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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Path: gap4r8 / doc / ref / algebra.xml
Views: ⁴¹⁸³⁴⁶
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<!-- %% -->
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<!-- %A  algebra.msk                  GAP documentation            Willem de Graaf -->
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<!-- %% -->
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<!-- %A  @(#)<M>Id: algebra.msk,v 1.35 2006/03/10 08:55:52 gap Exp </M> -->
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<!-- %% -->
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<!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->
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<!-- %Y  Copyright (C) 2002 The GAP Group -->
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<!-- %% -->
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<Chapter Label="Algebras">
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<Heading>Algebras</Heading>
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<#Include Label="[1]{algebra}">
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<!-- %%  The algebra functionality was designed and implemented by Thomas Breuer and -->
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<!-- %%  Willem de Graaf. -->
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="sect:InfoAlgebra">
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<Heading>InfoAlgebra (Info Class)</Heading>
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<#Include Label="InfoAlgebra">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Constructing Algebras by Generators">
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<Heading>Constructing Algebras by Generators</Heading>
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<!-- % AlgebraByGenerators( <A>F</A>, <A>gens</A>, <A>zero</A> ) Left out... -->
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<#Include Label="Algebra">
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<!-- %  AlgebraWithOneByGenerators( <A>F</A>, <A>gens</A>, <A>zero</A> ) Left out... -->
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<#Include Label="AlgebraWithOne">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Constructing Algebras as Free Algebras">
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<Heading>Constructing Algebras as Free Algebras</Heading>
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<#Include Label="FreeAlgebra">
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<#Include Label="FreeAlgebraWithOne">
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<#Include Label="FreeAssociativeAlgebra">
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<#Include Label="FreeAssociativeAlgebraWithOne">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Constructing Algebras by Structure Constants">
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<Heading>Constructing Algebras by Structure Constants</Heading>
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<#Include Label="[2]{algebra}">
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<#Include Label="AlgebraByStructureConstants">
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<#Include Label="StructureConstantsTable">
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<#Include Label="EmptySCTable">
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<#Include Label="SetEntrySCTable">
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<#Include Label="GapInputSCTable">
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<#Include Label="TestJacobi">
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<#Include Label="IdentityFromSCTable">
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<#Include Label="QuotientFromSCTable">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Some Special Algebras">
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<Heading>Some Special Algebras</Heading>
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<#Include Label="QuaternionAlgebra">
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<#Include Label="ComplexificationQuat">
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<#Include Label="OctaveAlgebra">
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<#Include Label="FullMatrixAlgebra">
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<#Include Label="NullAlgebra">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Subalgebras">
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<Heading>Subalgebras</Heading>
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<#Include Label="Subalgebra">
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<#Include Label="SubalgebraNC">
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<#Include Label="SubalgebraWithOne">
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<#Include Label="SubalgebraWithOneNC">
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<#Include Label="TrivialSubalgebra">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Ideals of Algebras">
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<Heading>Ideals of Algebras</Heading>
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For constructing and working with ideals in algebras the same functions
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are available as for ideals in rings. So for the precise description of
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these functions we refer to Chapter <Ref Chap="Rings"/>. Here we give examples
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demonstrating the use of ideals in algebras.
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For an introduction into the construction of quotient algebras
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we refer to Chapter <Ref Sect="Algebras" BookName="tut"/>
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of the user's tutorial.
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<P/>
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<Example><![CDATA[
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gap> m:= [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0] ];;
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gap> A:= AlgebraWithOne( Rationals, [ m ] );;
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gap> I:= Ideal( A, [ m ] );  # the two-sided ideal of `A' generated by `m'
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<two-sided ideal in <algebra-with-one of dimension 3 over Rationals>, 
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  (1 generators)>
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gap> Dimension( I );
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2
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gap> GeneratorsOfIdeal( I );
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[ [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0 ] ] ]
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gap> BasisVectors( Basis( I ) );
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[ [ [ 0, 1, 3/2 ], [ 0, 0, 2 ], [ 0, 0, 0 ] ], 
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  [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
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gap> A:= FullMatrixAlgebra( Rationals, 4 );;
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gap> m:= NullMat( 4, 4 );; m[1][4]:=1;;
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gap> I:= LeftIdeal( A, [ m ] );
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<left ideal in ( Rationals^[ 4, 4 ] ), (1 generators)>
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gap> Dimension( I );
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4
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gap> GeneratorsOfLeftIdeal( I );
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[ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ]
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gap> mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
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gap> A:= Algebra( Rationals, mats );;
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gap> # Form the two-sided ideal for which `mats[2]' is known to be
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gap> # the unique basis element.
