|
#############################################################################
##
## 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 elements of algebras given by structure
## constants (s.~c.).
##
## The family of s.~c. algebra elements has the following components.
##
## `sctable' :
## the structure constants table,
## `names' :
## list of names of the basis vectors (for printing only),
## `zerocoeff' :
## the zero coefficient (needed already for the s.~c. table),
## `defaultTypeDenseCoeffVectorRep' :
## the type of s.~c. algebra elements that are represented by
## a dense list of coefficients.
##
## If the family has *not* the category `IsFamilyOverFullCoefficientsFamily'
## then it has the component `coefficientsDomain'.
##
#T need for the norm of a quaternion?
#T (note: returns an element in the coefficients domain, not in the algebra!
#T f( a b[1] + b b[2] + c b[3] + d b[4] ) = a^2 +b^2 +c^2 + d^2.)
#T
#T NormQuat := function( quat )
#T if not IsQuaternion( quat ) then
#T Error( "<quat> must be a quaternion" );
#T fi;
#T return Sum( List( ExtRepOfObj( quat ), c -> c^2 ) );
#T end;
#############################################################################
##
#M IsWholeFamily( <V> ) . . . . . . . for s.~c. algebra elements collection
##
InstallMethod( IsWholeFamily,
"for s. c. algebra elements collection",
[ IsSCAlgebraObjCollection and IsLeftModule and IsFreeLeftModule ],
function( V )
local Fam;
Fam:= ElementsFamily( FamilyObj( V ) );
if IsFamilyOverFullCoefficientsFamily( Fam ) then
return IsWholeFamily( LeftActingDomain( V ) )
and IsFullSCAlgebra( V );
else
return LeftActingDomain( V ) = Fam!.coefficientsDomain
and IsFullSCAlgebra( V );
fi;
end );
#############################################################################
##
#M IsFullSCAlgebra( <V> ) . . . . . . for s.~c. algebra elements collection
##
InstallMethod( IsFullSCAlgebra,
"for s. c. algebra elements collection",
[ IsSCAlgebraObjCollection and IsAlgebra ],
V -> Dimension(V) = Length( ElementsFamily( FamilyObj( V ) )!.names ) );
#############################################################################
##
#R IsDenseCoeffVectorRep( <obj> )
##
## This representation uses a coefficients vector
## w.r.t. the basis that is known for the whole family.
##
## The external representation is the coefficients vector,
## which is stored at position 1 in the object.
##
if IsHPCGAP then
DeclareRepresentation( "IsDenseCoeffVectorRep",
IsAtomicPositionalObjectRep, [ 1 ] );
else
DeclareRepresentation( "IsDenseCoeffVectorRep",
IsPositionalObjectRep, [ 1 ] );
fi;
#############################################################################
##
#M ObjByExtRep( <Fam>, <descr> ) . . . . . . . . for s.~c. algebra elements
##
## Check whether the coefficients list <coeffs> has the right length,
## and lies in the correct family.
## If the coefficients family of <Fam> has a uniquely determined zero
## element, we need to check only whether the family of <descr> is the
## collections family of the coefficients family of <Fam>.
##
InstallMethod( ObjByExtRep,
"for s. c. algebra elements family",
[ IsSCAlgebraObjFamily, IsHomogeneousList ],
function( Fam, coeffs )
if IsFamilyOverFullCoefficientsFamily( Fam )
or not IsBound( Fam!.coefficientsDomain ) then
TryNextMethod();
elif Length( coeffs ) <> Length( Fam!.names ) then
Error( "<coeffs> must be a list of length ", Length( Fam!.names ) );
elif not ForAll( coeffs, c -> c in Fam!.coefficientsDomain ) then
Error( "all in <coeffs> must lie in `<Fam>!.coefficientsDomain'" );
fi;
return Objectify( Fam!.defaultTypeDenseCoeffVectorRep,
[ Immutable( coeffs ) ] );
end );
InstallMethod( ObjByExtRep,
"for s. c. alg. elms. family with coefficients family",
[ IsSCAlgebraObjFamily and IsFamilyOverFullCoefficientsFamily,
IsHomogeneousList ],
function( Fam, coeffs )
if not IsIdenticalObj( CoefficientsFamily( Fam ),
ElementsFamily( FamilyObj( coeffs ) ) ) then
Error( "family of <coeffs> does not fit to <Fam>" );
elif Length( coeffs ) <> Length( Fam!.names ) then
Error( "<coeffs> must be a list of length ", Length( Fam!.names ) );
fi;
return Objectify( Fam!.defaultTypeDenseCoeffVectorRep,
[ Immutable( coeffs ) ] );
end );
#############################################################################
##
#M ExtRepOfObj( <elm> ) . . . . . . . . . . . . for s.~c. algebra elements
##
InstallMethod( ExtRepOfObj,
"for s. c. algebra element in dense coeff. vector rep.",
[ IsSCAlgebraObj and IsDenseCoeffVectorRep ],
elm -> elm![1] );
#############################################################################
##
#M Print( <elm> ) . . . . . . . . . . . . . . . for s.~c. algebra elements
##
InstallMethod( PrintObj,
"for s. c. algebra element",
[ IsSCAlgebraObj ],
function( elm )
local F, # family of `elm'
names, # generators names
len, # dimension of the algebra
zero, # zero element of the ring
depth, # first nonzero position in coefficients list
one, # identity element of the ring
i; # loop over the coefficients list
F := FamilyObj( elm );
names := F!.names;
elm := ExtRepOfObj( elm );
len := Length( elm );
