division ring and an algebra-with-one",
[ IsDivisionRing, IsFLMLORWithOne ],
function( F, D )
local L, A;
if LeftActingDomain( D ) = F then
return D;
elif IsSubset( LeftActingDomain( D ), F ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( F, LeftActingDomain(D) ) ) );
L:= Concatenation( List( L, x -> List( GeneratorsOfAlgebra( D ),
y -> x * y ) ) );
A:= AlgebraWithOne( F, L );
elif IsSubset( F, LeftActingDomain( D ) ) then
# Make sure that the field change does not change the elements.
L:= BasisVectors( Basis( AsField( LeftActingDomain(D), F ) ) );
if ForAny( L, x -> ForAny( GeneratorsOfAlgebra( D ),
y -> not x * y in D ) ) then
return fail;
fi;
A:= AlgebraWithOne( F, GeneratorsOfAlgebra( D ) );
else
# Note that we need not use the isomorphism and subset relations
# (see below) because this is the task of the calls to
# `AsAlgebraWithOne'.
D:= AsAlgebraWithOne( Intersection( F, LeftActingDomain( D ) ), D );
return AsAlgebraWithOne( F, D );
fi;
UseIsomorphismRelation( D, A );
UseSubsetRelation( D, A );
return A;
end );
#############################################################################
##
#M ClosureLeftOperatorRing( <A>, <a> ) . . . . . . . closure with an element
##
InstallMethod( ClosureLeftOperatorRing,
"for a FLMLOR and a ring element",
#T why not a general function for any left operator ring?
#T (need `LeftOperatorRingByGenerators' ?)
IsCollsElms,
[ IsFLMLOR, IsRingElement ],
function( A, a )
# if possible test if the element lies in the ring already,
if HasGeneratorsOfLeftOperatorRing( A )
and a in GeneratorsOfLeftOperatorRing( A ) then
return A;
# otherwise make a new left operator ring
else
return FLMLOR( LeftActingDomain( A ),
Concatenation( GeneratorsOfLeftOperatorRing( A ), [ a ] ) );
fi;
end );
InstallMethod( ClosureLeftOperatorRing,
"for an FLMLOR with basis, and a ring element",
IsCollsElms,
[ IsFLMLOR and HasBasis, IsRingElement ],
function( A, a )
# test if the element lies in the FLMLOR already,
if a in A then
return A;
# otherwise make a new FLMLOR
else
return FLMLOR( LeftActingDomain( A ),
#T FLMLORByGenerators?
Concatenation( BasisVectors( Basis( A ) ), [ a ] ),
"basis" );
fi;
end );
InstallMethod( ClosureLeftOperatorRing,
"for a FLMLOR-with-one and a ring element",
IsCollsElms,
[ IsFLMLORWithOne, IsRingElement ],
function( A, a )
# if possible test if the element lies in the ring already,
if a in GeneratorsOfLeftOperatorRingWithOne( A ) then
