Ideals of Finite Dimensional Algebras

It is necessary to use algebras in the category of associative algebras.

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_ideal.FiniteDimensionalAlgebraIdeal(A, gens=None, given_by_matrix=False)

Bases: Ideal_generic

An ideal of a FiniteDimensionalAlgebra.

INPUT:

• A – a finite-dimensional algebra

• gens – the generators of this ideal

• given_by_matrix – boolean (default: False); whether the basis matrix is given by gens

EXAMPLES:

Sage

sage: cat = CommutativeAlgebras(GF(3)).FiniteDimensional().WithBasis()
sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]),
....:                                      Matrix([[0, 1], [0, 0]])],
....:                              category=cat)
sage: A.ideal(A([0,1]))
Ideal (e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3

Python

>>> from sage.all import *
>>> cat = CommutativeAlgebras(GF(Integer(3))).FiniteDimensional().WithBasis()
>>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                      Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
...                              category=cat)
>>> A.ideal(A([Integer(0),Integer(1)]))
Ideal (e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3

basis_matrix()

Return the echelonized matrix whose rows form a basis of self.

EXAMPLES:

Sage

sage: cat = CommutativeAlgebras(GF(3)).FiniteDimensional().WithBasis()
sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]),
....:                                      Matrix([[0, 1], [0, 0]])],
....:                              category=cat)
sage: I = A.ideal(A([1,1]))
sage: I.basis_matrix()
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> cat = CommutativeAlgebras(GF(Integer(3))).FiniteDimensional().WithBasis()
>>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                      Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
...                              category=cat)
>>> I = A.ideal(A([Integer(1),Integer(1)]))
>>> I.basis_matrix()
[1 0]
[0 1]

vector_space()

Return self as a vector space.

EXAMPLES:

Sage

sage: cat = CommutativeAlgebras(GF(3)).FiniteDimensional().WithBasis()
sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]),
....:                                      Matrix([[0, 1], [0, 0]])],
....:                              category=cat)
sage: I = A.ideal(A([1,1]))
sage: I.vector_space()
Vector space of degree 2 and dimension 2 over Finite Field of size 3
Basis matrix:
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> cat = CommutativeAlgebras(GF(Integer(3))).FiniteDimensional().WithBasis()
>>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                      Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
...                              category=cat)
>>> I = A.ideal(A([Integer(1),Integer(1)]))
>>> I.vector_space()
Vector space of degree 2 and dimension 2 over Finite Field of size 3
Basis matrix:
[1 0]
[0 1]