# Treat the case that the algebra is trivial.
if len = 0 then
Print( "<zero of trivial s.c. algebra>" );
return;
fi;
depth := PositionNonZero( elm );
if len < depth then
# Print the zero element.
# (Note that the unique element of a zero algebra has a name.)
Print( "0*", names[1] );
else
one:= One( elm[1] );
zero:= Zero( elm[1] );
if elm[ depth ] <> one then
Print( "(", elm[ depth ], ")*" );
fi;
Print( names[ depth ] );
for i in [ depth+1 .. len ] do
if elm[i] <> zero then
Print( "+" );
if elm[i] <> one then
Print( "(", elm[i], ")*" );
fi;
Print( names[i] );
fi;
od;
fi;
end );
#############################################################################
##
#M String( <elm> ) . . . . . . . . . . . . . . . for s.~c. algebra elements
##
InstallMethod( String,
"for s. c. algebra element",
[ IsSCAlgebraObj ],
function( elm )
local F, # family of `elm'
s, # string
names, # generators names
len, # dimension of the algebra
zero, # zero element of the ring
depth, # first nonzero position in coefficients list
one, # identity element of the ring
i; # loop over the coefficients list
F := FamilyObj( elm );
names := F!.names;
elm := ExtRepOfObj( elm );
len := Length( elm );
# Treat the case that the algebra is trivial.
if len = 0 then
return "<zero of trivial s.c. algebra>";
fi;
depth := PositionNonZero( elm );
s:="";
if len < depth then
# Print the zero element.
# (Note that the unique element of a zero algebra has a name.)
Append(s, "0*");
Append(s,names[1]);
else
one:= One( elm[1] );
zero:= Zero( elm[1] );
if elm[ depth ] <> one then
Add(s,'(');
Append(s,String(elm[ depth ]));
Append(s, ")*" );
fi;
Append(s, names[ depth ] );
for i in [ depth+1 .. len ] do
if elm[i] <> zero then
Add(s, '+' );
if elm[i] <> one then
Add(s,'(');
Append(s,String(elm[ i ]));
Append(s, ")*" );
fi;
Append(s, names[ i ] );
fi;
od;
fi;
return s;
end );
#############################################################################
##
#M One( <Fam> )
##
## Compute the identity (if exists) from the s.~c. table.
##
InstallMethod( One,
"for family of s. c. algebra elements",
[ IsSCAlgebraObjFamily ],
function( F )
local one;
one:= IdentityFromSCTable( F!.sctable );
if one <> fail then
one:= ObjByExtRep( F, one );
fi;
return one;
end );
#############################################################################
##
#M \=( <x>, <y> ) . . . . . . . . . . equality of two s.~c. algebra objects
#M \<( <x>, <y> ) . . . . . . . . . comparison of two s.~c. algebra objects
#M \+( <x>, <y> ) . . . . . . . . . . . . sum of two s.~c. algebra objects
#M \-( <x>, <y> ) . . . . . . . . . difference of two s.~c. algebra objects
#M \*( <x>, <y> ) . . . . . . . . . . product of two s.~c. algebra objects
#M Zero( <x> ) . . . . . . . . . . . . . . zero of an s.~c. algebra element
#M AdditiveInverse( <x> ) . . additive inverse of an s.~c. algebra element
#M Inverse( <x> ) . . . . . . . . . . . inverse of an s.~c. algebra element
##
InstallMethod( \=,
"for s. c. algebra elements",
IsIdenticalObj,
[ IsSCAlgebraObj, IsSCAlgebraObj ],
function( x, y ) return ExtRepOfObj( x ) = ExtRepOfObj( y ); end );
InstallMethod( \=,
"for s. c. algebra elements in dense vector rep.",
IsIdenticalObj,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep,
IsSCAlgebraObj and IsDenseCoeffVectorRep ],
function( x, y ) return x![1] = y![1]; end );
InstallMethod( \<,
"for s. c. algebra elements",
IsIdenticalObj,
[ IsSCAlgebraObj, IsSCAlgebraObj ],
function( x, y ) return ExtRepOfObj( x ) < ExtRepOfObj( y ); end );
InstallMethod( \<,
"for s. c. algebra elements in dense vector rep.",
IsIdenticalObj,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep,
IsSCAlgebraObj and IsDenseCoeffVectorRep ], 0,
function( x, y ) return x![1] < y![1]; end );
InstallMethod( \+,
"for s. c. algebra elements",
IsIdenticalObj,
[ IsSCAlgebraObj, IsSCAlgebraObj ],
function( x, y )
return ObjByExtRep( FamilyObj(x), ExtRepOfObj(x) + ExtRepOfObj(y) );
end );
InstallMethod( \+,
"for s. c. algebra elements in dense vector rep.",
IsIdenticalObj,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep,
IsSCAlgebraObj and IsDenseCoeffVectorRep ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), x![1] + y![1] );
end );
InstallMethod( \-,
"for s. c. algebra elements",
IsIdenticalObj,
[ IsSCAlgebraObj, IsSCAlgebraObj ],
function( x, y )
return ObjByExtRep( FamilyObj(x), ExtRepOfObj(x) - ExtRepOfObj(y) );