return A;
# otherwise make a new left operator ring-with-one
else
return FLMLORWithOne( LeftActingDomain( A ),
Concatenation( GeneratorsOfLeftOperatorRingWithOne( A ),
[ a ] ) );
fi;
end );
InstallMethod( ClosureLeftOperatorRing,
"for a FLMLOR-with-one with basis, and a ring element",
IsCollsElms,
[ IsFLMLORWithOne and HasBasis, IsRingElement ],
function( A, a )
# test if the element lies in the FLMLOR already,
if a in A then
return A;
# otherwise make a new FLMLOR-with-one
else
return FLMLORWithOne( LeftActingDomain( A ),
Concatenation( BasisVectors( Basis( A ) ), [ a ] ),
"basis" );
fi;
end );
InstallMethod( ClosureLeftOperatorRing,
"for a FLMLOR containing the whole family, and a ring element",
IsCollsElms,
[ IsFLMLOR and IsWholeFamily, IsRingElement ],
SUM_FLAGS, # this is better than everything else
ReturnFirst);
#############################################################################
##
#M ClosureLeftOperatorRing( <A>, <U> ) . closure of two left operator rings
##
InstallMethod( ClosureLeftOperatorRing,
"for two left operator rings",
IsIdenticalObj,
[ IsLeftOperatorRing, IsLeftOperatorRing ],
function( A, S )
local g; # one generator
for g in GeneratorsOfLeftOperatorRing( S ) do
A := ClosureLeftOperatorRing( A, g );
od;
return A;
end );
InstallMethod( ClosureLeftOperatorRing,
"for two left operator rings-with-one",
IsIdenticalObj,
[ IsLeftOperatorRingWithOne, IsLeftOperatorRingWithOne ],
function( A, S )
local g; # one generator
for g in GeneratorsOfLeftOperatorRingWithOne( S ) do
A := ClosureLeftOperatorRing( A, g );
od;
return A;
end );
InstallMethod( ClosureLeftOperatorRing,
"for a left op. ring cont. the whole family, and a collection",
IsIdenticalObj,
[ IsLeftOperatorRing and IsWholeFamily, IsCollection ],
SUM_FLAGS, # this is better than everything else
ReturnFirst);
#############################################################################
##
#M ClosureLeftOperatorRing( <A>, <list> ) . . . . closure of left op. ring
##
InstallMethod( ClosureLeftOperatorRing,
"for left operator ring and list of elements",
IsIdenticalObj,
[ IsLeftOperatorRing, IsCollection ],
function( A, list )
local g; # one generator
for g in list do
A:= ClosureLeftOperatorRing( A, g );
od;
return A;
end );
#############################################################################
##
#F MutableBasisOfClosureUnderAction( <F>, <Agens>, <from>, <init>, <opr>,
#F <zero>, <maxdim> )
