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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.
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.
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.
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.
Returns the idempotent of smallest rank that is a two sided identity for
the elements in the set S.
The centre of the basic algebra as a basic algebra
together with the inclusion homomorphism.
Returns the centralizer in the basic algebra A of the elements in the
sequence S, along with the homomorphism embedding the centralizer into A.
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.
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.
Returns a basis for the left annihilator of the sequence S of elements
in the basic algebra A.
Returns a basis for the right annihilator of the sequence S of elements
in the basic algebra A.
Returns a basis for the two-sided annihilator of the sequence of
elements S of the basic algebra A.
Returns true if the subspace spanned by the elements of S is a two-sided
ideal of the basic algebra A.
Returns true if the subspace spanned by the elements of S is a left
ideal of the basic algebra A.
Returns true if the subspace spanned by the elements of S is a two-sided
ideal of the basic algebra A.
Returns the ideal generated by n randomly selected elements in the
Jacobson radical of the basic algebra A.
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.
Constructs the maximal extension B as in [0 -> K -> B -> A -> 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.
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.
The quotient of the algebra by the nth power of the radical
of A. Returns also the quotient map.
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.
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.
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