end );
InstallMethod( \-,
"for s. c. algebra elements in dense vector rep.",
IsIdenticalObj,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep,
IsSCAlgebraObj and IsDenseCoeffVectorRep ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), x![1] - y![1] );
end );
InstallMethod( \*,
"for s. c. algebra elements",
IsIdenticalObj,
[ IsSCAlgebraObj, IsSCAlgebraObj ],
function( x, y )
local F;
F:= FamilyObj( x );
return ObjByExtRep( F, SCTableProduct( F!.sctable,
ExtRepOfObj( x ), ExtRepOfObj( y ) ) );
end );
InstallMethod( \*,
"for s. c. algebra elements in dense vector rep.",
IsIdenticalObj,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep,
IsSCAlgebraObj and IsDenseCoeffVectorRep ],
function( x, y )
local F;
F:= FamilyObj( x );
return ObjByExtRep( F, SCTableProduct( F!.sctable, x![1], y![1] ) );
end );
InstallMethod( \*,
"for ring element and s. c. algebra element",
IsCoeffsElms,
[ IsRingElement, IsSCAlgebraObj ],
function( x, y )
return ObjByExtRep( FamilyObj( y ), x * ExtRepOfObj( y ) );
end );
InstallMethod( \*,
"for ring element and s. c. algebra element in dense vector rep.",
IsCoeffsElms,
[ IsRingElement, IsSCAlgebraObj and IsDenseCoeffVectorRep ],
function( x, y )
return ObjByExtRep( FamilyObj( y ), x * y![1] );
end );
InstallMethod( \*,
"for s. c. algebra element and ring element",
IsElmsCoeffs,
[ IsSCAlgebraObj, IsRingElement ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), ExtRepOfObj( x ) * y );
end );
InstallMethod( \*,
"for s. c. algebra element in dense vector rep. and ring element",
IsElmsCoeffs,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep, IsRingElement ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), x![1] * y );
end );
InstallMethod( \*,
"for integer and s. c. algebra element",
[ IsInt, IsSCAlgebraObj ],
function( x, y )
return ObjByExtRep( FamilyObj( y ), x * ExtRepOfObj( y ) );
end );
InstallMethod( \*,
"for integer and s. c. algebra element in dense vector rep.",
[ IsInt, IsSCAlgebraObj and IsDenseCoeffVectorRep ],
function( x, y )
return ObjByExtRep( FamilyObj( y ), x * y![1] );
end );
InstallMethod( \*,
"for s. c. algebra element and integer",
[ IsSCAlgebraObj, IsInt ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), ExtRepOfObj( x ) * y );
end );
InstallMethod( \*,
"for s. c. algebra element in dense vector rep. and integer",
[ IsSCAlgebraObj and IsDenseCoeffVectorRep, IsInt ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), x![1] * y );
end );
InstallMethod( \/,
"for s. c. algebra element and scalar",
IsElmsCoeffs,
[ IsSCAlgebraObj, IsScalar ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), ExtRepOfObj( x ) / y );
end );
InstallMethod( \/,
"for s. c. algebra element in dense vector rep. and scalar",
IsElmsCoeffs,
[ IsSCAlgebraObj and IsDenseCoeffVectorRep, IsScalar ],
function( x, y )
return ObjByExtRep( FamilyObj( x ), x![1] / y );
end );
InstallMethod( ZeroOp,
"for s. c. algebra element",
[ IsSCAlgebraObj ],
x -> ObjByExtRep( FamilyObj( x ), Zero( ExtRepOfObj( x ) ) ) );
InstallMethod( AdditiveInverseOp,
"for s. c. algebra element",
[ IsSCAlgebraObj ],
x -> ObjByExtRep( FamilyObj( x ),
AdditiveInverse( ExtRepOfObj( x ) ) ) );
InstallOtherMethod( OneOp,
"for s. c. algebra element",
[ IsSCAlgebraObj ],
function( x )
local F, one;
F:= FamilyObj( x );
one:= IdentityFromSCTable( F!.sctable );
if one <> fail then
one:= ObjByExtRep( F, one );
fi;
return one;
end );
InstallOtherMethod( InverseOp,
"for s. c. algebra element",
[ IsSCAlgebraObj ],
function( x )
local one, F;
one:= One( x );
if one <> fail then
F:= FamilyObj( x );
one:= QuotientFromSCTable( F!.sctable, ExtRepOfObj( one ),
ExtRepOfObj( x ) );
if one <> fail then
one:= ObjByExtRep( F, one );
fi;
fi;
return one;
end );
#############################################################################
##
#M \in( <a>, <A> )
##
InstallMethod( \in,
"for s. c. algebra element, and full s. c. algebra",
IsElmsColls,
[ IsSCAlgebraObj, IsFullSCAlgebra ],
function( a, A )
return IsSubset( LeftActingDomain( A ), ExtRepOfObj( a ) );
end );
#############################################################################
##
#F AlgebraByStructureConstants( <R>, <sctable> )
#F AlgebraByStructureConstants( <R>, <sctable>, <name> )
#F AlgebraByStructureConstants( <R>, <sctable>, <names> )
#F AlgebraByStructureConstants( <R>, <sctable>, <name1>, <name2>, ... )
