Algebras
<#Include Label="[1]{algebra}">

InfoAlgebra (Info Class)
<#Include Label="InfoAlgebra">

Constructing Algebras by Generators
<#Include Label="Algebra"> <#Include Label="AlgebraWithOne">

Constructing Algebras as Free Algebras
<#Include Label="FreeAlgebra"> <#Include Label="FreeAlgebraWithOne"> <#Include Label="FreeAssociativeAlgebra"> <#Include Label="FreeAssociativeAlgebraWithOne">

Constructing Algebras by Structure Constants
<#Include Label="[2]{algebra}"> <#Include Label="AlgebraByStructureConstants"> <#Include Label="StructureConstantsTable"> <#Include Label="EmptySCTable"> <#Include Label="SetEntrySCTable"> <#Include Label="GapInputSCTable"> <#Include Label="TestJacobi"> <#Include Label="IdentityFromSCTable"> <#Include Label="QuotientFromSCTable">

Some Special Algebras
<#Include Label="QuaternionAlgebra"> <#Include Label="ComplexificationQuat"> <#Include Label="OctaveAlgebra"> <#Include Label="FullMatrixAlgebra"> <#Include Label="NullAlgebra">

Subalgebras
<#Include Label="Subalgebra"> <#Include Label="SubalgebraNC"> <#Include Label="SubalgebraWithOne"> <#Include Label="SubalgebraWithOneNC"> <#Include Label="TrivialSubalgebra">

Ideals

For constructing and working with ideals in algebras the same functions are available as for ideals in rings. So for the precise description of these functions we refer to Chapter . Here we give examples demonstrating the use of ideals in algebras. For an introduction into the construction of quotient algebras we refer to Chapter of the user's tutorial.

gap> m:= [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0] ];; gap> A:= AlgebraWithOne( Rationals, [ m ] );; gap> I:= Ideal( A, [ m ] ); # the two-sided ideal of `A' generated by `m' <two-sided ideal in <algebra-with-one of dimension 3 over Rationals>, (1 generators)> gap> Dimension( I ); 2 gap> GeneratorsOfIdeal( I ); [ [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0 ] ] ] gap> BasisVectors( Basis( I ) ); [ [ [ 0, 1, 3/2 ], [ 0, 0, 2 ], [ 0, 0, 0 ] ], [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ] gap> A:= FullMatrixAlgebra( Rationals, 4 );; gap> m:= NullMat( 4, 4 );; m[1][4]:=1;; gap> I:= LeftIdeal( A, [ m ] ); <left ideal in ( Rationals^[ 4, 4 ] ), (1 generators)> gap> Dimension( I ); 4 gap> GeneratorsOfLeftIdeal( I ); [ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ] gap> mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];; gap> A:= Algebra( Rationals, mats );; gap> # Form the two-sided ideal for which `mats[2]' is known to be gap> # the unique basis element. gap> I:= Ideal( A, [ mats[2] ], "basis" ); <two-sided ideal in <algebra of dimension 3 over Rationals>, (dimension 1)>
Categories and Properties of Algebras
<#Include Label="IsFLMLOR"> <#Include Label="IsFLMLORWithOne"> <#Include Label="IsAlgebra"> <#Include Label="IsAlgebraWithOne"> <#Include Label="IsLieAlgebra"> <#Include Label="IsSimpleAlgebra">

returns true (always) for a matrix algebra matalg, since matrix algebras are always finite dimensional.

gap> A:= MatAlgebra( Rationals, 3 );; gap> IsFiniteDimensional( A ); true
<#Include Label="IsQuaternion">

Attributes and Operations for Algebras
<#Include Label="GeneratorsOfAlgebra"> <#Include Label="GeneratorsOfAlgebraWithOne"> <#Include Label="ProductSpace"> <#Include Label="PowerSubalgebraSeries"> <#Include Label="AdjointBasis"> <#Include Label="IndicesOfAdjointBasis"> <#Include Label="AsAlgebra"> <#Include Label="AsAlgebraWithOne"> <#Include Label="AsSubalgebra"> <#Include Label="AsSubalgebraWithOne"> <#Include Label="MutableBasisOfClosureUnderAction"> <#Include Label="MutableBasisOfNonassociativeAlgebra"> <#Include Label="MutableBasisOfIdealInNonassociativeAlgebra"> <#Include Label="DirectSumOfAlgebras"> <#Include Label="FullMatrixAlgebraCentralizer"> <#Include Label="RadicalOfAlgebra"> <#Include Label="CentralIdempotentsOfAlgebra"> <#Include Label="DirectSumDecomposition"> <#Include Label="LeviMalcevDecomposition"> <#Include Label="Grading">

Homomorphisms of Algebras
<#Include Label="[1]{alghom}"> <#Include Label="AlgebraGeneralMappingByImages"> <#Include Label="AlgebraHomomorphismByImages"> <#Include Label="AlgebraHomomorphismByImagesNC"> <#Include Label="AlgebraWithOneGeneralMappingByImages"> <#Include Label="AlgebraWithOneHomomorphismByImages"> <#Include Label="AlgebraWithOneHomomorphismByImagesNC"> <#Include Label="NaturalHomomorphismByIdeal_algebras"> <#Include Label="OperationAlgebraHomomorphism"> <#Include Label="NiceAlgebraMonomorphism"> <#Include Label="IsomorphismFpAlgebra"> <#Include Label="IsomorphismMatrixAlgebra"> <#Include Label="IsomorphismSCAlgebra"> <#Include Label="RepresentativeLinearOperation">

Representations of Algebras
<#Include Label="[1]{algrep}"> <#Include Label="LeftAlgebraModuleByGenerators"> <#Include Label="RightAlgebraModuleByGenerators"> <#Include Label="BiAlgebraModuleByGenerators"> <#Include Label="LeftAlgebraModule"> <#Include Label="RightAlgebraModule"> <#Include Label="BiAlgebraModule"> <#Include Label="GeneratorsOfAlgebraModule"> <#Include Label="IsAlgebraModuleElement"> <#Include Label="IsLeftAlgebraModuleElement"> <#Include Label="IsRightAlgebraModuleElement"> <#Include Label="LeftActingAlgebra"> <#Include Label="RightActingAlgebra"> <#Include Label="ActingAlgebra"> <#Include Label="IsBasisOfAlgebraModuleElementSpace"> <#Include Label="MatrixOfAction"> <#Include Label="SubAlgebraModule"> <#Include Label="LeftModuleByHomomorphismToMatAlg"> <#Include Label="RightModuleByHomomorphismToMatAlg"> <#Include Label="AdjointModule"> <#Include Label="FaithfulModule"> <#Include Label="ModuleByRestriction"> <#Include Label="NaturalHomomorphismBySubAlgebraModule"> <#Include Label="DirectSumOfAlgebraModules"> <#Include Label="TranslatorSubalgebra">