##
## This function is used to compute bases of finite dimensional ideals $I$
## in *associative* algebras that are given by ideal generators $J$ and
## (generators of) the acting algebra $A$.
## An important special case is that of the algebra $A$ itself, given by
## algebra generators.
##
## The algorithm assumes that it is possible to deal with mutable bases of
## vector spaces generated by elements of $A$.
## It proceeds as follows.
##
## Let $A$ be a finite dimensional algebra over the ring $F$,
## and $I$ a two-sided ideal in $A$ that is generated (as a two-sided ideal)
## by the set $J$.
## (For the cases of one-sided ideals, see below.)
##
## Let $S$ be a set of algebra generators of $A$.
## The identity of $A$, if exists, need not be contained in $S$.
##
## Each element $x$ in $I$ can be written as a linear combination of
## products $j a_1 a_2 \cdots a_n$, with $j \in J$ and $a_i \in S$ for
## $1 \leq i \leq n$.
## Let $l(x)$ denote the minimum of the largest $n$ for involved words
## $a_1 a_2 \cdots a_n$, taken over all possible expressions for $x$.
##
## Define $I_i = \{ x \in I \mid l(x) \leq i \}$.
## Then $I_i$ is an $F$-space, $A_0 = \langle J \rangle_F$,
## and $I_0 \< I_1 \< I_2 \< \cdots I_k \< I_{k+1} \< \ldots$
## is an ascending chain that eventually reaches $I$.
## For $i > 0$ we have
## $I_{i+1} = \langle I_i\cup\bigcup_{s\in S} ( I_i s\cup s I_i )\rangle_F$.
##
## (*Note* that the computation of the $I_i$ gives us the smallest value $k$
## such that every element is a linear combination of words in terms of the
## algebra generators, of maximal length $k$.)
##
InstallGlobalFunction( MutableBasisOfClosureUnderAction,
function( F, Agens, from, init, opr, zero, maxdim )
local MB, # mutable basis, result
gen, # loop over generators
v, #
dim, # dimension of the actual left module
right, # `true' if we have to multiply from the right
left; # `true' if we have to multiply from the left
# Get the side(s) from where to multiply.
left := true;
right := true;
if from = "left" then
right:= false;
elif from = "right" then
left:= false;
fi;
# $I_0$
MB := MutableBasis( F, init, zero );
dim := 0;
while dim < NrBasisVectors( MB ) and dim < maxdim do
# `MB' is a mutable basis of $I_i$.
dim:= NrBasisVectors( MB );
if right then
# Compute $I^{\prime}_i = I_i + \sum_{s \in S} I_i s$.
for gen in Agens do
for v in BasisVectors( MB ) do
CloseMutableBasis( MB, opr( v, gen ) );
od;
od;
fi;
if left then
# Compute $I_i + \sum_{s \in S} s I_i$
# resp. $I^{\prime}_i + \sum_{s \in S} s I_i$.
for gen in Agens do
for v in BasisVectors( MB ) do
CloseMutableBasis( MB, opr( gen, v ) );
od;
od;
fi;
od;
# Return the mutable basis.
return MB;
end );
#############################################################################
##
#F MutableBasisOfNonassociativeAlgebra( <F>, <Agens>, <zero>, <maxdim> )
##
InstallGlobalFunction( MutableBasisOfNonassociativeAlgebra,
function( F, Agens, zero, maxdim )
local MB, # mutable basis, result
dim, # dimension of the current left module
bv, # current basis vectors
v, w; # loop over basis vectors found already
MB := MutableBasis( F, Agens, zero );
dim := 0;
while dim < NrBasisVectors( MB ) and dim < maxdim do
dim := NrBasisVectors( MB );
bv := BasisVectors( MB );
for v in bv do
for w in bv do
CloseMutableBasis( MB, v * w );
CloseMutableBasis( MB, w * v );
od;
od;
od;
# Return the mutable basis.
return MB;
end );
#############################################################################
##
#F MutableBasisOfIdealInNonassociativeAlgebra( <F>, <Vgens>, <Igens>,
#F <zero>, <from>, <maxdim> )
##
InstallGlobalFunction( MutableBasisOfIdealInNonassociativeAlgebra,
function( F, Vgens, Igens, zero, from, maxdim )
local MB, # mutable basis, result
dim, # dimension of the current left module
bv, # current basis vectors
right, # `true' if we have to multiply from the right
left, # `true' if we have to multiply from the left
v, gen; # loop over basis vectors found already
# Get the side(s) from where to multiply.
left := true;
right := true;
if from = "left" then
right:= false;
elif from = "right" then
left:= false;
fi;
dim := 0;
MB := MutableBasis( F, Igens, zero );
while dim < NrBasisVectors( MB ) and dim < maxdim do
dim := NrBasisVectors( MB );
bv := BasisVectors( MB );
for v in bv do
for gen in Vgens do
if left then
CloseMutableBasis( MB, gen * v );
fi;
if right then
CloseMutableBasis( MB, v * gen );
fi;
od;
od;
od;
# Return the mutable basis.
return MB;
end );