##
## is an algebra $A$ over the ring <R>, defined by the structure constants
## table <sctable> of length $n$.
##
## The generators of $A$ are linearly independent abstract space generators
## $x_1, x_2, \ldots, x_n$ which are multiplied according to the formula
## $ x_i x_j = \sum_{k=1}^n c_{ijk} x_k$
## where `$c_{ijk}$ = <sctable>[i][j][1][i_k]'
## and `<sctable>[i][j][2][i_k] = k'.
##
BindGlobal( "AlgebraByStructureConstantsArg",
function( arglist, filter, one_coeffs... )
local T, # structure constants table
n, # dimensions of structure matrices
R, # coefficients ring
zero, # zero of `R'
names, # names of the algebra generators
Fam, # the family of algebra elements
A, # the algebra, result
gens, # algebra generators of `A'
one; # multiplicative identity, if available
# Check the argument list.
if not 1 < Length( arglist ) and IsRing( arglist[1] )
and IsList( arglist[2] ) then
Error( "usage: AlgebraByStructureConstantsArg([<R>,<sctable>]) or \n",
"AlgebraByStructureConstantsArg([<R>,<sctable>,<name1>,...])" );
fi;
# Check the s.~c. table.
#T really do this?
R := arglist[1];
zero := Zero( R );
T := arglist[2];
if zero = Last(T) then
T:= Immutable( T );
else
if Last(T) = 0 then
T:= ReducedSCTable( T, One( zero ) );
else
Error( "<R> and <T> are not compatible" );
fi;
fi;
if Length( T ) = 2 then
n:= 0;
else
n:= Length( T[1] );
fi;
# Construct names of generators (used for printing only).
if Length( arglist ) = 2 then
names:= List( [ 1 .. n ],
x -> Concatenation( "v.", String(x) ) );
MakeImmutable( names );
elif Length( arglist ) = 3 and IsString( arglist[3] ) then
names:= List( [ 1 .. n ],
x -> Concatenation( arglist[3], String(x) ) );
MakeImmutable( names );
elif Length( arglist ) = 3 and IsHomogeneousList( arglist[3] )
and Length( arglist[3] ) = n
and ForAll( arglist[3], IsString ) then
names:= Immutable( arglist[3] );
elif Length( arglist ) = 2 + n then
names:= Immutable( arglist{ [ 3 .. Length( arglist ) ] } );
else
Error( "usage: AlgebraByStructureConstantsArg([<R>,<sctable>]) or \n",
"AlgebraByStructureConstantsArg([<R>,<sctable>,<name1>,...])" );
fi;
# If the coefficients know to be additively commutative then
# also the s.c. algebra will know this.
if IsAdditivelyCommutativeElementFamily( FamilyObj( zero ) ) then
filter:= filter and IsAdditivelyCommutativeElement;
fi;
# Construct the family of elements of our algebra.
# If the elements family of `R' has a uniquely determined zero element,
# then all coefficients in this family are admissible.
# Otherwise only coefficients from `R' itself are allowed.
Fam:= NewFamily( "SCAlgebraObjFamily", filter );
if Zero( ElementsFamily( FamilyObj( R ) ) ) <> fail then
SetFilterObj( Fam, IsFamilyOverFullCoefficientsFamily );
else
Fam!.coefficientsDomain:= R;
fi;
Fam!.sctable := T;
Fam!.names := names;
Fam!.zerocoeff := zero;
