The functions desribed in this section compute certain subalgebras
of a given algebra, e.g.,
Centre computes the centre of an algebra.
They return algebra records as described in Algebra Records for the computed subalgebras. Some functions may not terminate if the given algebra has an infinite set of elements, while other functions may signal an error in such cases.
Here the term ``subalgebra'' is used in a mathematical sense. But in GAP, every algebra is either a parent algebra or a subalgebra of a unique parent algebra. If you compute the centre C of an algebra U with parent algebra A then C is a subalgebra of U but its parent algebra is A (see Parent Algebras and Subalgebras).
Centralizer( A, x )
Centralizer( A, U ):
The centralizer of an element x in A is defined as the set C of elements c of A such that c and x commute.
The centralizer of an algebra U in A is defined as the set C of elements c of A such that c commutes with every element of U.
gap> a:= MatAlgebra( GF(2), 2 );; gap> a.name:= "a";; gap> m:= [ [ 1, 1 ], [ 0, 1 ] ] * Z(2);; gap> Centralizer( a, m ); UnitalSubalgebra( a, [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ], [ [ 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2) ] ] ] )
Centre( A ):
gap> c:= Centre( a ); UnitalSubalgebra( a, [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] ] )
Closure( U, a )
Closure( U, S )
Let U be an algebra with parent algebra A and let a be an element
of A. Then
Closure returns the closure C of U and a as
subalgebra of A. The closure C of U and a is the subalgebra
generated by U and a.
Let U and S be two algebras with a common parent algebra A. Then
Closure returns the subalgebra of A generated by U and S.
gap> Closure( c, m ); UnitalSubalgebra( a, [ [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ], [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] ] )
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