#############################################################################
##
#M IsSubset( <A>, <B> ) . . . . . . . . . . . . test for subset of FLMLORs
##
## These methods are preferable to that for free left modules because they
## use algebra generators.
##
## We assume that generators of an extension of the left acting domains can
## be computed if they are fields; note that infinite field extensions do
## not (yet) occur in {\GAP} as `LeftActingDomain' values.
##
InstallMethod( IsSubset,
"for two FLMLORs",
IsIdenticalObj,
[ IsFLMLOR, IsFLMLOR ],
function( D1, D2 )
local F1, F2;
F1:= LeftActingDomain( D1 );
F2:= LeftActingDomain( D2 );
if not ( HasIsDivisionRing( F1 ) and IsDivisionRing( F1 ) and
HasIsDivisionRing( F2 ) and IsDivisionRing( F2 ) ) then
TryNextMethod();
fi;
# catch trivial case
if IsSubset(GeneratorsOfLeftOperatorRing(D1),
GeneratorsOfLeftOperatorRing(D2)) then
return true;
fi;
return IsSubset( D1, GeneratorsOverIntersection( D2,
GeneratorsOfLeftOperatorRing( D2 ),
F2, F1 ) );
end );
InstallMethod( IsSubset,
"for two FLMLORs-with-one",
IsIdenticalObj,
[ IsFLMLORWithOne, IsFLMLORWithOne ],
function( D1, D2 )
local F1, F2;
F1:= LeftActingDomain( D1 );
F2:= LeftActingDomain( D2 );
if not ( HasIsDivisionRing( F1 ) and IsDivisionRing( F1 ) and
HasIsDivisionRing( F2 ) and IsDivisionRing( F2 ) ) then
TryNextMethod();
fi;
return IsSubset( D1, GeneratorsOverIntersection( D2,
GeneratorsOfLeftOperatorRingWithOne( D2 ),
F2, F1 ) );
end );
#############################################################################
##
#M ViewObj( <A> ) . . . . . . . . . . . . . . . . . . . . . . view a FLMLOR
##
## print left acting domain, if known also dimension or no. of generators
##
InstallMethod( ViewObj,
"for a FLMLOR",
[ IsFLMLOR ],
function( A )
Print( "<free left module over ", LeftActingDomain( A ), ", and ring>" );
end );
InstallMethod( ViewObj,
"for a FLMLOR with known dimension",
[ IsFLMLOR and HasDimension ], 1, # override method requiring gens.
function( A )
Print( "<free left module of dimension ", Dimension( A ),
" over ", LeftActingDomain( A ), ", and ring>" );
end );
InstallMethod( ViewObj,
"for a FLMLOR with known generators",
[ IsFLMLOR and HasGeneratorsOfAlgebra ],
function( A )
Print( "<free left module over ", LeftActingDomain( A ),
", and ring, with ",
Pluralize( Length( GeneratorsOfFLMLOR( A ) ), "generator" ), ">" );
end );
#############################################################################
##
#M PrintObj( <A> ) . . . . . . . . . . . . . . . . . . . . . print a FLMLOR
##
InstallMethod( PrintObj,
"for a FLMLOR",
[ IsFLMLOR ],
function( A )
Print( "FLMLOR( ", LeftActingDomain( A ), ", ... )" );
end );
InstallMethod( PrintObj,
"for a FLMLOR with known generators",
[ IsFLMLOR and HasGeneratorsOfFLMLOR ],
function( A )
if IsEmpty( GeneratorsOfFLMLOR( A ) ) then
Print( "FLMLOR( ", LeftActingDomain( A ), ", [], ", Zero( A ), " )" );
else
Print( "FLMLOR( ", LeftActingDomain( A ), ", ",
GeneratorsOfFLMLOR( A ), " )" );
fi;
end );
#############################################################################
##
#M ViewObj( <A> ) . . . . . . . . . . . . . . . . . view a FLMLOR-with-one
##
## print left acting domain, if known also dimension or no. of generators
##
InstallMethod( ViewObj,
"for a FLMLOR-with-one",
[ IsFLMLORWithOne ],
function( A )
Print( "<free left module over ", LeftActingDomain( A ),
", and ring-with-one>" );
end );
InstallMethod( ViewObj,
"for a FLMLOR-with-one with known dimension",
[ IsFLMLORWithOne and HasDimension ], 1, # override method requ. gens.
