Morphisms Between Finite Algebras

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism.FiniteDimensionalAlgebraHomset(R, S, category=None)

Bases: RingHomset_generic

Set of morphisms between two finite-dimensional algebras.

zero()

Construct the zero morphism of self.

EXAMPLES:

Sage

sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
....:                                   Matrix([[0, 1], [0, 0]])])
sage: H = Hom(A, B)
sage: H.zero()
Morphism from Finite-dimensional algebra of degree 1 over Rational Field to
 Finite-dimensional algebra of degree 2 over Rational Field given by matrix
[0 0]

Python

>>> from sage.all import *
>>> A = FiniteDimensionalAlgebra(QQ, [Matrix([Integer(1)])])
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                   Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])])
>>> H = Hom(A, B)
>>> H.zero()
Morphism from Finite-dimensional algebra of degree 1 over Rational Field to
 Finite-dimensional algebra of degree 2 over Rational Field given by matrix
[0 0]

Sage Live

A = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
B = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
                                  Matrix([[0, 1], [0, 0]])])
H = Hom(A, B)
H.zero()

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism.FiniteDimensionalAlgebraMorphism(parent, f, check=True, unitary=True)

Bases: RingHomomorphism_im_gens

Create a morphism between two finite-dimensional algebras.

INPUT:

• parent – the parent homset

• f – matrix of the underlying \(k\)-linear map

• unitary – boolean (default: True); if True and check is also True, raise a ValueError unless A and B are unitary and f respects unit elements

• check – boolean (default: True); check whether the given \(k\)-linear map really defines a (not necessarily unitary) \(k\)-algebra homomorphism

The algebras A and B must be defined over the same base field.

EXAMPLES:

Sage

sage: from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism import FiniteDimensionalAlgebraMorphism
sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
....:                                   Matrix([[0, 1], [0, 0]])])
sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
sage: H = Hom(A, B)
sage: f = H(Matrix([[1], [0]]))
sage: f.domain() is A
True
sage: f.codomain() is B
True
sage: f(A.basis()[0])
e
sage: f(A.basis()[1])
0

Python

>>> from sage.all import *
>>> from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism import FiniteDimensionalAlgebraMorphism
>>> A = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                   Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])])
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([Integer(1)])])
>>> H = Hom(A, B)
>>> f = H(Matrix([[Integer(1)], [Integer(0)]]))
>>> f.domain() is A
True
>>> f.codomain() is B
True
>>> f(A.basis()[Integer(0)])
e
>>> f(A.basis()[Integer(1)])
0

Sage Live

from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism import FiniteDimensionalAlgebraMorphism
A = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
                                  Matrix([[0, 1], [0, 0]])])
B = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
H = Hom(A, B)
f = H(Matrix([[1], [0]]))
f.domain() is A
f.codomain() is B
f(A.basis()[0])
f(A.basis()[1])

Todo

An example illustrating unitary flag.

inverse_image(I)

Return the inverse image of I under self.

INPUT:

• I – FiniteDimensionalAlgebraIdeal, an ideal of self.codomain()

OUTPUT: FiniteDimensionalAlgebraIdeal, the inverse image of \(I\) under self

EXAMPLES:

Sage

sage: cat = CommutativeAlgebras(QQ).FiniteDimensional().WithBasis()
sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
....:                                   Matrix([[0, 1], [0, 0]])],
....:                              category=cat)
sage: I = A.maximal_ideal()                                                 # needs sage.libs.pari
sage: q = A.quotient_map(I)                                                 # needs sage.libs.pari
sage: B = q.codomain()                                                      # needs sage.libs.pari
sage: q.inverse_image(B.zero_ideal()) == I                                  # needs sage.libs.pari
True

Python

>>> from sage.all import *
>>> cat = CommutativeAlgebras(QQ).FiniteDimensional().WithBasis()
>>> A = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                   Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
...                              category=cat)
>>> I = A.maximal_ideal()                                                 # needs sage.libs.pari
>>> q = A.quotient_map(I)                                                 # needs sage.libs.pari
>>> B = q.codomain()                                                      # needs sage.libs.pari
>>> q.inverse_image(B.zero_ideal()) == I                                  # needs sage.libs.pari
True

Sage Live

cat = CommutativeAlgebras(QQ).FiniteDimensional().WithBasis()
A = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
                                  Matrix([[0, 1], [0, 0]])],
                             category=cat)
I = A.maximal_ideal()                                                 # needs sage.libs.pari
q = A.quotient_map(I)                                                 # needs sage.libs.pari
B = q.codomain()                                                      # needs sage.libs.pari
q.inverse_image(B.zero_ideal()) == I                                  # needs sage.libs.pari

matrix()

Return the matrix of self.

EXAMPLES:

Sage

sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
....:                                   Matrix([[0, 1], [0, 0]])])
sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
sage: M = Matrix([[1], [0]])
sage: H = Hom(A, B)
sage: f = H(M)
sage: f.matrix() == M
True

Python

>>> from sage.all import *
>>> A = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
...                                   Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])])
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([Integer(1)])])
>>> M = Matrix([[Integer(1)], [Integer(0)]])
>>> H = Hom(A, B)
>>> f = H(M)
>>> f.matrix() == M
True

Sage Live

A = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]),
                                  Matrix([[0, 1], [0, 0]])])
B = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
M = Matrix([[1], [0]])
H = Hom(A, B)
f = H(M)
f.matrix() == M

Make live