# Construct the default type of the family.
Fam!.defaultTypeDenseCoeffVectorRep :=
NewType( Fam, IsSCAlgebraObj and IsDenseCoeffVectorRep );
SetCharacteristic( Fam, Characteristic( R ) );
SetCoefficientsFamily( Fam, ElementsFamily( FamilyObj( R ) ) );
# Make the generators and the algebra.
if 0 < n then
SetZero( Fam, ObjByExtRep( Fam, List( [ 1 .. n ], x -> zero ) ) );
gens:= Immutable( List( IdentityMat( n, R ),
x -> ObjByExtRep( Fam, x ) ) );
A:= FLMLORByGenerators( R, gens );
UseBasis( A, gens );
else
SetZero( Fam, ObjByExtRep( Fam, EmptyRowVector( FamilyObj(zero) ) ) );
gens:= Immutable( [] );
A:= FLMLORByGenerators( R, gens, Zero( Fam ) );
SetIsTrivial( A, true );
SetDimension( A, 0 );
fi;
if Length( one_coeffs ) = 1 then
# We want to construct an algebra-with_one.
one:= ObjByExtRep( Fam, one_coeffs[1] );
SetOne( Fam, one );
SetOne( A, one );
SetFilterObj( A, IsMagmaWithOne );
SetGeneratorsOfAlgebraWithOne( A, gens );
fi;
Fam!.basisVectors:= gens;
#T where is this needed?
# Store the algebra in the family of the elements,
# for accessing the full algebra, e.g., in `DefaultFieldOfMatrixGroup'.
Fam!.fullSCAlgebra:= A;
SetIsFullSCAlgebra( A, true );
# Return the algebra.
return A;
end );
InstallGlobalFunction( AlgebraByStructureConstants, function( arg )
return AlgebraByStructureConstantsArg( arg, IsSCAlgebraObj );
end );
InstallGlobalFunction( AlgebraWithOneByStructureConstants, function( arg )
local onecoeffs;
onecoeffs := Remove(arg);
return AlgebraByStructureConstantsArg( arg, IsSCAlgebraObj, onecoeffs );
end );
InstallGlobalFunction( LieAlgebraByStructureConstants, function( arg )
local A;
A:= AlgebraByStructureConstantsArg( arg, IsSCAlgebraObj and IsJacobianElement );
SetIsLieAlgebra( A, true );
return A;
end );
InstallGlobalFunction( RestrictedLieAlgebraByStructureConstants, function( arg )
local A, fam, pmap, i, j, v;
A := AlgebraByStructureConstantsArg( arg{[1..Length(arg)-1]}, IsSCAlgebraObj and IsRestrictedJacobianElement );
SetIsLieAlgebra( A, true );
SetIsRestrictedLieAlgebra( A, true );
fam := FamilyObj(Representative(A));
fam!.pMapping := [];
pmap := Last(arg);
while Length(pmap)<>Dimension(A) do
Error("Pth power images list should have length ",Dimension(A));
od;
for i in [1..Length(pmap)] do
v := List(pmap,i->fam!.zerocoeff);
for j in [2,4..Length(pmap[i])] do
v[pmap[i][j]] := One(v[1])*pmap[i][j-1];
od;
v := ObjByExtRep(fam,v);
# while AdjointMatrix(Basis(A),A.(i))^Characteristic(A)<>AdjointMatrix(Basis(A),v) do
# Error("p-mapping at position ",i," doesn't satisfy the axioms of a restricted Lie algebra");
# od;
Add(fam!.pMapping,v);
od;
SetPthPowerImages(Basis(A),fam!.pMapping);
return A;
end );
#############################################################################
##
#M \.( <A>, <n> ) . . . . . . . access to generators of a full s.c. algebra
##
InstallAccessToGenerators( IsSCAlgebraObjCollection and IsFullSCAlgebra,
"s.c. algebra containing the whole family",
GeneratorsOfAlgebra );
#############################################################################
##
#F QuaternionAlgebra( <F>[, <a>, <b>] )
##
InstallGlobalFunction( QuaternionAlgebra, function( arg )
local F, a, b, e, z, stored, filter, A;
if Length( arg ) = 1 and IsRing( arg[1] ) then
F:= arg[1];
a:= AdditiveInverse( One( F ) );
b:= a;
elif Length( arg ) = 1 and IsCollection( arg[1] ) then
F:= Field( arg[1] );
a:= AdditiveInverse( One( F ) );
b:= a;
elif Length( arg ) = 3 and IsRing( arg[1] ) then
F:= arg[1];
a:= arg[2];
b:= arg[3];
elif Length( arg ) = 3 and IsCollection( arg[1] ) then
F:= Field( arg[1] );
a:= arg[2];
b:= arg[3];
else
Error( "usage: QuaternionAlgebra( <F>[, <a>, <b>] ) for a ring <F>" );
fi;
e:= One( F );
if e = fail then
Error( "<F> must have an identity element" );
fi;
z:= Zero( F );
