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############################################################################
##
#W module.tst XModAlg test files Chris Wensley
##
#@local level,m3,A3,c3,Rc3,g3,mg3,Amg3,homg3,actg3,bdy3,X3,V3,M3,famM3,v3,genM3,u4,D3,T3,B3a,B3,M2B3,B2M3,act3,a3,X4,C4,A3Rc3,hom33a,hom33b,actMA3,act4,act5,A5,B5,em3,ea3
gap> START_TEST( "XModAlg package: module.tst" );
gap> level := InfoLevel( InfoXModAlg );;
gap> SetInfoLevel( InfoXModAlg, 0 );
## Section 2.2.4
gap> m3 := [ [0,1,0], [0,0,1], [1,0,0,] ];;
gap> A3 := Algebra( Rationals, [m3] );;
gap> SetName( A3, "A3" );;
gap> c3 := Group( (1,2,3) );;
gap> Rc3 := GroupRing( Rationals, c3 );;
gap> SetName( Rc3, "GR(c3)" );
gap> g3 := GeneratorsOfAlgebra( Rc3 )[2];;
gap> mg3 := RegularAlgebraMultiplier( Rc3, Rc3, g3 );;
gap> Amg3 := AlgebraByGenerators( Rationals, [ mg3 ] );;
gap> homg3 := AlgebraHomomorphismByImages( A3, Amg3, [ m3 ], [ mg3 ] );;
gap> actg3 := AlgebraActionByHomomorphism( homg3, Rc3 );
[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] ->
[ [ (1)*(), (1)*(1,2,3), (1)*(1,3,2) ] -> [ (1)*(1,2,3), (1)*(1,3,2), (1)*()
] ]
## Section 2.3
gap> V3 := Rationals^3;;
gap> M3 := LeftAlgebraModule( A3, \*, V3 );;
gap> SetName( M3, "M3" );
gap> famM3 := ElementsFamily( FamilyObj( M3 ) );;
gap> v3 := [3,4,5];;
gap> v3 := ObjByExtRep( famM3, v3 );
[ 3, 4, 5 ]
gap> m3*v3;
[ 4, 5, 3 ]
gap> genM3 := GeneratorsOfLeftModule( M3 );;
gap> u4 := 6*genM3[1] + 7*genM3[2] + 8*genM3[3];
[ 6, 7, 8 ]
gap> u4 := ExtRepOfObj( u4 );
[ 6, 7, 8 ]
## Section 2.3.1
gap> D3 := LeftActingDomain( M3 );;
gap> T3 := EmptySCTable( Dimension(M3), Zero(D3), "symmetric" );;
gap> B3a := AlgebraByStructureConstants( D3, T3 );
<algebra of dimension 3 over Rationals>
gap> GeneratorsOfAlgebra( B3a );
[ v.1, v.2, v.3 ]
gap> B3 := ModuleAsAlgebra( M3 );
A(M3)
gap> GeneratorsOfAlgebra( B3 );
[ [[ 1, 0, 0 ]], [[ 0, 1, 0 ]], [[ 0, 0, 1 ]] ]
## Section 2.3.2
gap> IsModuleAsAlgebra( B3 );
true
gap> IsModuleAsAlgebra( A3 );
false
## Section 2.3.3
gap> Set( KnownAttributesOfObject( B3 ) );
[ "AlgebraToModuleIsomorphism", "Dimension",
"GeneratorsOfLeftOperatorAdditiveGroup", "GeneratorsOfLeftOperatorRing",
"LeftActingDomain", "ModuleToAlgebraIsomorphism", "Name", "ZeroImmutable" ]
gap> M2B3 := ModuleToAlgebraIsomorphism( B3 );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] -> [ [[ 1, 0, 0 ]], [[ 0, \
1, 0 ]],
[[ 0, 0, 1 ]] ]
gap> Source( M2B3 ) = M3;
false
gap> Source( M2B3 ) = V3;
true
gap> B2M3 := AlgebraToModuleIsomorphism( B3 );
[ [[ 1, 0, 0 ]], [[ 0, 1, 0 ]], [[ 0, 0, 1 ]] ] ->
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
gap> Range( B2M3 ) = M3;
false
gap> Range( B2M3 ) = V3;
true
## Section 2.3.4
gap> act3 := AlgebraActionByModule( A3, M3 );
[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] ->
[ [ [[ 1, 0, 0 ]], [[ 0, 1, 0 ]], [[ 0, 0, 1 ]] ] ->
[ [[ 0, 0, 1 ]], [[ 1, 0, 0 ]], [[ 0, 1, 0 ]] ] ]
gap> a3 := 2*m3 + 3*m3^2;
[ [ 0, 2, 3 ], [ 3, 0, 2 ], [ 2, 3, 0 ] ]
gap> Image( act3, a3 );
Basis( A(M3), [ [[ 1, 0, 0 ]], [[ 0, 1, 0 ]], [[ 0, 0, 1 ]] ] ) ->
[ (3)*[[ 0, 1, 0 ]]+(2)*[[ 0, 0, 1 ]], (2)*[[ 1, 0, 0 ]]+(3)*[[ 0, 0, 1 ]],
(3)*[[ 1, 0, 0 ]]+(2)*[[ 0, 1, 0 ]] ]
gap> Image( act3 );
<algebra over Rationals, with 1 generator>
## Section 2.4.1
gap> A3Rc3 := DirectSumOfAlgebrasWithInfo( A3, Rc3 );;
gap> SetName( A3Rc3, Concatenation( Name(A3), "(+)", Name(Rc3) ) );
gap> DirectSumOfAlgebrasInfo( A3Rc3 );
rec( algebras := [ A3, GR(c3) ],
embeddings :=
[
Basis( A3, [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
[ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ] ) -> [ v.1, v.2, v.3 ],
CanonicalBasis( GR(c3) ) -> [ v.4, v.5, v.6 ] ], first := [ 1, 4 ],
projections :=
[ CanonicalBasis( A3(+)GR(c3) ) ->
[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
[ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
[ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
[ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ],
CanonicalBasis( A3(+)GR(c3) ) -> [ <zero> of ..., <zero> of ...,
<zero> of ..., (1)*(), (1)*(1,2,3), (1)*(1,3,2) ] ] )
## Section 2.4.2
gap> Embedding( A3Rc3, 1 );
Basis( A3, [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
[ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ] ) -> [ v.1, v.2, v.3 ]
gap> Projection( A3Rc3, 2 );
CanonicalBasis( A3(+)GR(c3) ) -> [ <zero> of ..., <zero> of ...,
<zero> of ..., (1)*(), (1)*(1,2,3), (1)*(1,3,2) ]
gap> SetInfoLevel( InfoXModAlg, level );;
gap> STOP_TEST( "module.tst", 10000 );
############################################################################
##
#E module.tst . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
[ Dauer der Verarbeitung: 0.12 Sekunden
(vorverarbeitet)
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