function( A )
Print( "<free left module of dimension ", Dimension( A ),
" over ", LeftActingDomain( A ), ", and ring-with-one>" );
end );
InstallMethod( ViewObj,
"for a FLMLOR-with-one with known generators",
[ IsFLMLORWithOne and HasGeneratorsOfFLMLORWithOne ],
function( A )
Print( "<free left module over ", LeftActingDomain( A ),
", and ring-with-one, with ",
Pluralize( Length( GeneratorsOfAlgebraWithOne( A ) ), "generator" ),
">" );
end );
#############################################################################
##
#M PrintObj( <A> ) . . . . . . . . . . . . . . . . . print a FLMLOR-with-one
##
InstallMethod( PrintObj,
"for a FLMLOR-with-one",
[ IsFLMLORWithOne ],
function( A )
Print( "FLMLORWithOne( ", LeftActingDomain( A ), ", ... )" );
end );
InstallMethod( PrintObj,
"for a FLMLOR-with-one with known generators",
[ IsFLMLORWithOne and HasGeneratorsOfFLMLOR ],
function( A )
if IsEmpty( GeneratorsOfFLMLORWithOne( A ) ) then
Print( "FLMLORWithOne( ", LeftActingDomain( A ), ", [], ",
Zero( A ), " )" );
else
Print( "FLMLORWithOne( ", LeftActingDomain( A ), ", ",
GeneratorsOfFLMLORWithOne( A ), " )" );
fi;
end );
#############################################################################
##
#M ViewObj( <A> ) . . . . . . . . . . . . . . . . . . . . . view an algebra
##
## print left acting domain, if known also dimension or no. of generators
##
InstallMethod( ViewObj,
"for an algebra",
[ IsAlgebra ],
function( A )
Print( "<algebra over ", LeftActingDomain( A ), ">" );
end );
InstallMethod( ViewObj,
"for an algebra with known dimension",
[ IsAlgebra and HasDimension ], 1, # override method requiring gens.
function( A )
Print( "<algebra of dimension ", Dimension( A ),
" over ", LeftActingDomain( A ), ">" );
end );
InstallMethod( ViewObj,
"for an algebra with known generators",
[ IsAlgebra and HasGeneratorsOfAlgebra ],
function( A )
Print( "<algebra over ", LeftActingDomain( A ), ", with ",
Pluralize( Length( GeneratorsOfAlgebra( A ) ), "generator" ), ">" );
end );
#############################################################################
##
#M PrintObj( <A> ) . . . . . . . . . . . . . . . . . . . . print an algebra
##
InstallMethod( PrintObj,
"for an algebra",
[ IsAlgebra ],
function( A )
Print( "Algebra( ", LeftActingDomain( A ), ", ... )" );
end );
InstallMethod( PrintObj,
"for an algebra with known generators",
[ IsAlgebra and HasGeneratorsOfAlgebra ],
function( A )
if IsEmpty( GeneratorsOfAlgebra( A ) ) then
Print( "Algebra( ", LeftActingDomain( A ), ", [], ", Zero( A ), " )" );
else
Print( "Algebra( ", LeftActingDomain( A ), ", ",
GeneratorsOfAlgebra( A ), " )" );
fi;
end );
#############################################################################
##
#M ViewObj( <A> ) . . . . . . . . . . . . . . . . view an algebra-with-one
##
## print left acting domain, if known also dimension or no. of generators
##
InstallMethod( ViewObj, "for an algebra-with-one", [ IsAlgebraWithOne ],
function( A )
if IsIdenticalObj(A,LeftActingDomain(A)) then
Print( "<algebra-with-one over itself>" );
else
Print( "<algebra-with-one over ", LeftActingDomain( A ), ">" );
fi;
end );
InstallMethod( ViewObj,
"for an algebra-with-one with known dimension",
[ IsAlgebraWithOne and HasDimension ], 1, # override method requ. gens.