# Generators in the right family may be already available.
stored := GET_FROM_SORTED_CACHE( QuaternionAlgebraData, [ a, b, FamilyObj( F ) ],
function()
# Construct a filter describing element properties,
# which will be stored in the family.
filter:= IsSCAlgebraObj and IsQuaternion;
if HasIsAssociative( F ) and IsAssociative( F ) then
filter:= filter and IsAssociativeElement;
fi;
if IsNegRat( a ) and IsNegRat( b )
#T it suffices if the parameters are real and negative
and IsCyclotomicCollection( F ) and IsField( F )
and ForAll( GeneratorsOfDivisionRing( F ),
x -> x = ComplexConjugate( x ) ) then
filter:= filter and IsZDFRE;
fi;
# Construct the algebra.
return AlgebraByStructureConstantsArg(
[ F,
[ [ [[1],[e]], [[2],[ e]], [[3],[ e]], [[4],[ e]] ],
[ [[2],[e]], [[1],[ a]], [[4],[ e]], [[3],[ a]] ],
[ [[3],[e]], [[4],[-e]], [[1],[ b]], [[2],[ -b]] ],
[ [[4],[e]], [[3],[-a]], [[2],[ b]], [[1],[-a*b]] ],
0, z ],
"e", "i", "j", "k" ],
filter, [ e, z, z, z ] );
end );
A:= AlgebraWithOne( F, GeneratorsOfAlgebra( stored ), "basis" );
SetGeneratorsOfAlgebra( A, GeneratorsOfAlgebraWithOne( A ) );
SetIsFullSCAlgebra( A, true );
# A quaternion algebra with negative parameters over a real field
# is a division ring.
if IsNegRat( a ) and IsNegRat( b )
and IsCyclotomicCollection( F ) and IsField( F )
and ForAll( GeneratorsOfDivisionRing( F ),
x -> x = ComplexConjugate( x ) ) then
SetFilterObj( A, IsDivisionRing );
#T better use `DivisionRingByGenerators'?
SetGeneratorsOfDivisionRing( A, GeneratorsOfAlgebraWithOne( A ) );
fi;
# Return the quaternion algebra.
return A;
end );
#############################################################################
##
#M OneOp( <quat> ) . . . . . . . . . . . . . . . . . . . . for a quaternion
##
InstallMethod( OneOp,
"for a quaternion",
[ IsQuaternion and IsSCAlgebraObj ],
quat -> ObjByExtRep( FamilyObj( quat ),
[ 1, 0, 0, 0 ] * One( ExtRepOfObj( quat )[1] ) ) );
#############################################################################
##
#M InverseOp( <quat> ) . . . . . . . . . . . . . . . . . . for a quaternion
##
## Let $a$ and $b$ be the parameters from which the algebra of <quat> was
## constructed.
## The inverse of $c_1 e + c_2 i + c_3 j + c_4 k$ is
## $c_1/z e - c_2/z i - c_3/z j - c_4/z k$
## where $z = c_1^2 - c_2^2 a - c_3^2 b + c_4^2 a b$.
##
InstallMethod( InverseOp,
"for a quaternion",
[ IsQuaternion and IsSCAlgebraObj ],
function( quat )
local data, z, a, b;
data:= ExtRepOfObj( quat );
a:= FamilyObj( quat )!.sctable[2][2][2][1];
b:= FamilyObj( quat )!.sctable[3][3][2][1];
z:= data[1]^2 - data[2]^2 * a - data[3]^2 * b + data[4]^2 * a * b;
if IsZero( z ) then
return fail;
fi;
return ObjByExtRep( FamilyObj( quat ),
[ data[1]/z, AdditiveInverse( data[2]/z ),
AdditiveInverse( data[3]/z ),
AdditiveInverse( data[4]/z ) ] );
end );
#############################################################################
##
#M ComplexConjugate( <quat> ) . . . . . . . . . . . . . . for a quaternion
##
InstallMethod( ComplexConjugate,
"for a quaternion",
[ IsQuaternion and IsSCAlgebraObj ],
function( quat )
local v;
v:= ExtRepOfObj( quat );
return ObjByExtRep( FamilyObj( quat ), [ v[1], -v[2], -v[3], -v[4] ] );
end );
#############################################################################
##
#M RealPart( <quat> ) . . . . . . . . . . . . . . . . . . for a quaternion