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gap> I:= Ideal( A, [ mats[2] ], "basis" );
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<two-sided ideal in <algebra of dimension 3 over Rationals>, 
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  (dimension 1)>
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]]></Example>
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Categories and Properties of Algebras">
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<Heading>Categories and Properties of Algebras</Heading>
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<#Include Label="IsFLMLOR">
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<#Include Label="IsFLMLORWithOne">
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<#Include Label="IsAlgebra">
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<#Include Label="IsAlgebraWithOne">
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<#Include Label="IsLieAlgebra">
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<#Include Label="IsSimpleAlgebra">
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<!-- % IsMatrixFLMLOR left out... -->
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<ManSection>
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<Meth Name="IsFiniteDimensional" Arg='matalg' Label="for matrix algebras"/>
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<Description>
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returns <K>true</K> (always) for a matrix algebra <A>matalg</A>, since
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matrix algebras are always finite dimensional.
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<P/>
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<Example><![CDATA[
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gap> A:= MatAlgebra( Rationals, 3 );;
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gap> IsFiniteDimensional( A );
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true
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]]></Example>
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</Description>
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</ManSection>
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<#Include Label="IsQuaternion">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Attributes and Operations for Algebras">
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<Heading>Attributes and Operations for Algebras</Heading>
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<!-- % GeneratorsOfLeftOperatorRing left out.... -->
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<!-- % GeneratorsOfLeftOperatorRingWithOne left out.... -->
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<#Include Label="GeneratorsOfAlgebra">
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<#Include Label="GeneratorsOfAlgebraWithOne">
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<#Include Label="ProductSpace">
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<#Include Label="PowerSubalgebraSeries">
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<#Include Label="AdjointBasis">
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<#Include Label="IndicesOfAdjointBasis">
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<#Include Label="AsAlgebra">
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<#Include Label="AsAlgebraWithOne">
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<#Include Label="AsSubalgebra">
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<#Include Label="AsSubalgebraWithOne">
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<#Include Label="MutableBasisOfClosureUnderAction">
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<#Include Label="MutableBasisOfNonassociativeAlgebra">
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<#Include Label="MutableBasisOfIdealInNonassociativeAlgebra">
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<#Include Label="DirectSumOfAlgebras">
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<#Include Label="FullMatrixAlgebraCentralizer">
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<#Include Label="RadicalOfAlgebra">
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<#Include Label="CentralIdempotentsOfAlgebra">
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<#Include Label="DirectSumDecomposition">
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<#Include Label="LeviMalcevDecomposition">
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<#Include Label="Grading">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Homomorphisms of Algebras">
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<Heading>Homomorphisms of Algebras</Heading>
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<#Include Label="[1]{alghom}">
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<#Include Label="AlgebraGeneralMappingByImages">
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<#Include Label="AlgebraHomomorphismByImages">
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<#Include Label="AlgebraHomomorphismByImagesNC">
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<#Include Label="AlgebraWithOneGeneralMappingByImages">
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<#Include Label="AlgebraWithOneHomomorphismByImages">
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<#Include Label="AlgebraWithOneHomomorphismByImagesNC">
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<#Include Label="NaturalHomomorphismByIdeal_algebras">
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<#Include Label="OperationAlgebraHomomorphism">
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<#Include Label="NiceAlgebraMonomorphism">
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<#Include Label="IsomorphismFpAlgebra">
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<#Include Label="IsomorphismMatrixAlgebra">
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<#Include Label="IsomorphismSCAlgebra">
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<#Include Label="RepresentativeLinearOperation">
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</Section>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<Section Label="Representations of Algebras">
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<Heading>Representations of Algebras</Heading>
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<#Include Label="[1]{algrep}">
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<#Include Label="LeftAlgebraModuleByGenerators">
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<#Include Label="RightAlgebraModuleByGenerators">
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<#Include Label="BiAlgebraModuleByGenerators">
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<#Include Label="LeftAlgebraModule">
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<#Include Label="RightAlgebraModule">
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<#Include Label="BiAlgebraModule">
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<#Include Label="GeneratorsOfAlgebraModule">
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<#Include Label="IsAlgebraModuleElement">
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<#Include Label="IsLeftAlgebraModuleElement">
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<#Include Label="IsRightAlgebraModuleElement">
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<#Include Label="LeftActingAlgebra">
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<#Include Label="RightActingAlgebra">
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<#Include Label="ActingAlgebra">
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<#Include Label="IsBasisOfAlgebraModuleElementSpace">
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<#Include Label="MatrixOfAction">
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<#Include Label="SubAlgebraModule">
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<#Include Label="LeftModuleByHomomorphismToMatAlg">
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<#Include Label="RightModuleByHomomorphismToMatAlg">
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<#Include Label="AdjointModule">
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<!-- % One would be tempted to call <C>W</C> a left ideal in <C>V</C>, -->
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<!-- % but in the current implementation, neither <C>V</C> nor <C>W</C> are themselves -->
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<!-- % algebras; note that the element <C>v</C>, although looking like a matrix, -->
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<!-- % cannot be multiplied with itself. -->
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<#Include Label="FaithfulModule">
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<#Include Label="ModuleByRestriction">
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<#Include Label="NaturalHomomorphismBySubAlgebraModule">
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<#Include Label="DirectSumOfAlgebraModules">
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<#Include Label="TranslatorSubalgebra">
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</Section>
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</Chapter>
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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
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<!-- %% -->
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<!-- %E -->
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