Subalgebras and Quotient Algebras

A subalgebra is also returned with the embedding homomorphism, and a quotient algebra is returned with the natural quotient map. These are needed for creating some standard subalgebras such as the centre of the algebra.

Subalgebras and their Constructions

sub<A | S>: AlgBas, SeqEnum -> AlgBas, Map

The subalgebra of \(A\) generated by the elements of the sequence \(S\), together with the inclusion map of the subalgebra into \(A\). The subalgebra contains the idempotent of minimal rank in \(A\) that acts as a multiplicative identity on the elements of \(S\).

SubalgebraFromBasis(A, V): AlgBas, SeqEnum -> AlgBas, Map

Given a basic algebra A and the basis \(V\) of a subspace of \(A\), the function returns the basic algebra which is the subalgebra spanned by the subspace and the inclusion matrix of the homomorphism embedding the subalgebra into \(A.\) Note that the space \(V\) might not contain the identity element of V and in that case the minimal possible identity element is added to the returned subalgebra.

MaximalIdempotent(A, S): AlgBas, SeqEnum -> AlgBasElt

Given a basic algebra \(A\), a subspace \(S\) of the vector space of \(A\) that is closed under multiplication, this function returns an idempotent in \(A\) which has maximal rank among all idempotents contained in \(S\).

MinimalIdentity(A, S): AlgBas, SeqEnum[AlgBasElt] -> AlgBasElt

Returns the idempotent of smallest rank that is a two sided identity for the elements in the set \(S.\)

Centre(A): AlgBas -> AlgBas, Map

The centre of the basic algebra as a basic algebra together with the inclusion homomorphism.

Centralizer(A, S): AlgBas, SeqEnum -> AlgBas, Map

Returns the centralizer in the basic algebra \(A\) of the elements in the sequence \(S\), along with the homomorphism embedding the centralizer into \(A.\)

MaximalCommutativeSubalgebra(A, S): AlgBas, SeqEnum -> AlgBas, Map

Returns a maximal commutative subalgebra of the basic algebra \(A\) that contains the elements of the sequence \(S\). An error occurs if the elements of \(S\) do not commute.

Ideals and their Construction

ideal< A | S>: AlgBas, SeqEnum[AlgBasElt] -> ModTupFld

ideal< A | S>: AlgBasGrpP, SeqEnum[AlgBasElt] -> ModTupFld

Returns the subspace of the vector space of the algebra \(A\) that is the ideal of the \(A\) generated by the given sequence of elements \(S\).

LeftAnnihilator(A, S): AlgBas, SeqEnum[AlgBasElt] -> SeqEnum[AlgBasElt]

Returns a basis for the left annihilator of the sequence \(S\) of elements in the basic algebra \(A.\)

RightAnnihilator(A, S): AlgBas, SeqEnum[AlgBasElt] -> SeqEnum[AlgBaselt]

Returns a basis for the right annihilator of the sequence \(S\) of elements in the basic algebra \(A.\)

Annihilator(A, S): AlgBas, SeqEnum[AlgBasElt] -> SeqEnum[AlgBasElt]

Returns a basis for the two-sided annihilator of the sequence of elements \(S\) of the basic algebra \(A.\)

IsIdeal(A, S): AlgBas, ModTupFld -> Bool

Returns true if the subspace spanned by the elements of \(S\) is a two-sided ideal of the basic algebra \(A.\)

IsLeftIdeal(A, S): AlgBas, ModTupFld -> Bool

Returns true if the subspace spanned by the elements of \(S\) is a left ideal of the basic algebra \(A.\)

IsRightIdeal(A, S): AlgBas, ModTupFld -> Bool

Returns true if the subspace spanned by the elements of \(S\) is a two-sided ideal of the basic algebra \(A.\)

RandomIdealGeneratedBy(A, n): AlgBas, RngIntElt -> ModTupFld

Returns the ideal generated by \(n\) randomly selected elements in the Jacobson radical of the basic algebra \(A.\)

Quotient Algebras

quo<A | S>: AlgBas, ModTupFld -> AlgBas, Map

Returns the quotient algebra of \(A\) by the ideal \(S\), which is a subspace of the vector space of \(A\), together with the quotient map.

CoverAlgebra(A): AlgBas -> AlgBas, ModMatFldElt

Constructs the maximal extension B as in [\(0 \rightarrow K \rightarrow B \rightarrow A \rightarrow 0\)] such that \(B\) acts trivially on \(K\) and \(B\) is an algebra with exactly the same minimal number of generators as \(A\). Returns \(B\) and the algebra homomorphism of \(B\) onto \(A\).

GradedCoverAlgebra(A): AlgBas -> AlgBas, ModMatFldElt

This assumes that we are given the truncated algebra of a graded algebra. It creates the basic algebra of the natural cover of \(A\) and also returns the matrix of the cover onto \(A.\)

TruncatedAlgebra(A, n): AlgBas, RngIntElt -> AlgBas, ModMatFldElt

The quotient of the algebra by the \(n^{th}\) power of the radical of \(A\). Returns also the quotient map.

Units

GeneratorsOfGroupOfUnits(A): AlgBas -> SeqEnum, SeqEnum

Returns a sequence of elements of the basic algebra \(A\) that generate the group of invertible elements in \(A\) and a sequence containing the inverses of those elements.

NoncentralGeneratorsOfGroupOfUnits(A): AlgBas -> SeqEnum, SeqEnum

Returns a sequence of elements of the basic algebra \(A\) that generate the quotient of the group of invertible elements in \(A\) by the subgroup of invertible central elements. The inverses of those elements is also returned as a sequence.