##
InstallMethod( RealPart,
"for a quaternion",
[ IsQuaternion and IsSCAlgebraObj ],
function( quat )
local v, z;
v:= ExtRepOfObj( quat );
z:= Zero( v[1] );
return ObjByExtRep( FamilyObj( quat ), [ v[1], z, z, z ] );
end );
#############################################################################
##
#M ImaginaryPart( <quat> ) . . . . . . . . . . . . . . . . for a quaternion
##
InstallMethod( ImaginaryPart,
"for a quaternion",
[ IsQuaternion and IsSCAlgebraObj ],
function( quat )
local v, z, a;
v:= ExtRepOfObj( quat );
z:= Zero( v[1] );
a:= FamilyObj( quat )!.sctable[2][2][2][1]; # the first parameter
return ObjByExtRep( FamilyObj( quat ), [ v[2], z, -v[4], -v[3]/a ] );
end );
#############################################################################
##
#F ComplexificationQuat( <vector> )
#F ComplexificationQuat( <matrix> )
##
InstallGlobalFunction( ComplexificationQuat, function( matrixorvector )
local result,
i, e,
M,
m,
n,
j, k,
v,
coeff;
result:= [];
i:= E(4);
e:= 1;
if IsQuaternionCollColl( matrixorvector ) then
M:= matrixorvector;
m:= Length( M );
n:= Length( M[1] );
for j in [ 1 .. 2*m ] do
result[j]:= [];
od;
for j in [ 1 .. m ] do
for k in [ 1 .. n ] do
coeff:= ExtRepOfObj( M[j][k] );
result[ j ][ k ]:= e * coeff[1] + i * coeff[2];
result[ j ][ n+k ]:= e * coeff[3] + i * coeff[4];
result[ m+j ][ k ]:= - e * coeff[3] + i * coeff[4];
result[ m+j ][ n+k ]:= e * coeff[1] - i * coeff[2];
od;
od;
elif IsQuaternionCollection( matrixorvector ) then
v:= matrixorvector;
n:= Length( v );
for j in [ 1 .. n ] do
coeff:= ExtRepOfObj( v[j] );
result[ j ]:= e * coeff[1] + i * coeff[2];
result[ n+j ]:= e * coeff[3] + i * coeff[4];
od;
else
Error( "<matrixorvector> must be a vector or matrix of quaternions" );
fi;
return result;
end );
#############################################################################
##
#F OctaveAlgebra( <F> )
##
InstallGlobalFunction( OctaveAlgebra, F -> AlgebraByStructureConstants(
F,
[ [ [[1],[1]],[[],[]],[[3],[1]],[[],[]],[[5],[1]],[[],[]],[[],[]],
[[8],[1]] ],
[ [[],[]],[[2],[1]],[[],[]],[[4],[1]],[[],[]],[[6],[1]],[[7],[1]],
[[],[]] ],
[ [[],[]],[[3],[1]],[[],[]],[[1],[1]],[[7],[1]],[[],[]],[[],[]],
[[6],[1]] ],
[ [[4],[1]],[[],[]],[[2],[1]],[[],[]],[[],[]],[[8],[1]],[[5],[1]],
[[],[]] ],
[ [[],[]],[[5],[1]],[[7],[-1]],[[],[]],[[],[]],[[1],[1]],[[],[]],
[[4],[-1]] ],
[ [[6],[1]],[[],[]],[[],[]],[[8],[-1]],[[2],[1]],[[],[]],[[3],[-1]],
[[],[]] ],
[ [[7],[1]],[[],[]],[[],[]],[[5],[-1]],[[],[]],[[3],[1]],[[],[]],
[[2],[-1]] ],
[ [[],[]],[[8],[1]],[[6],[-1]],[[],[]],[[4],[1]],[[],[]],[[1],[-1]],
[[],[]] ],
0, 0 ],
"s1", "t1", "s2", "t2", "s3", "t3", "s4", "t4" ) );
#############################################################################
##
#M NiceFreeLeftModuleInfo( <V> )
#M NiceVector( <V>, <v> )
#M UglyVector( <V>, <r> )
##
InstallHandlingByNiceBasis( "IsSCAlgebraObjSpace", rec(
detect := function( R, gens, V, zero )
return IsSCAlgebraObjCollection( V );
end,
NiceFreeLeftModuleInfo := ReturnTrue,
NiceVector := function( V, v )
return ExtRepOfObj( v );
end,
UglyVector := function( V, r )
local F;
F:= ElementsFamily( FamilyObj( V ) );
if Length( r ) <> Length( F!.names ) then
return fail;
fi;
return ObjByExtRep( F, r );
end ) );
#############################################################################
##
#M MutableBasis( <R>, <gens> )
#M MutableBasis( <R>, <gens>, <zero> )