function( A )
Print( "<algebra-with-one of dimension ", Dimension( A ),
" over ", LeftActingDomain( A ), ">" );
end );
InstallMethod( ViewObj,
"for an algebra-with-one with known generators",
[ IsAlgebraWithOne and HasGeneratorsOfAlgebraWithOne ],
function( A )
Print( "<algebra-with-one over ", LeftActingDomain( A ), ", with ",
Pluralize( Length( GeneratorsOfAlgebraWithOne( A ) ), "generator" ),
">" );
end );
#############################################################################
##
#M PrintObj( <A> ) . . . . . . . . . . . . . . . . print an algebra-with-one
##
InstallMethod( PrintObj,
"for an algebra-with-one",
[ IsAlgebraWithOne ],
function( A )
Print( "AlgebraWithOne( ", LeftActingDomain( A ), ", ... )" );
end );
InstallMethod( PrintObj,
"for an algebra-with-one with known generators",
[ IsAlgebraWithOne and HasGeneratorsOfAlgebra ],
function( A )
if IsEmpty( GeneratorsOfAlgebraWithOne( A ) ) then
Print( "AlgebraWithOne( ", LeftActingDomain( A ), ", [], ",
Zero( A ), " )" );
else
Print( "AlgebraWithOne( ", LeftActingDomain( A ), ", ",
GeneratorsOfAlgebraWithOne( A ), " )" );
fi;
end );
#############################################################################
##
#M ViewObj( <A> ) . . . . . . . . . . . . . . . . . . view a Lie algebra
##
## print left acting domain, if known also dimension or no. of generators
##
InstallMethod( ViewObj,
"for a Lie algebra",
[ IsLieAlgebra ],
function( A )
Print( "<Lie algebra over ", LeftActingDomain( A ), ">" );
end );
InstallMethod( ViewObj,
"for a Lie algebra with known dimension",
[ IsLieAlgebra and HasDimension ], 1, # override method requ. gens.
function( A )
Print( "<Lie algebra of dimension ", Dimension( A ),
" over ", LeftActingDomain( A ), ">" );
end );
InstallMethod( ViewObj,
"for a Lie algebra with known generators",
[ IsLieAlgebra and HasGeneratorsOfAlgebra ],
function( A )
Print( "<Lie algebra over ", LeftActingDomain( A ), ", with ",
Pluralize( Length( GeneratorsOfAlgebra( A ) ), "generator" ), ">" );
end );
#############################################################################
##
#M AsSubalgebra(<A>, <U>) . view an algebra as subalgebra of another algebra
##
InstallMethod( AsSubalgebra,
"for two algebras",
IsIdenticalObj,
[ IsAlgebra, IsAlgebra ],
function( A, U )
local samecoeffs, S;
if not IsSubset( A, U ) then
return fail;
fi;
# Construct the generators list.
samecoeffs:= LeftActingDomain( A ) = LeftActingDomain( U );
if not samecoeffs then
U:= AsAlgebra( LeftActingDomain( A ), U );
fi;
# Construct the subalgebra.
S:= SubalgebraNC( A, GeneratorsOfAlgebra( U ) );
# Maintain useful information.
UseIsomorphismRelation( U, S );
UseSubsetRelation( U, S );
if samecoeffs and HasDimension( U ) then
SetDimension( S, Dimension( U ) );
fi;
# Return the subalgebra.
return S;
end );
InstallMethod( AsSubalgebra,
"for an algebra and an algebra-with-one",
IsIdenticalObj,
[ IsAlgebra, IsAlgebraWithOne ],
function( A, U )
local samecoeffs, S;
if not IsSubset( A, U ) then
return fail;
fi;
# Construct the generators list.
samecoeffs:= LeftActingDomain( A ) = LeftActingDomain( U );
if not samecoeffs then
U:= AsAlgebraWithOne( LeftActingDomain( A ), U );
fi;
# Construct the subalgebra.
S:= SubalgebraWithOneNC( A, GeneratorsOfAlgebraWithOne( U ) );
# Maintain useful information.
UseIsomorphismRelation( U, S );
UseSubsetRelation( U, S );
if samecoeffs and HasDimension( U ) then
SetDimension( S, Dimension( U ) );
fi;
# Return the subalgebra.
return S;
end );
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