59.7 Categories and Properties of Algebras

A FLMLOR (``free left module left operator ring'') in GAP is a ring that is also a free left module.

Note that this means that being a FLMLOR is not a property a ring can get, since a ring is usually not represented as an external left set.

Examples are magma rings (e.g. over the integers) or algebras.

gap> A:= FullMatrixAlgebra( Rationals, 2 );;
gap> IsFLMLOR ( A );
true

A FLMLOR-with-one in GAP is a ring-with-one that is also a free left module.

Note that this means that being a FLMLOR-with-one is not a property a ring-with-one can get, since a ring-with-one is usually not represented as an external left set.

Examples are magma rings-with-one or algebras-with-one (but also over the integers).

gap> A:= FullMatrixAlgebra( Rationals, 2 );;
gap> IsFLMLORWithOne ( A );
true

An algebra in GAP is a ring that is also a left vector space. Note that this means that being an algebra is not a property a ring can get, since a ring is usually not represented as an external left set.

gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsAlgebra( A );       
true

An algebra-with-one in GAP is a ring-with-one that is also a left vector space. Note that this means that being an algebra-with-one is not a property a ring-with-one can get, since a ring-with-one is usually not represented as an external left set.

gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsAlgebraWithOne( A );
true

An algebra A is called Lie algebra if a * a = 0 for all a in A and ( a * ( b * c ) ) + ( b * ( c * a ) ) + ( c * ( a * b ) ) = 0 for all a, b, c in A (Jacobi identity).

gap> A:= FullMatrixLieAlgebra( Rationals, 3 );;
gap> IsLieAlgebra( A );                        
true

is true if the algebra A is simple, and false otherwise. This function is only implemented for the cases where A is an associative or a Lie algebra.

gap> A:= FullMatrixLieAlgebra( Rationals, 3 );;
gap> IsSimpleAlgebra( A );
false
gap> A:= MatAlgebra( Rationals, 3 );;          
gap> IsSimpleAlgebra( A );           
true

Matrix algebras are always finite dimensional.

gap> A:= MatAlgebra( Rationals, 3 );;
gap> IsFiniteDimensional( A );
true

IsQuaternion is the category of elements in an algebra constructed by QuaternionAlgebra. A collection of quaternions lies in the category IsQuaternionCollection. Finally, a collection of quaternion collections (e.g., a matrix) lies in the category IsQuaternionCollColl.

gap> A:= QuaternionAlgebra( Rationals );;
gap> b:= BasisVectors( Basis( A ) ); 
[ e, i, j, k ]
gap> IsQuaternion( b[1] );
true
gap> IsQuaternionCollColl( [ [ b[1], b[2] ], [ b[3], b[4] ] ] );
true

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