##
## We choose a mutable basis that stores a mutable basis for a nice module.
##
InstallMethod( MutableBasis,
"for ring and collection of s. c. algebra elements",
[ IsRing, IsSCAlgebraObjCollection ],
MutableBasisViaNiceMutableBasisMethod2 );
InstallOtherMethod( MutableBasis,
"for ring, (possibly empty) list, and zero element",
[ IsRing, IsList, IsSCAlgebraObj ],
MutableBasisViaNiceMutableBasisMethod3 );
#############################################################################
##
#M Coefficients( <B>, <v> ) . . . . . . coefficients w.r.t. canonical basis
##
InstallMethod( Coefficients,
"for canonical basis of full s. c. algebra",
IsCollsElms,
[ IsBasis and IsCanonicalBasisFullSCAlgebra, IsSCAlgebraObj ],
function( B, v )
return ExtRepOfObj( v );
end );
#############################################################################
##
#M LinearCombination( <B>, <coeffs> ) . . . . . . . . . for canonical basis
##
InstallMethod( LinearCombination,
"for canonical basis of full s. c. algebra",
[ IsBasis and IsCanonicalBasisFullSCAlgebra, IsRowVector ],
function( B, coeffs )
return ObjByExtRep( ElementsFamily( FamilyObj( B ) ), coeffs );
end );
#############################################################################
##
#M BasisVectors( <B> ) . . . . . . for canonical basis of full s.~c. algebra
##
InstallMethod( BasisVectors,
"for canonical basis of full s. c. algebra",
[ IsBasis and IsCanonicalBasisFullSCAlgebra ],
B -> ElementsFamily( FamilyObj(
UnderlyingLeftModule( B ) ) )!.basisVectors );
#############################################################################
##
#M Basis( <A> ) . . . . . . . . . . . . . . . basis of a full s.~c. algebra
##
InstallMethod( Basis,
"for full s. c. algebra (delegate to `CanonicalBasis')",
[ IsFreeLeftModule and IsSCAlgebraObjCollection and IsFullSCAlgebra ],
CANONICAL_BASIS_FLAGS,
CanonicalBasis );
#############################################################################
##
#M CanonicalBasis( <A> ) . . . . . . . . . . . basis of a full s.~c. algebra
##
InstallMethod( CanonicalBasis,
"for full s. c. algebras",
[ IsFreeLeftModule and IsSCAlgebraObjCollection and IsFullSCAlgebra ],
function( A )
local B;
B:= Objectify( NewType( FamilyObj( A ),
IsCanonicalBasisFullSCAlgebra
and IsAttributeStoringRep
and IsFiniteBasisDefault
and IsCanonicalBasis ),
rec() );
SetUnderlyingLeftModule( B, A );
SetStructureConstantsTable( B,
ElementsFamily( FamilyObj( A ) )!.sctable );
return B;
end );
#############################################################################
##
#M IsCanonicalBasisFullSCAlgebra( <B> )
##
InstallMethod( IsCanonicalBasisFullSCAlgebra,
"for a basis",
[ IsBasis ],
function( B )
local A;
A:= UnderlyingLeftModule( B );
return IsSCAlgebraObjCollection( A )
and IsFullSCAlgebra( A )
and IsCanonicalBasis( B );
end );
#T change implementation: bases of their own right, as for Gaussian row spaces,
#T if the algebra is Gaussian
#############################################################################
##
#M Intersection2( <V>, <W> )
##
## Contrary to the generic case that is handled by `Intersection2Spaces',
## we know initially a (finite dimensional) common coefficient space,
## so we can avoid the intermediate construction of such a space.
##
InstallMethod( Intersection2,
"for two spaces in a common s.c. algebra",
IsIdenticalObj,
[ IsVectorSpace and IsSCAlgebraObjCollection,
IsVectorSpace and IsSCAlgebraObjCollection ],
function( V, W )
local F, # coefficients field
gensV, # list of generators of 'V'
gensW, # list of generators of 'W'
Fam, # family of an element
inters; # intersection, result
F:= LeftActingDomain( V );
if F <> LeftActingDomain( W ) then
# The generic method is good enough for this.
TryNextMethod();
fi;
gensV:= GeneratorsOfLeftModule( V );
gensW:= GeneratorsOfLeftModule( W );
if IsEmpty( gensV ) or IsEmpty( gensW ) then
inters:= [];
else
gensV:= List( gensV, ExtRepOfObj );
gensW:= List( gensW, ExtRepOfObj );
if not ( ForAll( gensV, v -> IsSubset( F, v ) )
and ForAll( gensW, v -> IsSubset( F, v ) ) ) then
# We are not in a Gaussian situation.
TryNextMethod();
fi;
Fam:= ElementsFamily( FamilyObj( V ) );
inters:= List( SumIntersectionMat( gensV, gensW )[2],
x -> ObjByExtRep( Fam, x ) );
fi;
# Construct the intersection space, if possible with a parent,
# and with as much structure as possible.
if IsEmpty( inters ) then
inters:= TrivialSubFLMLOR( V );
elif IsFLMLOR( V ) and IsFLMLOR( W ) then
inters:= FLMLOR( F, inters, "basis" );
else
inters:= VectorSpace( F, inters, "basis" );
fi;
if HasParent( V ) and HasParent( W )
and IsIdenticalObj( Parent( V ), Parent( W ) ) then
SetParent( inters, Parent( V ) );
fi;
# Run implications by the subset relation.
UseSubsetRelation( V, inters );
UseSubsetRelation( W, inters );
# Return the result.
return inters;
end );
# analogous for closure?
[ Dauer der Verarbeitung: 0.25 Sekunden
(vorverarbeitet)
]
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